oscillons in the planar ginzburg{landau equation with 2:1...
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Oscillons in the planar Ginzburg–Landau equation with 2:1 forcing
Kelly McQuighanDivision of Applied Mathematics
Brown UniversityProvidence, RI 02912, USA
Bjorn SandstedeDivision of Applied Mathematics
Brown UniversityProvidence, RI 02912, USA
September 11, 2014
Abstract
Oscillons are spatially localized, time-periodic structures that have been observed in many natural processes,
often under temporally periodic forcing. Near Hopf bifurcations, such systems can be formally reduced
to forced complex Ginzburg–Landau equations, with oscillons then corresponding to stationary localized
patterns. In this manuscript, stationary localized structures of the planar 2:1 forced Ginzburg–Landau
equation are investigated analytically and numerically. The existence of these patterns is proved in regions
where two spatial eigenvalues collide at zero. A numerical study complements these analytical results away
from onset.
1 Introduction
The formation of localized structures has been of interest to the scientific community for many years. One reason
for the continued study of localized solutions is that they have been observed and found in many experiments,
such as vegetation patches in the desert [41] and solitons in ferromagnetic fluids [36]. In this paper, we focus on
oscillons, which are spatially localized, temporally oscillating structures that emerge from a uniform background
state in parametrically forced nonlinear systems. The frequency of the pattern oscillation is often close to
the forcing frequency or half the forcing frequency, and we refer to these patterns as 1:1 and 2:1 resonance
oscillons, respectively. Oscillons have been observed in a variety of experimental settings including Newtonian
fluid [2, 40, 52, 53], chemical reactions [35, 48], granular particles [44], and colloidal suspensions [30]. In chemical
reactions, oscillons have also been observed in autonomous systems [45] and in systems subjected to global
feedback [47].
A variety of models have been proposed to study pattern formation in temporally forced systems. The experi-
ments in granular media pose particular challenges for theorists: unlike fluid flow and chemical reactions, there
do not exist any equations based on first principles that regulate the evolution of granular particles. While
molecular-dynamics type simulations have been successful at reproducing experimental observations in granular
media [5, 55], such simulations provide limited insight into the mechanism behind oscillon formation. As a result,
several continuum equations have been proposed as models of granular media [9, 17, 37]. Continuum equations
have also been proposed to model fluids [7], chemical reactions [45, 46], and plasmas [42]. A third approach to
modeling such systems has been phenomenological. One class of equations utilizes a temporally discrete, but
spatially continuous, evolution operator to account for the time periodicity of oscillons [26, 27, 50, 51]. A second
class are amplitude equations that are designed to capture the essential features of pattern-forming systems.
Such equations include variations of Ginzburg–Landau [6, 12, 24, 43], Swift–Hohenberg [13, 23], and nonlinear
Schrodinger equations [3, 25]. In all cases, numerical computations revealed that these equations support the
wealth of patterns observed in experiments.
1
Analytical results on oscillons are scarce. It was shown that a nonlinear Schrodinger equation in two and three
spatial dimensions, proposed in [34] to model Newtonian fluids, has two exact ring shaped oscillon-like solutions
that are also stable [4]. We are aware of two analytical results in the dissipative regime. A coupled Ginzburg–
Landau model, proposed in [43] to model granular media, was shown to support localized solutions in one spatial
dimension [14]. Oscillons were also found, along with other experimentally observed 2:1 resonant patterns,
in the forced complex Ginzburg–Landau equation in one dimension [8]. Both works used a weakly nonlinear
analysis.
In this work, we consider the planar forced complex Ginzburg–Landau equation with 2:1 forcing (CGL)1
ut “ p1` iαq∆u` p´µ` iωqu´ p1` iβq|u|2u` γu, x P R2. (1.1)
It has been argued, see for instance [12, 18], that this equation is the amplitude equation for a periodically forced
reaction-diffusion equation near onset of a temporal Hopf instability, where the parameter ω is related to the
offset of the forcing frequency, γ is related to the forcing amplitude, and µ is related to the offset from onset.
In the framework of (1.1), oscillons correspond to stationary localized radially symmetric solutions to (1.1). It
therefore suffices to construct localized solutions to
0 “ p1` iαq´
urr `urr
¯
` p´µ` iωqu´ p1` iβq|u|2u` γu. (1.2)
It was shown in [8] that the one-dimensional CGL
0 “ p1` iαquxx ` p´µ` iωqu´ p1` iβq|u|2u` γu (1.3)
supports two types of localized solutions. The first localized solution, referred to as a standard oscillon, can
be thought of as a homoclinic orbit connecting to the trivial background state u “ 0 in the limits x Ñ ˘8 as
shown in Figures 1a-1b. The second localized solution, referred to as a reciprocal oscillon, can be thought of
as a homoclinic orbit connecting to a nontrivial background state u`unif ‰ 0 in the limits x Ñ ˘8 as shown in
Figures 1c-1d. Reciprocal oscillon solutions to the CGL were originally reported in [6], and were referred to as
“superoscillons.” In both cases, the localized solution may have monotone or oscillatory tails.
(a) standard oscillon:
monotone tails
(b) standard oscillon:
oscillatory tails
(c) reciprocal oscillon:
monotone tails
(d) reciprocal oscillon:
oscillatory tails
Figure 1: Localized solutions to equation (1.3). The dotted line represents u “ 0; the dashed line represents the
nontrivial solution u “ u`unif .
Our goal is to see to what extent the results in [8] for the one-dimensional CGL (1.3) carry over to the planar
case. In particular, we will prove the existence of planar standard oscillons analytically. We will also provide
numerical studies to compare our predictions and expectations for the planar case with the numerical results
published previously in [6, 25, 54] for the one-dimensional and planar CGL with 1:1 and 2:1 forcing.
We begin by stating our main analytical result, which gives the existence of localized solutions of the stationary
radial CGL (1.2). Since (1.2) respects the gauge symmetry pγ, uq ÞÑ pγeiφ, ueiφ{2q, we can restrict our analysis
to the case γ ą 0. Using a one-dimensional spatial eigenvalue analysis of (1.3) around u “ 0, we will see that
these planar oscillons emerge for each fixed µ P pαω, βωq into the region γ ăa
µ2 ` ω2; see Figure 2 for an
illustration.
1Note that the abbreviation CGL traditionally refers to the unforced complex Ginzburg–Landau equation. We also note that
there exists a family of CGL given by integer ratios of the forcing frequency to the pattern frequency.
2
⇥2 ⇥4
subcritical supercritical
front
⇥2
� �p
µ2 + !2
µ � ↵!!(� � ↵)
Figure 2: Shown is the pµ, γq-plane with α, β, and ω fixed. The inlays are the spatial eigenvalues associated with
the linearization of equation (1.3) about u “ 0 plotted in the complex plane. The bifurcation from γ “a
µ2 ` ω2
changes from subcritical to supercritical at the point µ “ ωβ. The shaded region is the expected existence region
of oscillons; oscillons are seen numerically to terminate at the dotted line in a stationary 1D front solution.
Theorem 1 Fix αω ă µ ă βω and let γ “a
µ2 ` ω2´ε2. Then there is an ε0 ą 0 so that (1.2) has a nontrivial
stationary localized radial solution of amplitude Opεq for each ε P p0, ε0q.
We will show that the condition µ ą αω ensures that all four spatial eigenvalues at the origin u “ 0 are real. We
will also show that signpµ ´ ωβq equals the sign of the leading-order nonlinear term in an appropriate center-
manifold reduction of both (1.3) and (1.2) near ε “ 0: the condition µ ă ωβ ensures that the bifurcation from
the curve γpω, µq “a
µ2 ` ω2 is subcritical and leads to localized patterns.
The remainder of this paper is organized as follows. In section 2, we give a brief informal outline of the proof
of Theorem 1. The details of the proof are contained in the next sections: we construct oscillons for 0 ă r ! 1
in section 3 and for r " 1 in section 4; also contained in section 4 is the matching of these solutions at some
finite value of r. The necessary expansions that allow us to match these solutions are proved in section 5 using
geometric blow-up techniques. In section 6, we summarize our numerical investigations of planar oscillons and
compare them with earlier theoretical and numerical results in 1D and 2D. Finally, we comment on planned
extensions of our work in section 7.
2 Outline
We begin by outlining the ideas behind our proof. We model our work after a similar analysis of the planar
Swift–Hohenberg equation in [32]. The idea is to construct the core manifold as the set of solutions of (1.2) that
stay bounded as r Ñ 0 for r P r0, r0s for some r0 ă 8. We will separately construct the far-field manifold as
the set of all solutions of (1.2) that decay to zero exponentially as r Ñ8. A solution lying in both sets will, by
definition, be a bounded localized solution. A schematic is shown in Figure 3.
In the proof of Theorem 1 we are confronted with two main difficulties, both of which are related to the far field.
Firstly, in the parameter region we are interested in, the linearization of (1.2) about u “ 0 in the far field has
two eigenvalues close to the origin and, in addition, one strong stable and one strong unstable direction. Our
far-field analysis should therefore be performed on an appropriate center manifold. On the other hand, the final
matching between the core and far-field manifolds is performed in the original four-dimensional space: hence, in
order to carry out the final matching analysis, we need to employ a stable foliation over solutions in the center
manifold, denoted by Fs in Figure 3, and derive appropriate expansions of all foliations and manifolds.
Secondly, we encounter difficulties when analysing the flow on the center manifold. The small-amplitude solutions
on the center manifold are, to leading order, of the form Aprq “ εAexppεrq for some bounded function Aexppsq.
Thus, we can expect uniform expansions of an exponentially decaying solution A˚prq to be possible on the
3
!!!!!!
far-field center-manifold
far-field stable manifold
core manifold
0r0
r1A⇤(r)
Fs(A⇤)
Figure 3: Schematic overview of our strategy: we construct two sets of solutions to (1.2), the core manifold
(solutions that are bounded as r Ñ 0) and the far-field stable manifold (solutions that decay as r Ñ 8). Any
solution lying in the intersection of these manifolds is a localized solution. For fixed r, both manifolds have a
two-dimensional parametrization in four-dimensional space. The far-field stable manifold is identified with the
strong stable foliation associated with solutions on the center manifold that converge to zero as r Ñ8.
interval s P rs0,8q, or r ě s0{ε. In particular, we expect to lose control over bounds on A˚prq at r “ r0 for ε
small enough. To resolve this issue and obtain meaningful bounds near the matching point at r “ r0, we employ
the blow-up coordinates from [32] to mediate between exponential decay in the far field and algebraic behavior
in the core. Such geometric blow-up techniques were originally developed in [16, 28] to desingularize hyperbolic
equilibria.
3 Bounded solutions near the core
In this section we construct the set of all bounded solutions of the planar, radial, stationary forced Ginzburg–
Landau equation
0 “ p1` iαq´
urr `urr
¯
` p´µ` iωqu´ p1` iβq|u|2u` γu (3.1)
on r P r0, r0s with r0 ă 8 fixed.
3.1 The linearization about u “ 0
We use a far-field spatial eigenvalue analysis of u “ 0 to predict the standard-oscillon bifurcation curve. By
setting r “ 8 we reduce (3.1) to the one-dimensional CGL
0 “ p1` iαquxx ` p´µ` iωqu´ p1` iβq|u|2u` γu. (3.2)
Thus, a far-field spatial eigenvalue analysis of (3.1) is equivalent to the 1D case. The existence of localized
solutions to the one-dimensional CGL (3.2) was studied extensively in [8]. We briefly review the relevant results
from [8], translated into our notation2, and refer to Figure 4 for a visual schematic representation of these
results.
We decompose u into its real and imaginary parts via u “ u1` iu2 with u1, u2 P R and see that the linearization
of (3.2) about u “ 0, given in these coordinates by
0 “
¨
˚
˝
1 ´α
α 1
˛
‹
‚
¨
˚
˝
u1
u2
˛
‹
‚
xx
`
¨
˚
˝
´µ` γ ´ω
ω ´µ´ γ
˛
‹
‚
¨
˚
˝
u1
u2
˛
‹
‚
, (3.3)
2We remark that [8] uses the parameter `µ rather than ´µ. Hence their critical points and bifurcation diagrams are flipped
across the µ “ 0 axis relative to ours.
4
⇥2 ⇥2⇥2
⇥2
⇥2 ⇥4 ⇥2µ � ↵!
� �p
µ2 + !2
!(� � ↵)
�a �a
�0�0
�b
u±unif
Figure 4: Plotted is the pµ, γq-plane with α, β, and ω fixed. The inlays are the spatial eigenvalues associated with
the linearization of (3.2) about u “ 0 plotted in the complex plane. There are two bifurcation cases to consider:
(i) into the region below Γ0 (purple dashed line) for µ ą αω, and (ii) into the region below Γa (dash-dotted line)
for µ ă αω. Non-trivial uniform solutions u˘unif emerge at a fold bifurcation at Γb; the fold curve Γb intersects
Γ0 at µ “ βω where u`unif “ u´unif “ 0; we refer to [8] for details on the behavior of u˘unif away from the fold.
has four spatial eigenvalues. These eigenvalues k satisfy
p1` α2qk4 ` 2pαω ´ µqk2 ` µ2 ` ω2 ´ γ2 “ 0 (3.4)
and obey the symmetry k ÞÑ ´k. In Figure 4 we illustrate the spatial eigenvalues k, plotted in the complex
plane, for various values of µ and γ. In the autonomous case of (3.2), localized solutions may bifurcate as small-
amplitude solutions from the rest state at u “ 0 only if the latter is not hyperbolic, that is, if the linearization of
(3.2) about u “ 0 has purely imaginary eigenvalues. Furthermore, we want this rest state to be hyperbolic after
the bifurcation to ensure that the stable and unstable manifolds are as high dimensional as possible. Given the
symmetry of the spatial eigenvalues, there are then two generic cases to consider.
In case (i), two purely imaginary eigenvalues collide at the origin of the complex plane and split so that all four
eigenvalues are real; this bifurcation occurs for µ ą αω into the region below the curve Γ0 shown in Figure 4. In
particular, we define Γ0 :“ tpµ, γq : γ “a
µ2 ` ω2u, set µm :“ αω ` m2
2 , and pick γm so that pµm, γmq P Γ0; a
straightforward computation shows that, with pµ, γq “ pµm, γm ´ ε2q, k1,2 split into Opεq eigenvalues along the
real axis, whilst k3,4 “ ˘m ` Opεq. This bifurcation gives rise to standard oscillons with monotone tails: its
generalization to the radial equation (3.1) is the focus of this work.
In case (ii), two pairs of spatial eigenvalues collide on the imaginary axis and split into the complex plane in a
Turing bifurcation along the curve Γa in Figure 4. This bifurcation results in standard oscillons with oscillatory
tails. We will not discuss its extension to the planar case here, though we will, in section 6, report on preliminary
numerical results and also state the precise form of Γa there.
In Figure 4, we also indicate the existence region of the nontrivial uniform states u˘unif , which bifurcate through
a fold bifurcation along the curve Γb. The existence of these states is necessary for the existence of the reciprocal
oscillons. Further details on these rest states can be found in [8]; we will also review the relevant one-dimensional
results in section 6.
This ends our review of [8] and we now return to the planar case. Motivated by the far-field spatial dynamics
discussed above, we use from now on the parameter values µ “ µm :“ αω ` m2
2 and γ “ γm ´ ε2 in (3.1), where
m ą 0 is an order one constant. We write the resulting differential equation as a real second-order system usingrU “ pu1, u2q
T P R2 and rescale the radial variable r via r ÞÑ r{?α2 ` 1. The resulting equation is
0 “ rUrr `1
rrUr ´ C1
rU ´ ε2C2rU ´ |rU |2C3
rU (3.5)
5
where
C1 :“
¨
˚
˝
m2
2 ´ γm ω ` αµm ` αγm
´ω ´ αµm ` αγmm2
2 ` γm
˛
‹
‚
,
C2 :“
¨
˚
˝
1 ´α
´α ´1
˛
‹
‚
, C3 “
¨
˚
˝
1` αβ α´ β
β ´ α 1` αβ
˛
‹
‚
.
Equation (3.5) can also be written as the first-order system
BrU “ Ap1{rqU ` FpU, ε2q (3.6)
in U “ prU, rV qT P R4, where
Apκq :“
¨
˚
˝
0 I
C1 ´κI
˛
‹
‚
, F´
prU, rV qT ; ν¯
“
¨
˚
˝
F1
F2
˛
‹
‚
´
prU, rV qT ; ν¯
:“
¨
˚
˝
0
νC2rU ` |rU |2C3
rU
˛
‹
‚
.
We linearize (3.6) at ε “ 0 about U ” 0 and obtain
Vr “ Ap1{rqV. (3.7)
Before giving a set of linearly independent solutions to this equation, we remark that Ap1{rq converges to Ap0qas r Ñ8. We therefore first compute the eigenvalues and eigenvectors of Ap0q, which are related to those of the
matrix C1. The matrix C1 has eigenvalues λ0 “ 0 and λ1 “ m2 with associated eigenvectors
rU0 “
¨
˚
˝
αγm ` ω ` αµm
γm ´m2
2
˛
‹
‚
, rU1 “
¨
˚
˝
αγm ` ω ` αµm
γm `m2
2
˛
‹
‚
,
respectively. The eigenvalues of the matrix Ap0q coincide with the far-field spatial eigenvalues for equation (3.1)
about u “ 0 and are therefore equal to νc “ 0 (with multiplicity two), νu “ m, and νs “ ´m with associated
generalized eigenspaces
Ecr` “ span
$
’
&
’
%
¨
˚
˝
rU0
0
˛
‹
‚
,
¨
˚
˝
0
rU0
˛
‹
‚
,
/
.
/
-
, Eur` “ span
$
’
&
’
%
¨
˚
˝
rU1
mrU1
˛
‹
‚
,
/
.
/
-
, Esr` “ span
$
’
&
’
%
¨
˚
˝
rU1
´mrU1
˛
‹
‚
,
/
.
/
-
. (3.8)
Four linearly independent solutions tVjprqu4j“1 of the linearization (3.7) are given by
V1 “
¨
˚
˝
rU0
0
˛
‹
‚
, V2 “
¨
˚
˝
rU0 ln r
rU01r
˛
‹
‚
, V3 “
¨
˚
˝
rU1I0pmrq
rU1mI1pmrq
˛
‹
‚
, V4 “
¨
˚
˝
rU1K0pmrq
´rU1mK1pmrq
˛
‹
‚
, (3.9)
where I0pzq and K0pzq are the zeroth-order modified Bessel functions [1, §9.6]. The asymptotic behaviors of
I0pzq and K0pzq will be important in the analysis and are displayed in Table 1. Note that only V1prq and V3prq
are bounded as r Ñ 0.
For r large enough, we will see that the far-field stable and unstable manifolds will remain close, in an appropriate
sense, to the stable and unstable subspaces Esr` and Eur`, respectively. We will also see that the space Ecr` gives
appropriate center-manifold coordinates. Finally, we note that normalized solution vectors V3prq{|V3prq| and
V4prq{|V4prq| converge to unit vectors in Eur` and Esr`, respectively. Furthermore, the center subspace Ecr` is
actually invariant under the linearization (3.7), as discussed in the following remark.
Remark 3.1 We have span tV1prq, V2prqu “ Ecr` for all r since c1V1prq ` c2V2prq “ pc1 ` c2 ln rqV 01 `
c2r V
02 for
all c1, c2 P R, where V 01 :“ V1 and V 0
2 :“ p0, rU0qT .
6
z Ñ 0 z Ñ8
I0pzq 1`Opz2q 1?2πz
ez`
1`O`
1z
˘˘
K0pzq ´ ln`
z2
˘
I0pzq ´ γ `Opz2qa
π2z e
´z`
1`O`
1z
˘˘
Table 1: The asymptotic behavior of the zeroth-order modified Bessel functions for small and large arguments
quoted from [1, (9.6.12)-(9.6.13), (9.7.1)-(9.7.2)], respectively, where γ “ limnÑ8
´
řnj“1
1n´ lnn
¯
is the Euler-
Mascheroni constant.
3.2 Construction of the core manifold
We now construct the set of small-amplitude solutions to (3.6) that stay bounded on intervals of the form r0, r0s
with 0 ă r0 ă 8 fixed. We call this set the “core manifold” and denote it by ĂWcu´ pεq. We define P cur´pr0q to be
the projection onto span tV1pr0q, V3pr0qu along span tV2pr0q, V4pr0qu.
Throughout this paper, we say that fpx; r0q “ Or0pxq if, for each fixed r0, there are positive constants C “ Cpr0q
and δ “ δpr0q such that |fpx; r0q| ď Cpr0q|x| for all x with |x| ă δpr0q. A similar convention holds for all other
Landau symbols used in this paper.
Lemma 3.2 Fix 0 ă r0 ă 8 and let d “ pd1, d3q P R2. Then there exist constants ρ1, ρ2, ε0 ą 0 so that, for
ε ď ε0,
ĂWcu´ pεq :“
"
pUprq, rq : Uprq satisfies p3.6q for r P r0, r0s with sup0ďrďr0
|Uprq| ă ρ1,ˇ
ˇP cur´pr0qUpr0qˇ
ˇ ă ρ2
*
is a smooth three-dimensional submanifold of R5. Moreover, there are smooth functions pg2, g4qpd1, d3, εq with
pg2, g4qpd1, d3; εq “ Or0pε2|d| ` |d|3q so that U P ĂWcu
´ pεq if and only if
Upr0q “ d1V1pr0q ` g2pd1, d3; εqV2pr0q ` d3V3pr0q ` g4pd1, d3; εqV4pr0q (3.10)
with |d| “ |pd1, d3q| ă ρ2.
Proof. The proof follows from a standard application of the variation-of-constants formula on a bounded
interval: see, for instance, [31, Proof of Lemma 1] for details in a similar case.
Due to Remark 3.1, equation (3.10) is equivalent to
Upr0q “ pd1 ` g2pd1, d3; εq ln r0qV01 `
1
r0g2pd1, d3; εqV 0
2 ` d3V3pr0q ` g4pd1, d3; εqV4pr0q. (3.11)
We remark that we will consider the fiber ĂWcu´ pεq
ˇ
ˇ
r“r1for each fixed r1 P r0, r0s as a two-dimensional submanifold
of R4.
4 Far-field dynamics, and matching with the core
We have constructed the core manifold on bounded intervals. We now turn our attention to the far field. We
recall (3.6), the forced complex Ginzburg–Landau equation,
BrU “ Ap1{rqU ` FpU, ε2q, (4.1)
where
Apκq “
¨
˚
˝
0 I
C1 ´κI
˛
‹
‚
, F´
prU, rV qT ; ν¯
“
¨
˚
˝
0
νC2rU ` |rU |2C3
rU
˛
‹
‚
.
7
!!!!!!
Esr+
Ecr+
= 1 = 2
p1
p2
Wcr+
Fs✏ (p1,1)
Fs✏ (p2,2)
q1
q2
Figure 5: Shown is Wcsr` with foliation Fs
ε pp, κq over base points in Wcr`pεq. An example fiber is drawn in each
of the two Poincare sections κ “ κ1 and κ “ κ2 with associated base points p1, p2 PWcr`pεq. Solutions associated
with initial data q1 P Fsε pp1, κ1q approach the solution associated with the initial condition p1 on the center
manifold exponentially as r Ñ8.
We augment (4.1) near r “ 8 with κ “ 1{r. The resulting vector field
¨
˚
˝
U
κ
˛
‹
‚
r
“
¨
˚
˝
ApκqU ` FpU ; ε2q
´κ2
˛
‹
‚
(4.2)
is autonomous with linearization about the fixed point pU, κq “ p0, 0q given by the system Vr “ Ap0qV and
ρr “ 0. As discussed in section 3.1, the equation for V has two center, one unstable, and one stable direction
given by the subspaces Ecr`, Eur`, and Esr` from (3.8), respectively. Taking the additional κ-direction into
account, we therefore expect to find a four-dimensional center-stable and a three-dimensional center manifold
near U “ 0.
In this section, we prove the existence of these manifolds. As illustrated in Figure 5, we can write the center-
stable manifold Wcsr`pεq as the stable foliation tFsε pp, κqupPWc
r`pεq|κ“1{rwith base points in the center manifold
Wcr`pεq ĂWcs
r`pεq: a trajectory with initial data pq, κq converges to zero as r Ñ 8 if, and only if, its associated
base point with initial data pp, κq on the center manifold does. The far-field stable manifold, consisting by
definition of all solutions for which Uprq Ñ 0 as r Ñ 8, is therefore given by the union of the stable fibers
associated with decaying solutions on the center manifold. The remaining steps for finding localized solutions
are therefore to (i) derive an expansion for the vector field restricted to the center manifold, (ii) analyze the
flow on the center manifold, and (iii) match the resulting far-field stable manifold with the core manifold. In
this section, we will carry out steps (i) and (iii), anticipating the results for step (ii) which will be carried out in
section 5.
Throughout this section, all invariant manifolds will be considered as subsets of R4 ˆ R; we will consider their
restrictions to κ “ κ1, for each fixed κ1, as submanifolds of R4.
4.1 Existence of a center-stable manifold
The existence of center-stable and center manifolds are standard; see, for example, [49]. However, since we will
need specific properties of these manifolds, we show here briefly how the results of [49] apply. First, we control
the nonlinear terms via a cutoff function: let χpzq be a smooth cutoff function with χpzq “ 1 for z ď 1 and
8
χpzq “ 0 for z ě 2 and define, for ρ small enough, the modified vector field
¨
˚
˚
˚
˚
˝
rU
rV
κ
˛
‹
‹
‹
‹
‚
r
“
¨
˚
˚
˚
˚
˝
0 I 0
C1 0 0
0 0 0
˛
‹
‹
‹
‹
‚
¨
˚
˚
˚
˚
˝
rU
rV
κ
˛
‹
‹
‹
‹
‚
`
¨
˚
˚
˚
˚
˝
0
p´κrV ` F2pU, ε2qqχ
´
|U |2
ρ2
¯
χ´
ε4
ρ2
¯
χ´
|κ|2
ρ2
¯
´κ2χ´
|κ|2
ρ2
¯
˛
‹
‹
‹
‹
‚
, (4.3)
which then coincides with (4.2) for |U |, |κ|, ε2 ď ρ and also satisfies the hypotheses of the invariant-manifold
theorems in [49]. It remains to show that these manifolds satisfy certain properties which we will need later on.
In particular, we are interested in the tangent space of these manifolds: we denote the tangent spaces of the center
and center-stable manifolds at a solution pU, κq by TpU,κqWcsr`pεq and TpU,κqWc
r`pεq, respectively. Similarly, we use
TpU,κqWcsr`pεq
ˇ
ˇ
κ“κ1to denote the tangent space of the center-stable manifold restricted to κ “ κ1 and considered
as a submanifold of R4 for every fixed κ1; an analogous notation is used for the center manifold. Let SεrpU0, κ0q
denote the solution to the original vector field (4.2) at time r starting from pU, κqp0q “ pU0, κ0q; similarly,rSεrpU0, κ0q is the solution to the modified vector field (4.3) at time r starting from pU, κqp0q “ pU0, κ0q.
Proposition 4.1 Fix ` ě 2. Then there exist ε0 ą 0, ρ0 ą 0 so that, for all 0 ď ε ď ε0 and 0 ă ρ ď ρ0, equation
(4.2) possesses a flow-invariant four-dimensional center-stable manifoldWcsr`pεq near the equilibrium pU, κ, εq “ 0.
The manifold Wcsr`pεq is C`, depends C` on ε2 and contains all solutions to (4.2) with suprě0t|Uprq|, |κprq|, |ε|u ď
ρ. Furthermore, Wcsr`pεq satisfies the following properties:
(i) there exists a smooth, monotonically decreasing function κρprq with |κρprq| ď 2ρ so that for every 0 ď ε ď ε0,
U˚prqˇ
ˇ
rě1{ρ:“ tpU, κq “ p0, κρprqqu
ˇ
ˇ
|κ|ďρP Wcs
r`pεq, and κρprqˇ
ˇ
rě1{ρ“ 1{r.
In particular, 0 PWcsr`pεq
ˇ
ˇ
κ“1{r; and
(ii) TU˚prqWcsr`p0q
ˇ
ˇ
κ“1{r“ span
V 01 , V
02 , V4prq
(
for all 0 ď κ ď ρ, where we recall from (3.9) that the functions
tVjprqu4j“1 are the solutions to Vr “ ApκqV :
V1prq “
¨
˚
˝
rU0
0
˛
‹
‚
, V2prq “
¨
˚
˝
rU0 ln |r|
rU01r
˛
‹
‚
, V3prq “
¨
˚
˝
rU1I0pmrq
rU1mI1pmrq
˛
‹
‚
, V4prq “
¨
˚
˝
rU1K0pmrq
´rU1mK1pmrq
˛
‹
‚
, (4.4)
and where V 01 :“
¨
˚
˝
rU0
0
˛
‹
‚
, V 02 :“
¨
˚
˝
0
rU0
˛
‹
‚
were defined in Remark 3.1 so that span tV1prq, V2prqu “ span
V 01 , V
02
(
“
Ecr` with V 01 and V 0
2 independent of r.
Proof. Let ρ0 be such that (4.3) satisfies the hypotheses of [49, Theorem 5.3]. Then, for every 0 ă ρ ď ρ0 there
exists a smooth global center-stable manifold near the equilibrium U “ 0, denoted ĂWcsr`pεq, with
ĂWcsr`pεq :“
"
pU0, κ0qT P R5 : sup
rě0|Pur`
rSεrpU0, κ0q| ă 8
*
,
where the operators P jr` are the complementary projections onto the subspaces Ejr` for j P ts, u, cu. For every
ε ď ρ define Wcsr`pεq :“ ĂWcs
r`pεq X tpU, κq : |U | ď ρ, |κ| ď ρu to obtain a center-stable manifold for (4.2). It
remains to verify properties (i) and (ii).
(i) Direct substitution shows that U ” 0 solves the evolution equation in U for the modified equation (4.3); it
remains to solve κr “ ´κ2χp|κ|2{ρ2q for κρprq. First intersect with the set κρprq ď ρ to show κρprq
ˇ
ˇ
rě1{ρ“ 1{r.
Then observe that κρprq is monotone since χp¨q is nonnegative. Smoothness follows from the smoothness of χp¨q.
Monotonicity and χp2q “ 0 together show |κρprq| ď 2ρ for all r. Boundedness of κρprq shows U˚prq P ĂWcsr`pεq.
9
(ii) We solve the linearization of (4.3) about U˚prq with ε “ 0
¨
˚
˝
qU
qV
˛
‹
‚
r
“
¨
˚
˝
0 I
C1 ´κρprqχ`
|κρprq|2{ρ2
˘
˛
‹
‚
¨
˚
˝
qU
qV
˛
‹
‚
(4.5a)
qκr “ ´2“
κρprqχ`
|κρprq|2{ρ2
˘
` κ3ρprq{ρ2χ1
`
|κρprq|2{ρ2
˘‰
qκ. (4.5b)
Since qκprq decouples from the rest of the system we set qκprq “ 0 and solve (4.5a). Since κρprq is positive and
monotonically decreasing we solve forward until κρprq ď ρ so that χp¨q “ 1. Now (4.5a) reduces to Vr “ rApm,κqV
solved by span tVjprqu4j“1. Observing that V3prq grows exponentially as r Ñ8 completes the proof.
4.2 Existence of a center manifold
Next we show the existence of a center manifold Wcr` ĂWcs
r` for (4.2) with U˚prq PWcr`.
Proposition 4.2 Fix ` ě 2. Then there exist 0 ă ε1 ď ε0 and 0 ă ρ1 ď ρ0 so that, for every 0 ď ε ď ε1 and
0 ă ρ ď ρ1, equation (4.2) possesses a flow-invariant three-dimensional C`-center manifold Wcr`pεq near the
equilibrium pU, κ, εq “ 0, which contains all solutions with suprPRt|Uprq|, |κprq|, |ε|u ď ρ. The center manifold
depends C` on ε2 and has the following additional properties:
(i) for all 0 ď ε ď ε1, Wcr`pεq ĂWcs
r`pεq;
(ii) for all 0 ď ε ď ε1, U˚prq PWcr`pεq;
(iii) TU˚prqWcr`p0q
ˇ
ˇ
κ“1{r“ Ecr` for r ě 1{ρ; and
(iv) the flow on Wcr`pεq respects the actions
• Z2 symmetry: prU, rV , κ, rq ÞÑ p´rU,´rV , κ, rq;
• reverser: prU, rV , κ, rq ÞÑ prU,´rV ,´κ,´rq.
Remark 4.3 We emphasize that property (iii) states in particular that the tangent space TU˚pκqWcr`p0q is in-
dependent of κ “ 1{r and hence Ecr` is invariant under the linearization about U˚prq.
Before we prove Proposition 4.2, we first state and prove the following result, which will be used in the proof of
property (iii).
Lemma 4.4 There exist functions ajprq, bjprq, j P t2, 3, 4u, so that all solutions to (4.5a) are given by linear
combinations of trVju4j“1 where
rV1prq :“ V1 “
¨
˚
˝
rU0
0
˛
‹
‚
, rV2prq :“
¨
˚
˝
a2prqrU0
b2prqrU0
˛
‹
‚
, rV3prq :“
¨
˚
˝
a3prqrU1
b3prqrU1
˛
‹
‚
, rV4prq :“
¨
˚
˝
a4prqrU1
b4prqrU1
˛
‹
‚
.
Moreover, rV2prq grows at most exponentially with rate m{4 as r Ñ ˘8, whereas rV3prq and rV4prq grow exponen-
tially with rate at least m{2 as r Ñ8 and r Ñ ´8 respectively.
Proof. Note that (4.5a) is a small perturbation of the constant-coefficient problem
¨
˚
˝
qU
qV
˛
‹
‚
r
“
¨
˚
˝
0 I
C1 0
˛
‹
‚
¨
˚
˝
qU
qV
˛
‹
‚
.
The claims can be proved relatively easily using exponential dichotomies and the particular structure of (4.5a).
We omit the details.
10
We are now ready to prove Proposition 4.2. We recall that rSεrpU0, κ0q represents the solution to (4.3) at time r
starting from pU, κqp0q “ pU0, κ0q. We also recall that we need to verify the following properties:
(i) for all 0 ď ε ď ε1, Wcr`pεq ĂWcs
r`pεq;
(ii) for all 0 ď ε ď ε1, U˚prq PWcr`pεq;
(iii) TU˚prqWcr`p0q
ˇ
ˇ
κ“1{r“ Ecr` for r ě 1{ρ; and
(iv) the flow on Wcr`pεq respects the Z2 and reverser symmetries.
Proof of Proposition 4.2 As in the proof of Proposition 4.1, we apply [49, Theorem 2.1] to (4.3) near
pU, κq “ 0 for ρ small enough to show the existence of a smooth global center manifold ĂWcr`pεq with
ĂWcr`pεq :“
"
pU0, κ0qT P R5 : sup
rPR|P sur`
rSεrpU0, κ0q| ă 8
*
.
We then define Wcr`pεq :“ ĂWc
r`pεq X tpU, κq : |U | ď ρ, |κ| ď ρu to obtain a center manifold for (4.2). It remains
to verify properties (i)-(iv).
(i) Comparing the definitions of the center-stable and center manifolds constructed in [49] shows that ĂWcr`pεq Ă
ĂWcsr`pεq. The containment remains true after intersection with tpU, κq : |U |, |κ| ď ρu.
(ii) By the same argument as in Lemma 4.1(a) we see that U˚prq remains bounded for all r P R.
(iii) We again linearize (4.3) about U˚prq with ε “ 0 to obtain (4.5). Again, since qκprq decouples from the rest
of the system we set qκprq “ 0 and solve (4.5a). We then apply Lemma 4.4 to obtain TU˚prqĂWcr`p0q
ˇ
ˇ
κ“1{r“
span!
rV1, rV2
)
. Intersection with |κρ| ă ρ shows TU˚Wcr`p0q “ span tV1, V2u, since we know from the proof of
Lemma 4.4 that rV2prqˇ
ˇ
|κρ|ďρ“ V2.
(iv) Equation (4.2) respects the Z2 symmetry prU, rUr, κ, rq ÞÑ p´rU,´rUr, κ, rq and the reverser prU, rUr, κ, rq ÞÑ
prU,´rUr,´κ,´rq. The cutoff χp|z|q is symmetric in z so that (4.3) respects these actions as well. Then
prUprq, rV prq, κprqq P ĂWcr`pεq ðñ p´rUprq,´rV prq, κprqq P ĂWc
r`pεq
since |U | “ | ´ U | and
prUprq, rV prq, κprqq P ĂWcr`pεq ðñ prUp´rq,´rV p´rq,´κp´rqq P ĂWc
r`pεq
since the hyperbolic projections of solutions in ĂWcr`pεq are bounded in both forward and backward r. Together
this implies that (4.3) respects the symmetries on ĂWcr`pεq. These symmetries are preserved after intersection
with tpU, κq : |U | ď ρ, |κ| ď ρu, again because |U | “ | ´ U | and |κ| “ | ´ κ|.
4.3 Strong stable foliations
In this section we show Wcsr`pεq is given as the union of strong stable fibers over base points in Wc
r`pεq. We
refer to [11] for background on stable foliations and to Figure 5 for an illustration. Let SεrpU0, κ0q represent the
solution to (4.2)¨
˚
˚
˚
˚
˝
rU
rV
κ
˛
‹
‹
‹
‹
‚
r
“
¨
˚
˚
˚
˚
˝
0 I 0
C1 ´κI 0
0 0 0
˛
‹
‹
‹
‹
‚
¨
˚
˚
˚
˚
˝
rU
rV
κ
˛
‹
‹
‹
‹
‚
`
¨
˚
˚
˚
˚
˝
0
ε2C2rU ` |rU |2C3
rU
´κ2
˛
‹
‹
‹
‹
‚
at time r with initial data pU, κqp0q “ pU0, κ0q. We recall from (3.8) and Remark 3.1 the definitions
V 01 “
¨
˚
˝
rU0
0
˛
‹
‚
, V 02 “
¨
˚
˝
0
rU0
˛
‹
‚
, Esr` “
¨
˚
˝
rU1
´mrU1
˛
‹
‚
, Eur` “
¨
˚
˝
rU1
mrU1
˛
‹
‚
.
11
Lemma 4.5 Fix ` ě 1 and a decay rate ν with 0 ă ν ă m. Define ε1, ρ1 as in Lemma 4.2 so that the manifolds
Wcsr`pεq
ˇ
ˇ
κ“1{rand Wc
r`pεqˇ
ˇ
κ“1{rexist. For every ε ď ε1, any κ “ 1{r ď ρ1, and each p P Wc
r`pεqˇ
ˇ
κ“1{r, there
exists a one-dimensional strong stable fiber Fsε pp, κq in R4 so that the following are true:
(i) p P Fsε pp, κq for all p PWcr`pεq
ˇ
ˇ
κ“1{r;
(ii) Wcsr`pεq
ˇ
ˇ
κ“1{r“ 9Ť
pPWcr`pεq|κ“1{r
Fsε pp, κq;(iii) Fsε p¨, κq depends C` on ε2 and κ;
(iv) for every ε, κ, there exists a C` ˆ C` function
φsεp¨, κ, ¨q :Wcr`pεq
ˇ
ˇ
κ“1{rˆ RV4prq Ñ RV 0
1 ‘ RV 02 ‘ RV3prq
so that Fsε pp, κq “ graphpφsεpp, κ, ¨qq;
(v) |Sεrpq1, κq ´ Sεrpq2, κq| “ Ope´νtq for r ě 0 and all q1, q2 P Fsε pp, κq; and
(vi) TFs0 p1{r, U˚prqq “ V4prq.
Proof. Recall that equation (4.3) is the autonomous vector field (4.2) with an appropriate cutoff applied near
pU, κq “ p0, 0q. For every ε and ρ small enough, we can apply [11, Theorem 4.3] to (4.3): we therefore know the
existence of stable fibers rFsε pp1q, with p1 “ pp, κq PWcr`pεq, which satisfy
(i’) p1 P rFsε pp1q for all p1 PWcr`pεq;
(ii’) Wcsr`pεq “
9Ť
p1PWcr`pεq
rFsε pp1q;
(iii’) rFsε p¨q depends C` on ε2;
(iv’) for every ε there exists a C`ˆC` function rφsε :Wcr`pεqˆE
sr` Ñ pR‘Ecur`q so that rFsε pp1q “ graphprφsεpp
1, ¨qq;
(v’) |Sεrpq11q ´ S
εrpq
12q| “ Ope´νtq for r ě 0 and all q11, q
12 P
rFsε pp1q; and
(vi’) T rFs0 p0, 0q “ Esr`.
It remains to show that conditions (i’-vi’) imply (i-vi).
First, we show that each fiber is completely contained within the Poincare section κ “ 1{r. We then define
tFsε p¨, κqu :“ t rFsε p¨quˇ
ˇ
κ“1{rfor every κ “ 1{r small enough; conditions (i’-v’) are then equivalent to (i-v). We
argue by contradiction: assume that there exists a fiber rFsε pp0q with base point p0 “ px0, κ0q and furthermore
assume that there exists some p1 “ px1, κ1q P rFsε pp0q with κ1 ‰ κ0. We use the evolution equation for κ to see
that |Sεrpp0q ´ Sεrpp1q| ě |κ0prq ´ κ1prq| ě C{r2 for some C ą 0 as r Ñ8 in contradiction to condition (v’).
Next we show that condition (vi’) extends to (vi) for κ ą 0. First, we have U˚prq “ tpU, κq “ p0, 1{rqu PWcsr`pεq
for every ε from Lemma 4.1(a). Next, we again linearize about U˚prq with ε “ 0 to get (4.5). We solve forward
in κ until κ ă ρ so that (4.5a) reduces to Vr “ ApκqV , which is solved by linear combinations of Vjprq for
j P t1, 2, 3, 4u where
V1prq “ V 01 , V2prq “
¨
˚
˝
ln |r|rU0
1rrU0
˛
‹
‚
, V3prq “ I0pmrq
¨
˚
˝
rU1
mrU1I1pmrqI0pmrq
˛
‹
‚
, V4prq “ K0pmrq
¨
˚
˝
rU1
´mrU1K1pmrqK0pmrq
˛
‹
‚
.
We define qVjprq :“ Vjprq{|Vjprq| and observe that qV3prq and qV4pmrq converge to unit vectors in Eur` and Esr`,
respectively, as r Ñ8. We define the normalized tangent vector aprqV prq :“ Tφs0pU˚prq, ¨qˇ
ˇ
κ“1{rwhere
qV prq “ pa1 ` a2 ln |r|qV 01 `
a2rV 02 ` a3I0pmrq
qV3prq ` a4K0pmrqqV4prq
is a solution to the linear flow (4.5a). By condition (vi’) we must have aprqV prq Ñ Esr` as r Ñ 8. Taking
aprq “ 1K0pmrq
we find
Tφs0pU˚prq, ¨qˇ
ˇ
κ“1{r“a1 ` a2 ln |r|
K0pmrqV 01 `
a2rK0pmrq
V 02 ` a3
I0pmrq
K0pmrqqV3prq ` a4 qV4prq.
12
Using the asymptotic expansion for K0pmrq in the limit r Ñ 8 given in Table 1 we see that a1 “ a2 “ a3 “ 0
and a4 “ 1. The proof now follows by uniqueness.
4.4 Parametrization of Wcsr`pεq, Wc
r`pεq, and F sεWe use the properties ofWc
r`pεq,Wcsr`pεq, and Fsε listed in Sections 4.1-4.3 to parametrize each of these manifolds
as graphs. Each parametrization will be performed near U˚prq. We first collect the relevant results, where
0 ă ρ1 ď ρ0 ! 1:
(i) For each ε, U˚prq :“ tpU, κq “ p0, 1{rqu PWcr`pεq ĂWcs
r`pεq [Propositions 4.1(i) and 4.2(i-ii)];
(ii) TU˚prqWcsr`p0q
ˇ
ˇ
κ“1{r“ span
V 01 , V
02 , V4prq
(
for all 0 ď κ ď ρ0 [Proposition 4.1(ii)];
(iii) TU˚prqWcr`p0q
ˇ
ˇ
κ“1{r“ span
V 01 , V
02
(
for all 0 ď κ ď ρ1 [Proposition 4.2(iii)]; and
(iv) TU˚prqFs0 p¨, κq “ V4prq [Lemma 4.5(vi)].
We begin with Wcsr`pεq. For every κ ď ρ0, Wcs
r`pεqˇ
ˇ
κ“1{rcan be written as graph of a function h3p¨;κ, εq :
RV 01 ‘ RV 0
2 ‘ RV4prq Ñ RV3prq so that
Wcsr`pεq
ˇ
ˇ
κ“1{r“ td1V
01 ` d2V
02 ` h3pd1, d2, d4;κ, εqV3prq ` d4V4prq : d1, d2, d4 smallu. (4.6a)
Property (i) implies h3p0, 0, 0;κ, ε2q “ 0 for all ε and property (ii) implies h3pd1, d2, d4;κ, 0q “ Oκp|d1|2` |d2|
2`
|d4|2q. Hence,
h3pd1, d2, d4;κ, εq “ Oκ
`
p|d1| ` |d2| ` |d4|qp|d1| ` |d2| ` |d4| ` ε2q˘
, (4.6b)
where we recall that the Landau symbol Oκp¨q is interpreted in the usual sense except that the subscript κ means
that the bounding constant and the region where the estimate is valid may depend on κ.
Similarly, for every κ ď ρ1 we can write Wcr`pεq
ˇ
ˇ
κ“1{ras
Wcr`pεq
ˇ
ˇ
κ“1{r“ td1V
01 ` d2V
02 ` h3pd1, d2;κ, εqV3prq ` h4pd1, d2;κ, εqV4prq : d1, d2 smallu (4.7a)
where hjp¨;κ, εq : V 01 ‘ V 0
2 Ñ Vjprq. Property (i) shows hjp0, 0;κ, ε2q “ 0 for all ε and property (iii) implies
hjpd1, d2;κ, 0q “ Oκp|d1|2 ` |d2|
2q. Therefore,
hjpd1, d2;κ, εq “ Oκ
`
p|d1| ` |d2|qp|d1| ` |d2| ` ε2q˘
, j P t3, 4u. (4.7b)
Since Wcr`pεq ĂWcs
r`pεq, we have in particular
h3pd1, d2;κ, εq “ h3pd1, d2, h4pd1, d2;κ, εq;κ, εq.
Lastly, we can parametrize the fibers Fsε pp, κq using appropriate functions hjp¨, ¨;κ, εq :Wcr`pεq
ˇ
ˇ
κ“1{r‘RV4prq Ñ
RVjprq for j P t1, 2, 3u so that
Fsε pp, κq “ tp` h1pp, ds;κ, εqV 01 ` h2pp, ds;κ, εqV
02 ` h3pp, ds;κ, εqV3prq ` dsV4prq : ds smallu. (4.8a)
In the parametrizations of Wcsr`pεq and Wc
r`pεq, the coordinates dj were taken relative to U “ 0. In contrast, in
our parametrization of Fsε , the offset is measured from the base point p so that hjpp, 0;κ, εq “ 0; see Figure 6.
From condition (iv) we also know that hjp0, ds;κ, 0q “ Oκp|ds|2q. Taking these properties together we arrive at
the expansion
hjpp, ds;κ, εq “ Oκ
`
|ds|p|p| ` |ds| ` ε2q˘
(4.8b)
of the function hjp¨, ¨;κ, εq.
13
pWc
r+(✏)��=1/r
V4(r)
V 01 � V 0
2
ds
(h1, h2)(p, d2;, ✏)
Fs✏ (p,)
Figure 6: Parametrization of a strong stable fiber within Wcsr`
ˇ
ˇ
κ“1{rfor some κ ď ρ1 fixed.
Our goal now is to write the fibers Fsε relative to U˚prq. Let P c1 , Pc2 , P
uprq, and P sprq be the complementary
projection operators onto the subspaces V 01 , V
02 , V3prq, and V4prq respectively. We define A :“ P c1p,B :“ P c2p
as the center-manifold coordinates of p and compute Puprqp and P sprqp using (4.7). Lastly, we use (4.8) to
compute the foliation expansion
Fsε pp, κq “ tdc1V 01 ` d
c2V
02 ` d
uV3prq ` dsV4prqu (4.9a)
where
dc1 :“P c1p` h1pp, ds;κ, εq “ A`Oκ
`
|ds|p|A| ` |B| ` |ds| ` ε2q˘
dc2 :“P c2p` h2pp, ds;κ, εq “ B `Oκ
`
|ds|p|A| ` |B| ` |ds| ` ε2q˘
du :“Puprqp` h3pp, ds;κ, εq “ Oκ
`
p|A| ` |B| ` |ds|qp|A| ` |B| ` |ds| ` ε2q˘
ds :“P sprqp` ds “ Oκ
`
p|A| ` |B|qp|A| ` |B| ` ε2q˘
` ds. (4.9b)
Since Fsε pp, κq ĂWcsr`pεq
ˇ
ˇ
κ“1{r, we could also have used (4.6) to determine du “ h3pd
c1, d
c2, d
s;κ, εq once the other
three components had been computed. The result is the same.
4.5 Reduction of the vector field to the center manifold
Next we derive a convenient expansion for the vector field (4.2)
¨
˚
˚
˚
˚
˝
rU
rV
κ
˛
‹
‹
‹
‹
‚
r
“
¨
˚
˚
˚
˚
˝
0 I 0
C1 ´κI 0
0 0 0
˛
‹
‹
‹
‹
‚
¨
˚
˚
˚
˚
˝
rU
rV
κ
˛
‹
‹
‹
‹
‚
`
¨
˚
˚
˚
˚
˝
0
ε2C2rU ` |rU |2C3
rU
´κ2
˛
‹
‹
‹
‹
‚
restricted to the center manifold. In Proposition 4.2(iv) we showed that the flow on the center manifold respects
the Z2 symmetry prU, rUr, κ, rq ÞÑ p´rU,´rUr, κ, rq and the reverser symmetry prU, rUr, κ, rq ÞÑ prU,´rUr,´κ,´rq.
We use these symmetries in the proof of the following lemma.
Lemma 4.6 Using the coordinates P cr`U “ AV 01 ` BV 0
2 from the center-manifold parametrization (4.7), with
A “ d1 and B “ d2, the vector field (4.2) restricted to Wcr`pεq can be written as
Ar “ B `RApA,B, κ; εq
Br “ ´κB ` ε2A` c03A
3 `RBpA,B, κ; εq
κr “ ´κ2 (4.10a)
where sgnpc03q “ sgnpµm ´ βωq. The remainder terms RA and RB satisfy
RApA,B, κ; εq “ O`
|A|pε2 `A2q ` |B|pε2 ` |A| ` |B|q˘
(4.10b)
RBpA,B, κ; εq “ O`
|ε2κ2A| ` pε2 ` κ2q|A|3 ` |A|5 ` pε2 ` κ2 `A2q|κB| ` |AB2| ` |B|3˘
. (4.10c)
14
Proof. The derivation of (4.10) follows from a modification of the proof of [38, Lemma 3.9 in Chapter 3]. We
define uc :“ AV 01 `BV
02 and ψpuc, κ; εq : EcˆRˆRÑ Es‘Eu so thatWc
r` “ tpuc, κ, ψpuc, κ; εqq; |uc|, |κ|, ε2 ď
ρ1u. By equation (4.7), ψpA,B, κ; εq “ Oκ
`
pA`BqpA`B ` ε2q˘
. We recall from section 3.1 that FpU ; εq “
Opε2|U | ` U2q. By Remark 4.3, Ecprq “ Ec is independent of r and is invariant under the non-autonomous
linearization Vr “ ApκqV . Therefore, the reduced vector field on Wcr`pεq, projected onto Ecr`, is
Bruc “ Apκq
ˇ
ˇ
Ecuc ` P cFpuc ` ψpuc, κ; εq; εq, rApκq
ˇ
ˇ
Ec“
¨
˚
˝
0 1
0 ´κ
˛
‹
‚
, P c :“ P c1 ` Pc2 (4.11)
where rApκq was defined in (4.2), and P cj were defined in section 4.4. Equation (4.11) shows, in particular, that
Ar “ B `AγApκ; ε2q `BγBpκ; ε2q ` rRApA,B, κ; εq (4.12)
with γApκ; 0q “ γBpκ; 0q “ 0. We next use the reverser action and Z2 symmetry; by Proposition 4.2(iv), the
remainder terms RA and RB must respect both these symmetries. The Z2 symmetry shows that rRApA, 0, 0; 0q “
OpA3q. This shows that the leading-order terms in RA are given by (4.10b).
For RB we start with the remainder terms from [38, Lemma 3.9 in Chapter 3]. In that lemma, a new variablerB :“ B ` RApA,B, κ; εq was introduced. However, because RA is higher order in B and because it respects
the Z2 and reverser symmetries, no new terms are introduced into RB through undoing this transformation.
Therefore, Br “ ´κB ` rRBpA,B, κ; εq with
rRBpA,B, κ; εq “ γ1pκ; ε2qA` γ2pκ; ε2qA2 ` γ3pκ; ε2qA3 `Oppε2 ` κ2 ` |A|q|κB| ` |AB2| ` |A|4 ` |B|4q, (4.13)
and γ1pκ; 0q “ 0 and γ2pκ; 0q ” γ2p0; 0q. In the derivation of (4.13) the reverser symmetry was enforced. We now
drop all terms which do not also respect the Z2 symmetry and obtain Br “ ´κB`c01ε
2A`c03A3`RBpA,B, κ; εq
with RB given in (4.10c). Although the term B3 does not respect the reverser symmetry, it is a higher-order
term relative to AB2 and serves to bound non-algebraic terms in B.
To determine the sign of c01 and c03 we use a weakly nonlinear analysis near the curve Γ0 : tpµm, γmq : γ2m “
µ2m ` ω2u as in [8, (A.7) pp. 698-699]. The computations are similar for the planar radial case, so we omit the
details.
4.6 Matching core and far-field stable manifolds
Finally, we use the results of the preceding sections to prove the following result.
Theorem 2 Fix µ ą αω and µ ă βω so that c03 ă 0 in the vector field (4.10) on the center manifold and let
γ “a
µ2 ` ω2 ´ ε2. Then there is an ε0 ą 0 so that (3.5) has a nontrivial stationary localized radial solution of
amplitude Opεq for each ε P p0, ε0s.
We will prove Theorem 2 by showing that there exists a nontrivial intersection of the core manifold ĂWcu´ pεq
and the far-field stable manifold ĂWs`pεq. Recall that the core manifold consists of all solutions which remain
bounded as r Ñ 0, while the far-field stable manifold is the set of all solutions that decay to zero as r Ñ 8.
Any nontrivial solution laying in the intersection of these two sets is, by definition, a localized solution to (3.5).
The core manifold was constructed in section 3.2 on bounded intervals r P r0, r0s, whereas the far-field stable
manifold relies on an analysis of the far-field center-stable and center manifolds; the existence proofs of these
far-field manifolds are only valid for r ě 1{ρ1. In the proof of Theorem 2 we will choose r0 large enough so that
1{ρ1 ă r0, which ensures that both Wcr`pεq and Wcs
r`pεq exist at the matching point r “ r0.
In order to complete the construction of ĂWs`pεq it remains to find solutions on the center manifold that decay
to zero as r Ñ 8. The following result, which we will prove in section 5 using the blow-up coordinates of [32],
characterizes such solutions:
15
Proposition 4.7 (proven in section 5) Fix c03 ă 0, pη0 ą 0 and r0 ą 1{ρ1, where ρ1 ď ρ0 was chosen in the
proof of Proposition 4.2. Then there exists an a0 ą 0 so that for every fixed δ0 ą 0 there is an ε0 ą 0 so that for
each pη P r0, pη0s and each ε with 0 ă ε ď ε0 there exists an initial condition of the form
Apr0q “ ε`
a0 `Opδ20 ` 1{r20q˘
` ε2Oδ0p1` |pη| ln εq
Bpr0q “ ´ε2pηpa0 `Opδ20 ` 1{r20qq ` ε
3Oδ0,r0
`
p1` |pη|2q ln ε` |pη|˘
(4.14)
so that the corresponding solution of (4.10) decays to zero exponentially as r Ñ8.
The Landau symbol Oδ0,r0 is interpreted in the usual sense, except that the subscript pδ0, r0q means that the
bounding constant and region of validity may depend on δ0 and r0.
We can now complete the proof of Theorem 2. As shown in (4.9), the strong stable fiber associated with solutions
(4.14) can be parametrized as
Fsε p1{r0, pq “ tdc1V 01 ` d
c2V
02 ` d
uV3pr0q ` dsV4pr0qu (4.15a)
where, due to (4.9b) and (4.14), we have
dc1 “ A`Or0
`
|ds|p|A| ` |B| ` |ds| ` ε2q˘
“ ε`
a0 `Opδ20 ` 1{r20q˘
`Oδ0,r0
`
ε2p1` |pη| ln εq ` |ds|pε` ε2|pη| ` |ds|q
˘
dc2 “ B `Or0
`
|ds|p|A| ` |B| ` |ds| ` ε2q˘
“ ´ε2pη`
a0 `Opδ20 ` 1{r20q˘
`Oδ0,r0
`
ε3pp1` |pη|2q ln ε` |pη|q ` |ds|pε` ε2|pη| ` |ds|q
˘
du “ Or0
`
p|A| ` |B| ` |ds|qp|A| ` |B| ` |ds| ` ε2q˘
“ Oδ0,r0
`
ε2 ` ε4|pη|2 ` |ds|2˘
ds “ Or0
`
p|A| ` |B|qp|A| ` |B| ` ε2q˘
` ds
“ Oδ0,r0
`
ε2 ` ε4|pη|2˘
` ds. (4.15b)
Proof of Theorem 2 We find a nontrivial solution solution contained in the intersection ĂWcu´ pεq X
ĂWs`pεq by
matching the coefficients of each manifold at r “ r0 in the directions V 01 , V
02 , V3pr0q, and V4pr0q. The manifold
ĂWs`pεq
ˇ
ˇ
r“r0is parametrized by (4.15), whilst the manifold ĂWcu
´ pεqˇ
ˇ
r“r0is parametrized by (3.11),
ĂWcu´ pεq
ˇ
ˇ
r“r0“ pd1 ` ln r0g2pd1, d3; εqqV 0
1 `1
r0g2pd1, d3; εqV 0
2 ` d3V3pr0q ` g4pd1, d3; εqV4pr0q,
where pg2, g4qpd1, d3; εq “ Or0pε2|d| ` |d|3q and d “ pd1, d3q. Collecting the expansion of ĂWcu
´ pεqˇ
ˇ
r“r0on the
left-hand side and of ĂWs`pεq
ˇ
ˇ
r“r0on the right-hand side, we get the system
V 01 : d1 `Or0
`
ε2|d| ` |d|3˘
“ ε`
a0 `Opδ20 ` 1{r20q˘
`Oδ0,r0
`
ε2p|pη| ln ε` 1q ` |ds|pε` ε2|pη| ` |ds|q
˘
V 02 : Or0
`
ε2|d| ` |d|3˘
“ ´ε2pη`
a0 `Opδ20 ` 1{r20q˘
`Oδ0,r0
`
ε3pp1` |pη|2q ln ε` |pη|q ` |ds|pε` ε2|pη| ` |ds|q
˘
V3pr0q : d3 “ Oδ0,r0
`
ε2 ` ε4|pη|2 ` |ds|2˘
V4pr0q : Or0
`
ε2|d| ` |d|3˘
“ Oδ0,r0
`
ε2 ` ε4|pη|2˘
` ds
of equations. We solve the first, third, and fourth equation for pd1, d3, dsq near zero as functions of pε, pηq near
zero by finding zeros of Fpd1, d3, ds; ε, pηq for all ε ă ε0 and |pη| ă pη0, where
Fpd1, d3, ds; ε, pηq “
¨
˚
˚
˚
˚
˝
d1 ´ ε`
a0 `Opδ20 ` 1{r20q˘
`Oδ0,1{r0
`
ε2|d| ` |d3| ` ε2p|pη| ln ε` 1q ` |ds|pε` ε2|pη| ` |ds|q
˘
d3 `Oδ0,r0
`
ε2 ` ε4|pη|2 ` |ds|2˘
Oδ0,r0
`
ε2|d| ` |d|3 ` ε2 ` ε4|pη|2˘
´ ds
˛
‹
‹
‹
‹
‚
.
16
Note that Fp0; 0q “ 0 and that DFp0; 0q is invertible for all sufficiently small δ0 and sufficiently large r0 since
a0 ‰ 0: hence, we can apply the implicit function theorem near the origin and, afterwards, match orders in ε to
find the expansions
d1 “ ε`
a0 `Opδ20 ` 1{r20q `Oδ0,r0pε ln εq˘
, d3 “ Oδ0,r0pε2q, ds “ Oδ0,r0pε
2q.
Using these expansions, we see that the remaining equation for the V 02 coordinate becomes ε2Gppη; εq “ 0 where
Gppη; εq “ pη`
a0 `Opδ20 ` 1{r20q˘
`Oδ0,r0
`
ε ln ε` ε|pη| ` |pη|2ε2 ln ε˘
.
We again apply the implicit function theorem to find zeros of Gp¨ ; εq and, matching orders in ε, we arrive at
pη “ Oδ0,r0pε ln εq.
5 Dynamics on the center manifold
We complete the proof of Theorem 2 by proving Proposition 4.7. To do so, it will be convenient to transform the
vector field (4.10) on the center manifold. We introduce rB :“ B `RApA,B, κ; εq and note that we can invert
this transformation by the implicit function theorem for pA,B, κ, εq near zero so that B “ rB ` rRApA, rB, κ; εq,
where the remainder term satisfies
rRApA, rB, κ; εq “ O´
|A|pε2 `A2q ` | rB|pε2 ` |A| ` | rB|q¯
.
The vector field (4.10) written in the pA, rBq coordinates becomes
Ar “ rB
rBr “ ´κ rB ` ε2A` c03A3 ` rRBpA, rB, κ; εq (5.1)
κr “ ´κ2,
where
rRBpA, rB, κ; εq :“ RB´
A, rB ` rRApA, rB, κ; εq, κ; ε¯
“ O´
ε2κ2|A| ` pε2 ` κ2q|A|3 ` |A|5 ` pε2 ` κ2 `A2q|κ|| rB| ` |A| rB2 ` | rB|3¯
.
We now state the following proposition, whose proof will occupy the remainder of this section and which will
implies Proposition 4.7.
Proposition 5.1 Fix c03 ă 0, pη0 ą 0 and r0 ą 1{ρ1, where ρ1 ď ρ0 was chosen in the proof of Proposition 4.2.
Then there exists an a0 ą 0 so that, for every fixed δ0 ą 0, there is an ε0 ą 0 so that for each pη P r0, pη0s and
each ε with 0 ă ε ď ε0 there exists an initial condition of the form
Apr0q “ ε`
a0 `Opδ20 ` 1{r20q˘
` ε2Oδ0 p|pη| ln ε` 1q
rBpr0q “ ´ε2pη`
a0 `Opδ20 ` 1{r20q˘
` ε3Oδ0,r0
`
p1` |pη|2q ln ε` |pη|˘
(5.2)
so that the corresponding solution of (5.1) decays to zero exponentially as r Ñ8.
Proof of Proposition 4.7 Assuming Proposition 5.1, we need to derive the expansion for B:
Bpr0q “ rBpr0q ` rRA`
Apr0q, rBpr0q, 1{r0; ε2˘
“ rBpr0q `O´
pε2 `A2q|A| ` pε2 ` |A| ` | rB|q| rB|¯
“ rBpr0q `Opε3q,
and Proposition 4.7 follows.
17
It remains to prove Proposition 5.1. With an abuse of notation, we write B in place of rB from now on. We
construct solutions to the vector field (5.1) that decay exponentially. We append the evolution equation for the
parameter ε, given by εr “ 0, and use the geometric blow-up coordinates of [32] in our analysis.
Before giving the details, we discuss the intuition behind our specific choice of blow-up coordinates. We make the
ansatz that decaying solutions on the far-field center manifold are of the form Aprq “ εA2pεrq for some function
A2psq. Using this scaling and s “ εr we find that equation (5.1) at ε “ 0 reduces to the real Ginzburg–Landau
equation
BssA2 `BsA2
s“ A2 ` c
03A
32 (5.3)
with sgnpc03q “ sgnpµ ´ ωβq. We will see in section 5.2 that (5.3) has non-trivial bounded solutions if, and
only if, c03 ă 0, which explains why we imposed this hypothesis in Proposition 5.1 and Theorem 2. Solutions to
(5.3) have dominant exponential behavior with rate ˘1 as sÑ8. Exponentially decaying solutions of the form
A2 „ expp˘sq can be expanded for s P rδ0,8q, hence for r ě δ0{ε. Thus, at the matching point r “ r0 we will
lose control over bounds on the solution A2 as εÑ 0. This widening gap is represented as the space between the
two Poincare sections in Figure 7. We therefore introduce two coordinate transformations: one to handle the
exponential behavior of solutions for r ě δ0{ε and another to maintain control over a parametrization of such
solutions for r0 ď r ď δ0{ε.
First consider r ě δ0{ε. We introduce z2 :“ BsA2{A2 and observe that, heuristically, solutions of the form
A2 „ expp˘sq, correspond to z2 „ ˘1 as s Ñ 8. Therefore, intuitively, the set of exponentially decaying
solutions in the original coordinates corresponds to the center-stable manifold of pA2, z2q “ p0,´1q. We will
show that the center-stable manifold of pA2, z2q “ p0,´1q is two-dimensional for every fixed ε. Following [32],
we call the pA2, z2q coordinates the “rescaling chart” coordinates.
Next, consider r0 ď r ď δ0{ε. We expect the solution behavior in this intermediate region to be, in general,
algebraic. For example, radially symmetric localized solutions to the planar Swift–Hohenberg equation solutions
in the intermediate region are Opr˘12 q [32]. Heuristically, intermediate algebraic solution behavior serves to
mediate between bounded behavior near the core and exponential behavior in the tail. In order to capture
the transitional algebraic behavior, we introduce the blow-up coordinates A1 :“ rAprq, z1 :“ rBrA{A. Then
solutions of the form A “: 1rA1 „ rρ as r Ñ 0 correspond with z1 „ ρ so that algebraic solutions correspond
with pA1, z1q “ p0, ρq in the limit r Ñ 0. Again following [32], we call the pA1, z1q coordinates the “transition
chart” coordinates.
This concludes the discussion of the rationale behind our choice of blow-up coordinates pA2, z2q and pA1, z1q.
5.1 Rescaling and transition charts
We first define z :“ B{A “ Ar{A. Next, we augment (5.1) by the evolution equation for ε, given by εr “ 0. We
then blow-up the vector field and all four variables in two different directions. First, we blow-up the vector field
(5.1) in the κ direction using the coordinates
A1 :“A
κ, z1 :“
z
κ, κ1 :“ κ, ε1 :“
ε
κ, τ :“ ln r, (5.4)
called the “transition chart” coordinates. Using the rescaled time eτ :“ r, we obtain
BτA1 “ A1pz1 ` 1q
Bτz1 “ ε21 ` c03A
21 ´ z
21 ` κ
21R1pA1, z1, ε1q
Bτκ1 “ ´κ1
Bτ ε1 “ ε1 (5.5)
18
P�2
P+2
P1 q0
far-field center-manifold
r = r0
r = �0/✏
1/r
1/s✏r
core manifold
A2A1
z2z1
Figure 7: Schematic overview of our proof including the blow-up coordinates of the far field. The transition
and rescaling chart are blow-up coordinates of the center-manifold coordinates, fixed points of which capture
algebraically and exponentially behaving solutions, respectively.
with R1pA1, z1, ε1q “ O`
ε21 `A21 ` z1
˘
. The vector field (5.5) has the fixed point P1 “ p0, 0, 0, 0q. The lineariza-
tion of (5.5) about P1 is
¨
˚
˚
˚
˚
˚
˚
˚
˚
˝
qA1
qz1
qκ1
qε1
˛
‹
‹
‹
‹
‹
‹
‹
‹
‚
τ
“
¨
˚
˚
˚
˚
˚
˚
˚
˚
˝
1 0 0 0
0 0 0 0
0 0 ´1 0
0 0 0 1
˛
‹
‹
‹
‹
‹
‹
‹
‹
‚
¨
˚
˚
˚
˚
˚
˚
˚
˚
˝
qA1
qz1
qκ1
qε1
˛
‹
‹
‹
‹
‹
‹
‹
‹
‚
(5.6)
with eigenvalues t1, 0,´1, 1u. Therefore, the strong unstable manifold Wuτ´ of P1 is two-dimensional and given
to leading order near P1 by the pA1, ε1q plane.
Next, we blow-up the vector field (5.1) in the ε-direction using the coordinates
A2 :“A
ε, z2 :“
z
ε, κ2 :“
κ
ε, ε2 :“ ε, s :“ εr “
1
κ 2, (5.7)
called the “rescaling chart” coordinates. We remark that solutions in the coordinate systems (5.4) and (5.7) are
related through the relationships
A1 “ sA2, z1 “ sz2, κ1ε1 “ ε2, ε1 “ εeτ “1
κ2“ s. (5.8)
Using (5.7) and the rescaled time s :“ εr, we obtain
BsA2 “ z2A2
Bsz2 “ ´κ2z2 ` c03A
22 ´ z
22 ` 1` ε22R2pA2, z2, κ2q
Bsκ2 “ ´κ22
Bsε2 “ 0 (5.9)
with R2pA2, z2, κ2q “ O`
κ22 `A22 ` κ2z2
˘
. We first set ε “ ε2 “ 0. Then the vector field (5.9)ε2“0 has the two
fixed points P˘2 “ p0,˘1, 0, 0q. We will show at the end of this section that P´2 is associated with exponentially
19
decaying solutions to (5.1) in the original coordinates. The linearization of (5.9) about P´2 is
¨
˚
˚
˚
˚
˚
˚
˚
˚
˝
qA2
qz2
qκ2
qε2
˛
‹
‹
‹
‹
‹
‹
‹
‹
‚
s
“
¨
˚
˚
˚
˚
˚
˚
˚
˚
˝
´1 0 0 0
0 2 1 0
0 0 0 0
0 0 0 0
˛
‹
‹
‹
‹
‹
‹
‹
‹
‚
¨
˚
˚
˚
˚
˚
˚
˚
˚
˝
qA2
qz2
qκ2
qε2
˛
‹
‹
‹
‹
‹
‹
‹
‹
‚
. (5.10)
The linearization (5.10) has eigenvalues t´1, 2, 0, 0u with associated eigenvectors
v1 “ p1, 0, 0, 0qT , v2 “ p0, 1, 0, 0q
T , v3 “ p0,´1, 2, 0qT , v4 “ p0, 0, 0, 1qT .
For ε “ 0, the two-dimensional center-stable manifold Wcss` of P´2 is given to leading order by span tv1, v3u. We
remark that ε2psq “ ε2 “ ε is constant for all s; therefore, the v4 direction is neutral and not contained in Wcss`,
to leading order.
For all ε small enough, the fixed points P˘2 pεq and all invariant manifolds Wuτ´pεq, Wcs
s`pεq persist and depend
smoothly on ε. For simplicity of notation, all fixed points and invariant manifolds are evaluated at ε “ 0 unless
we explicitly indicate their dependence on ε.
We conclude this section by arguing that a small-amplitude solution Aprq “ Opεq in the original center-manifold
coordinates decays exponentially for ε ą 0 if, and only if, it is contained in the center-stable manifold Wcss`pεq
of P´2 pεq. Indeed, fix ε ą 0 and let Aprq be a solution such that pA2, z2, κ2, ε2q Ñ P´2 pεq as r Ñ 8. The
evolution equation for A2psq in (5.9) and a standard fixed-point argument show that A2psq “ A0eşss0z2pρqdρ for
some s0 " 1 with z2psq Ñ ´1 as s Ñ 8. It is straightforward to show that, for s0 large enough, there exists a
0 ă C ă 8 and a δ ! 1 so that supsěs0 |A2psq| ď Cep´1`δqs. Transforming back into the original coordinates
Aprq “ εA2pεrq, we see that |Aprq| ď Ce´εr as r Ñ 8. Conversely, fix ε ą 0 and consider only solutions for
r " 1{ε. Then, using a fixed-point argument in an appropriate exponentially weighted space, one can show
that Aprq “ εr´1{2e´εr p1`Op1{rqq for the nonlinear problem (5.1), and it is straightforward to show that the
corresponding solution in the rescaling chart coordinates converges to P´2 pεq as sÑ8.
5.2 Singular connecting orbit between transition and rescaling charts
We first set ε “ 0 so that ε2 “ 0. Inspecting (5.8) shows that κ1 “ 0 as well. The rescaling-chart vector field
(5.9) then reduces to
BsA2 “ z2A2
Bsz2 “ ´κ2z2 ` c03A
22 ´ z
22 ` 1
Bsκ2 “ ´κ22 (5.11)
and the transition chart vector field (5.5) reduces to
BτA1 “ A1pz1 ` 1q
Bτz1 “ ε21 ` c03A
21 ´ z
21
Bτ ε1 “ ε1. (5.12)
Note that the equations for κ2 and ε1 decouple from the rest of the system. Using κ2psq “ 1{s we can rewrite
(5.11) as a second-order equation with a :“ A2
ass `ass“ a` c03a
3, s ą 0. (5.13)
We have the following result.
20
Lemma 5.2 For c03 ă 0, (5.13) has a unique monotonically decreasing, nontrivial, bounded solution q0psq.
Furthermore, there exists constants a0, a2, c1 ą 0 such that the solution q0psq satisfies q0psq “ a0´a2s2`Ops4 ln sq
as s Ñ 0 and q0psq “ K0psq`
c1 `Ope´2sq˘
as s Ñ 8. Lastly, the linearization of (5.13) about q0psq does not
have a nontrivial solution that is bounded uniformly on R`. For c03 ą 0, the only bounded solution to (5.13) is
apsq ” 0.
Proof. First consider c03 ă 0. Without loss of generality, let c03 “ ´1, otherwise rescale a. Then equation (5.13)
reduces to
ass ` as{s “ ap1´ a2q. (5.14)
Equation (5.14) arises in many applications and has been well studied; it is related, for example, to the nonlinear
Schrodinger equation. In two dimensions, the existence of a unique ground state solution to (5.14) is due to
[33], a result which was later made more general in [29]. The nonexistence of a nontrivial bounded solution of
the linearization is shown in [10, Lemma 2.1]. The asymptotics of q0psq follow from the variation-of-constants
formula and a standard fixed-point argument in each limit; the sign of a2 then follows from monotonicity.
Next consider c03 ą 0. We multiply (5.13) by sapsq and integrate over s P p0,8q as in [31, Lemma 4]; integration
by parts shows that the only possible localized solution is apsq ” 0.
We now show that q0psq gives a connecting orbit between the fixed points P1 and P´2 .
Lemma 5.3 Assume that c03 ă 0 and set ε “ ε2 “ κ1 “ 0. Then Q0, given in the transition and rescaling chart
coordinates by
Q01pτq “
"
pA1, z1, ε1qpτq “
ˆ
eτq0peτ q, eτ
q10peτ q
q0peτ q, eτ
˙*
and
Q02psq “
"
pA2, z2, κ2qpsq “
ˆ
q0psq,q10psq
q0psq,
1
s
˙*
(5.15)
respectively, forms a connecting orbit between P1 of (5.12) and P´2 of (5.11), which lies in the intersection
Wuτ´ XWcs
s`. Moreover, the intersection Wuτ´ JXWcs
s` is transverse.
Proof. The proof is essentially the same as in [32, Lemma 2.4]. Using the asymptotic expansions for q0psq given
in Lemma 5.2 it is easy to check that Q01pτq converges to P1 exponentially as τ Ñ ´8 and that Q0
2psq converges
to P´2 as sÑ8. Since Q01pτq and Q0
2psq satisfy (5.12) and (5.11), respectively, we have that Q0 PWuτ´ XWcs
s`.
To show that the intersection of Wu´ and Wcs
` along Q0 is transverse we invoke the nondegeneracy condition in
Lemma 5.2 and argue by contradiction. Assume that the intersection Wuτ´ XWcs
s` is not transverse, then there
exists a nonzero solution pQ P TQ01pτqWuτ´ X TQ0
2psqWcss`. Let pQ1pτq and pQ2psq denote the solution pQ written in
the transition and rescaling chart coordinates respectively; we write pQ2psq “: p pA2, pz2, pκ2qpsq. Linearizing (5.9)
about Q02psq, we find that pκ2 “ 0 and that pA2 and pz2 satisfy
Bs pA2 “q10q0
pA2 ` q0pz2
Bspz2 “ ´1
spz2 ` 2c03q0
pA2 ´ 2q10q0pz2. (5.16)
Letting pa :“ pA2, a straightforward computation shows that (5.16) is equivalent to
pass `pass“ pap1` 3c03q
20q, (5.17)
the linearization of (5.13) about q0psq. In particular, pA2psq is a nonzero bounded solution of the linearization of
(5.14) about q0psq, in contradiction to Lemma 5.2.
21
5.3 The dynamics near P1 in the transition chart coordinates
Our goal is to use Lemma 5.3 to traceWcss`pεq backwards in time to the equilibrium P1 and beyond for all ε ă ε0.
We will find it convenient to first transform the vector field (5.5)
BτA1 “ A1pz1 ` 1q
Bτz1 “ ´z21 ` ε
21 ` c
03A
21 ` κ
21Opε21 `A
21 ` z1q
Bτκ1 “ ´κ1
Bτ ε1 “ ε1 (5.18)
in the transition chart coordinates into a more convenient form by straightening the center, stable, and unstable
manifolds near P1 “ p0, 0, 0, 0q as well as the strong stable and strong unstable fibers.
Lemma 5.4 There is a smooth change of coordinates of the form
rz1 “ z1 `Opz1κ21 `A
21 ` ε
21q (5.19)
that transforms equation (5.18) near P1 into
BτA1 “ A1p1`Op|rz1| ` κ21 `A
21 ` ε
21qq
Bτrz1 “ ´rz21 ` κ
21OpA2
1 ` ε21q
Bτκ1 “ ´κ1
Bτ ε1 “ ε1. (5.20)
The inverse transformation is given by
z1 “ rz1 `Oprz1κ21 `A
21 ` ε
21q. (5.21)
Proof. We first note that the center-stable and center-unstable manifolds of P1 are given by Wcsτ´ “ tA1 “ ε1 “
0u and Wcuτ´pεq “ tκ1 “ 0u, respectively. Furthermore,
Wsτ´ ĂWcs
τ´pεq “ tA1 “ z1 “ ε1 “ 0u and
Wcτ´ :“Wcs
τ´pεq XWcuτ´pεq “ tA1 “ κ1 “ ε1 “ 0u.
We also know that solutions on the center manifold are given by z1 “ z˚1 pτq :“ 1{pc1 ` τq with c1 “ 1{z˚1 p0q.
We first show that there exists a smooth change of coordinates of the form
qz1 “ z1 `OpA21 ` ε
21q (5.22)
which transforms the evolution of z1 in equation (5.18) near P1 into
Bτqz1 “ ´qz21 ` κ
21OpA2
1 ` ε21 ` qz1q. (5.23)
To achieve this, we straighten out the strong unstable fibers within the center-unstable manifold: this guarantees
that the evolution of qz1 must be of the form (5.23) so that, within the center-unstable manifold, qz1 evolves
independently of A1 and ε1. It remains to show that this transformation is of the form (5.22). Let pA1, z1, ε1q “
p0, qz1, 0q be a point on the center manifold within the center-unstable manifold. Then, for every A1, qz1, ε1 ! 1,
the strong unstable fiber associated with base point z1 “ qz1 can be written as a graph
z1 “ z1pqz1;A1, ε1q “ qz1 `Opp|A1| ` |ε1|qp|A1| ` |ε1| ` |qz1|qq. (5.24)
Next we claim that the graph (5.24) is actually of the form
z1 “ z1pqz1;A1, ε1q “ qz1 `OpA21 ` ε
21q. (5.25)
22
1 A1
✏1
ez1
⌃�0
transition chart
q0(s)
P+2
P�2P1
fWcss+
2
A2
z2
rescaling chart
(a) The manifold ĂWcss`pεq for fixed ε ă ε0.
fWu⌧�
fWcss+
eq0(�0)A1
ez1
�⌘
⌃�0
O(⌘)
(b) Parametrization of ĂWcss` in
the Poincare section Σδ0 .
Figure 8: Schematic of our use of the blow-up coordinates to parametrize Wcss`pεq.
Letting z1 be the linearized z1 coordinate, we have that |z1pτq´z˚1 pτq| “ Opeντ q for some ν ą 0 in the linearized
strong unstable fiber with τ ď 0. Solving the linearized vector field
Bτz1 “ ´2z˚1 pτqz1
for z1, we have z1 “ c2{pc1 ` τq2 so that c2 “ 0. The expansion for the strong unstable fiber which has tangent
space z1 “ 0 is given by (5.25). By the implicit function theorem we invert (5.25) near pA1, z1, ε1; qz1q “ p0, 0, 0; 0q
to get (5.22).
Next we straighten the strong stable fibers within the center-stable manifold. We let pqz1, κ1q “ prz1, 0q be a point
on the center manifold within the center-stable manifold after transformation (5.22). A completely analogous
argument shows that the strong stable fibers are given by graphs
qz1 “ qz1prz1;κ1q “ rz1p1`Opκ21qq. (5.26)
By combining (5.25) and (5.26) we obtain the inverse transformation (5.21). By inverting (5.26) near pqz1, κ1; rz1q “
p0, 0; 0q we find
rz1 “ qz1p1`Opκ21qq. (5.27)
The evolution of rz1 is independent of κ1 within the center-stable manifold. It is also still independent of A1 and
ε1 within the center-unstable manifold since (5.26) has no effect when κ1 “ 0. Therefore, the evolution of rz1
must be of the form given in (5.20). Composing equations (5.22) and (5.27) gives the transformation (5.19).
5.4 Passage through transition chart coordinates
We use the transversality of q0psq stated in Lemma 5.3 to parametrize Wcss`pεq near P1pεq in the coordinates
of (5.19). Throughout this section we use a tilde to denote any object which has been transformed into the
coordinates (5.19). We refer to Figure 8 for a visualization.
Lemma 5.5 For each sufficiently small δ0 ą 0, there exists constants η0, ε0 ą 0 such that, for all 0 ď ε ď ε0, the
following is true. Define the Poincare section Σδ0 :“ tε1 “ δ0u and let ĂWcss`pεq denote the center-stable manifold
transformed into the coordinates of (5.19) . Then
ĂWcss`pεq X Σδ0 “ tpA1, rz1, κ1, ε1q “ pδ0q0pδ0q `Op|η| ` εq,´η, ε{δ0, δ0q : η P p´η0, η0qu. (5.28)
23
Proof. First consider ε “ 0. Lemma 5.3 shows that the connecting orbit Q01pτq is contained in the unstable
manifold of P1 so that, after transformation into the coordinates of (5.4), the rz1 component of Q01pτq is zero for
all τ . Therefore, the connecting orbit in the transformed coordinates is given by
rQ01pτq X Σδ0 “ tpA1, rz1, κ1, ε1q “ pδ0q0pδ0q, 0, 0, δ0qu .
In the transformed coordinates, the strong unstable manifold ĂWuτ´ of P1 is the A1-axis. Using the transversality
of the intersection ĂWcss` JX
ĂWuτ´ we can parametrize rQ0
1 X Σδ0 by η, a small offset in the rz1 direction, so that
ĂWcss`p0q X Σδ0 “ tpA1, rz1, κ1, ε1q “ pδ0q0pδ0q `Op|η|q,´η, 0, δ0qu.
The fixed points, invariant manifolds, and connecting orbit are all smooth in ε so that for 0 ď ε ď ε0 the
parametrization is given by (5.28).
Lastly, we propagate the initial data (5.28) under vector field (5.20) backwards until the matching point κ1 “
1{r0.
Lemma 5.6 Fix pη0 ą 0 and r0 ą 1{ρ1, where ρ1 was determined in the proof of Proposition 4.2. Then, for each
fixed δ0 ą 0 there is an ε0 ą 0 so that for all η of the form η “ εpη with pη P r0, pη0s and all 0 ă ε ď ε0, we can
solve (5.20) with initial data given by (5.28) at time τ “ 0 back to τ˚ “ ln εr0δ0
. The associated solution at τ “ τ˚
are given by
A1pτ˚q “ εr0`
a0 `Opδ20 ` 1{r20q˘
` r0Oδ0,r0pε2 ln ε|pη| ` ε2q
rz1pτ˚q “ ´εpη `Oδ0,r0pε2 ln εq
ε1pτ˚q “ εr0
κ1pτ˚q “ 1{r0. (5.29)
In the original coordinates the solution becomes
Apτ˚q “ ε`
a0 `Opδ20 ` 1{r20q˘
`Oδ0p|pη|ε2 ln ε` ε2q
Bpτ˚q “ ´ε2pη`
a0 `Opδ20 ` 1{r20q˘
`Oδ0,r0
`
ε2p|pη|2ε ln ε` ε|pη| ` ε ln εq˘
, (5.30)
where the Landau symbol Oδ0,r0 means that the bounding constant and the region of validity may depend on δ0
and r0.
Proof. We set τ “ 0 at ε1 “ δ0, which is possible because the vector field (5.20) is autonomous. A formal
analysis shows that, as a first approximation, A1pτq „ eτ , η “ Opε2q, and z1pτ˚q “ Opηq. We therefore define
η “: εpη, rz1 “: εpz1, and A1 “: eτ pA1 and consider the fixed point system
eτ pA1pτq “ rδ0q0pδ0q ` εOp|pη| ` 1qseτeşτ0Opεpz1`κ2
1`e2σ
pA21`ε
21qdσ
εpz1pτq “ ´εpη `
ż τ
0
”
´ε2pz21 `O´
κ21ppA21e
2σ ` ε21q¯ı
dσ (5.31)
in the p pA1, pz1q coordinates for τ P rτ˚, 0s. A pair of smooth functions pA1, z1q “ peτpA1, εpz1q satisfies (5.20) with
initial data given by (5.28) if, and only if, p pA1, pz1q is a fixed point of (5.31). We show that (5.31) has a unique
fixed point.
We observe that κ1 and ε1 decouple from the rest of the system with κ1 “ ε{δ0e´τ and ε1 “ δ0e
τ . We substitute
these expressions into (5.31) and explicitly integrate the Opκ21 ` ε21q terms in the equation for pA1. Then (5.31)
is equivalent to the system
pA1pτq “ rδ0q0pδ0q ` εOp|pη| ` 1qseOp1{r20`δ
20q`
şτ0Opεpz1`e2σ pA2
1qdσ
pz1pτq “ ´pη ` ε
ż τ
0
”
´pz21 `O´
pA21{δ
20 ` 1
¯ı
dσ. (5.32)
24
Using a standard contraction mapping principal argument on the space of continuous functions one can show
that (5.32) has a unique fixed point. By uniqueness, the fixed point can be written
pA1pτq “ rδ0q0pδ0q ` εOp|pη| ` 1qseOp1{r20`δ
20`εpητ`ε
2τ2q
“ δ0q0pδ0q `Orεp|pη| ` 1q ` δ0p1{r20 ` δ
20 ` ε|pη|τ ` ε
2τ2qs
pz1pτq “ ´pη ` ετOp|pη|2 ` 1q.
We transform back into pA1, rz1q and evaluate at τ “ τ˚ to get
A1pτ˚q “εr0δ0
„
δ0q0pδ0q `O
ˆ
ε|pη| ` ε` δ0p1{r20 ` δ
20q ` δ0|pη|ε ln
εr0δ0` δ0ε
2 ln2 εr0δ0
˙
“ εr0`
a0 `Opδ20 ` 1{r20q˘
` r0Oδ0,r0pε2 ln ε|pη| ` ε2q
rz1pτ˚q “ ´εpη ` ε2 ln
εr0δ0
Op|pη|2 ` 1q
“ ´εpη `Oδ0,r0pε2 ln εq,
which proves (5.29). We invert the transformation rz1 from Lemma 5.4 to recover
z1 “ rz1pτ˚q `O`
κ21pτ˚q `A21pτ˚q ` ε
21pτ˚q
˘
“ ´εpη`
1`Op1{r20q˘
`Oδ0,r0pε2 ln εq.
Finally, we write pA1, z1q in the original pA,Bq coordinates by inverting the transition chart transformation (5.4).
Then Apτ˚q “ A1pτ˚q{r0 and Bpτ˚q “ A1pτ˚qz1pτ˚q{r20, which are given by (5.30).
This concludes the proof of Proposition 4.7.
6 Numerical results, and comparison with earlier findings
In this section, we summarize our numerical results on the existence and stability of oscillons of the planar forced
complex Ginzburg–Landau equation (CGL)
ut “ p1` iαq∆u` p´µ` iωqu´ p1` iβq|u|2u` γu, x P R2. (6.1)
We also compare our numerical findings with those obtained in [8] and [6, 54] for the one-dimensional and planar
CGL with 2:1 forcing, respectively. We remark that we did not carry out an exhaustive numerical investigation,
and further study is required to systematically characterize all of the possible behavior of oscillons far from
onset.
Radial oscillons can be found as solutions of the equation
0 “ p1` iαq´
urr `urr
¯
` p´µ` iωqu´ p1` iβq|u|2u` γu. (6.2)
The following symmetries of (6.2) simplify our computations. First, we can restrict our computations to the
case γ ą 0, because (6.2) respects the gauge symmetry pα, β, γ, ω, µ, uq ÞÑ pα, β, γeiφ, ω, µ, ueiφ{2q. We similarly
restrict our computations to β ą 0, because (6.2) respects the symmetry pα, β, γ, ω, µ, uq ÞÑ p´α,´β, γ,´ω, µ, uq.
Finally, note that (6.2) is equivariant under u ÞÑ ´u so that all nontrivial solutions come in pairs; whenever we
discuss solution counts, we mean the number of group orbits of solutions.
Computing radial oscillons numerically: We performed our numerical investigation by using the contin-
uation package AUTO07p [15] applied to the steady-state problem (6.2) on the interval r0, Ls with Neumann
boundary conditions at r “ 0 and r “ L. We choose L “ 50 for most computations; however, we occasionally
find that solutions broaden so that L must be taken larger. Our results are robust under changes in L, pro-
vided that the interval length is large enough so that the solution does not interact through its tails with the
boundary.
25
Determining 2D stability of oscillons: To determine whether radial oscillons are stable with respect to
planar (radial and nonradial) perturbations, we study their spectral stability as steady states of (6.1) numerically.
In (6.1), we write u “ u1 ` iu2 to get
¨
˚
˝
u1
u2
˛
‹
‚
t
“
»
—
–
¨
˚
˝
1 ´α
α 1
˛
‹
‚
∆`
¨
˚
˝
´µ` γ ´ω
ω ´µ´ γ
˛
‹
‚
´ pu21 ` u22q
¨
˚
˝
1 ´β
β 1
˛
‹
‚
fi
ffi
fl
¨
˚
˝
u1
u2
˛
‹
‚
, (6.3)
where ∆ is the two-dimensional Laplacian. Linearizing this system about a radial oscillon pu1, u2qprq and seeking
solutions to the linearization, written in polar coordinates pr, ϕq, of the form
¨
˚
˝
u1
u2
˛
‹
‚
pr, ϕ, tq “ÿ
kPZeλteikϕ
¨
˚
˝
upkq1 prq
upkq2 prq
˛
‹
‚
,
we arrive at the eigenvalue problems
λ
¨
˚
˝
upkq1
upkq2
˛
‹
‚
“ Lk
¨
˚
˝
upkq1
upkq2
˛
‹
‚
, k P Z,
where
Lk :“
¨
˚
˝
1 ´α
α 1
˛
‹
‚
ˆ
Brr `Br
r´k2
r2
˙
`
¨
˚
˝
´µ` γ ´ω
ω ´µ´ γ
˛
‹
‚
´
¨
˚
˝
1 ´β
β 1
˛
‹
‚
¨
˚
˝
3u21 ` u22 2u1u2
2u1u2 u21 ` 3u22
˛
‹
‚
. (6.4)
We then discretize the operators Lk using finite differences and compute the spectra of the resulting matrices
using Matlab’s sparse eigenvalue solver eigs. We initially compute these spectra for k “ 0: if the solution is
found to be stable, we repeat the computation for k “ 1, 2, 6.
6.1 Theoretical results in 1D and 2D
To prepare for the numerical investigations, we will briefly review the analytical predictions presented in [8]
for bifurcations of standard and reciprocal oscillons for the 1D CGL. Afterwards, we compare them with our
analytical results in 2D.
Standard oscillons bifurcate from the rest state u “ 0, while reciprocal oscillons bifurcate from a different non-
zero rest state u`unif that we will describe in more detail below. For each of these rest states, there are two
bifurcation scenarios, which are also outlined in Figure 9: (i) two spatial eigenvalues change from real to purely
imaginary upon colliding at the origin, whilst the remaining two eigenvalues are on opposite sides of the origin
on the real axis; (ii) two non-real complex conjugate pairs of eigenvalues collide on and split along the imaginary
axis. We refer to the latter case as a Turing bifurcation. In each case, additional conditions on the sub- or
super-criticality of the bifurcation may be needed to guarantee that oscillons bifurcate. Corresponding to these
cases, we define several bifurcation curves Γj as outlined further below and in Figure 9. For each bifurcation
curve Γj , we also define γ “ γj “ γjpωq such that
pω, γjq P Γj for all other parameters fixed. (6.5)
For example, γ “ γ0 means that pω, γ0q P Γ0, where we defined Γ0 in section 3.1 as the bifurcation curve for the
localized solutions in Theorem 2. We now discuss the different bifurcations obtained in [8] in 1D separately for
standard and reciprocal oscillons.
26
linearization aboutu = 0
�0
�a
linearization about
does not exist
�d
withlinearization about
do not existu±unif
u = u±unif
�b
�d
linearization about
linearization about
with
�⇤
u(r)
r
↵! > ↵!↵↵! > ↵!↵
↵! < ↵!↵
↵! < ↵!↵
�0
Note: these conditions are the same at ! since is equivalent to
Figure 9: Color and line style legend for the various bifurcation curves shown in Figure 12 in the pω, γq plane.
The insets for Γ0, Γa, Γb, and Γd represent the spatial eigenvalues of the linearization of (6.2) at r “ 8 about
u “ 0, u “ u`unif , or u “ u´unif . The spatial eigenvalues are shown for the parameter regions indicated; for
parameter values in the opposite parameter region, the spatial spectra should be rotated by 90 degrees. The two
top-left panels correspond to the linearization about u “ 0, whereas the middle and right panels correspond to
the linearization about u “ u˘unif and u “ u´unif . For ω ą ωβ, the solutions u “ u˘unif do not exist below the line
Γb, and it therefore does not make sense to linearize equation (6.2) about u “ u˘unif in this region. For ω ă ωβ,
neither the solution u´unif nor the bifurcation curve Γb exist, and the solution u`unif bifurcates instead from the
trivial solution along Γ0: since u`unif does not exist below Γ0 it does not make sense to linearize equation (6.2)
about this solution for γ ă γ0. Finally, the curve Γ˚ corresponds to the existence of stationary 1D fronts that
connect u “ 0 and u “ u`unif .
27
Standard oscillons in 1D ([8]): We use the spatial eigenvalues of (6.2) linearized about u “ 0 at r “ 8 to
predict bifurcations of standard oscillons. The spatial eigenvalues k satisfy
p1` α2qk4 ` 2pαω ´ µqk2 ` pµ2 ` ω2 ´ γ2q “ 0.
There are two cases to consider. First, we consider bifurcations along
Γ0 :“ tpµ, γq : γ2 “ µ2 ` ω2u,
which is the subject of Theorem 2. Along Γ0, two of the four spatial eigenvalues are zero; the large modulus
spatial eigenvalues are real provided that also αω ă µ. In this case, the small modulus spatial eigenvalues slide
in along the imaginary axis and split along the real axis as γ transitions from γ ą γ0 to γ ă γ0. Along Γ0, 1D
oscillons bifurcate only when µ ă βω, a condition which also arises for 2D oscillons in Theorem 2. Secondly,
along the curve
Γa :“ tpω, γq : p1` α2qγ2 “ pω ` αµq2u,
spatial eigenvalues exist in pairs. These pairs are on the imaginary axis provided that also µ ă αω; in this case,
a Turing bifurcation is expected as γ transitions from γ ą γa to γ ă γa. Oscillons in 1D occur along Γa only
when ηapβ ´ αq ą 0, where ηa :“ α ` sgnpω ` αµq?
1` α2. A schematic of the spatial eigenvalues near Γ0 for
αω ă αωα and near Γa for αω ą αωα, with ωα :“ µ{α, is shown in the left panels of Figure 9.
Reciprocal oscillons in 1D ([8]): Reciprocal oscillons have a nonzero background state. Equation (6.2)
admits non-trivial uniform solutions u˘unif :“ R˘eiφ˘
where
`
R˘˘2
:“ωβ ´ µ˘
a
p1` β2qγ2 ´ pω ` βµq2
1` β2(6.6)
and φ˘ solves
cos 2φ˘ “pR˘q
2` µ
γ, sin 2φ˘ “
ω ´ β pR˘q2
γ. (6.7)
There are two cases to consider depending on ωβ :“ µ{β, as shown schematically in Figure 10.
(i) ω ă ωβ : pR`q2 “ 0 precisely when γ “ γ0 and pR´q2 ă 0 for all γ so that the lower brach solution u´unifdoes not exist. The solution u`unif continues to exist for all γ ě γ0.
(ii) ω ą ωβ : pR´q2 “ 0 precisely when γ “ γ0. Furthermore, the upper and lower branch solutions annihilate
in a saddle node along the curve
Γb :“ tpω, γq : p1` β2qγ2 “ pω ` βµq2u.
Thus, the solution u`unif exists for all γ ě γb and the solution u´unif exists for all γb ď γ ď γ0.
As with standard oscillons, we use the spatial eigenvalues of (6.2) at r “ 8, now linearized about u “ u˘unif , to
predict bifurcations of reciprocal oscillons. The spatial eigenvalues k satisfy
p1` α2qk4 ` 2k2r´µ` αω ´ 2p1` αβq|u˘unif |2s ` 4rp´µ` βωq|u˘unif |
2 ` γ2 ´ γ20 s “ 0.
We focus first on the case ω ă ωβ . In this case, the spatial eigenvalues of the rests state u`unif on Γ0 are exactly
as for the linearization about u ” 0: two of the four spatial eigenvalues are zero while the large modulus spatial
eigenvalues are real provided that also αω ă µ. As γ increases, the small modulus eigenvalues split along the
real axis. Thus, near Γ0 all four spatial eigenvalues associated with the linearization about u`unif are real for
γ ą γ0, provided that also αω ă αωα. The results in [8] predict that reciprocal oscillons will bifurcate generically.
28
(a) The case ω ă ωβ . (b) The case ω ą ωβ .
Figure 10: Existence of the nontrivial uniform solutions u˘unif . The upper branch solution u`unif is shown in dark
magenta whereas the lower branch solution u´unif is shown in light blue.
u
!
�
!��b
z! < z!z
u+unif
u�unif
Figure 11: Nontrivial uniform solutions to (6.2) bifurcate along the curve Γb. The insets show the spatial
eigenvalues of the linearization of (6.2) at r “ 8 about the upper and lower branches plotted in the complex plane
for zω ă zωz; for zω ą zωz, the spatial spectrum is rotated by 90 degrees. At ω “ ωβ, we have u˘unif “ 0; for
ω ă ωβ, the fold ceases to exist.
Along a second curve, which we denote by Γd, the spatial eigenvalues of the linearization about the upper branch
solution u`unif exist in pairs: these eigenvalues lie on the imaginary axis provided that αω ą αωα. A schematic of
the spatial eigenvalues from the linearization about uunif` with ω ă ωβ is shown in the center panels of Figure 9.
We remark that Γd can be computed only numerically.
If ω ą ωβ , we compute the spatial eigenvalues near Γb. We will find it convenient to define the variables3
z :“ αp1´ β2q ´ 2β and ωz :“ ´µp1´ β2 ` 2αβq{z.
On Γb, two spatial eigenvalues equal zero, whilst the two remaining nonzero eigenvalues are real provided zω ă
zωz. The spatial eigenvalues for the upper and lower branches near the fold are indicated in the insets in
Figure 11. In particular, the small modulus spatial eigenvalues for the linearization about the upper branch
solution u “ u`unif are real, and [8] predicts the bifurcation of reciprocal oscillons. Along the curve Γd, the spatial
eigenvalues of the linearization about the upper branch solution exist again in pairs. These pairs are purely
imaginary provided that also zω ą zωz. The right panels of Figure 9 show a schematic of the spatial eigenvalues
from the linearization about uunif` , with ω ą ωβ .
We make two final remarks. Firstly, the spatial eigenvalues associated with the linearization about the lower
branch solution u´unif near Γb are such that two are real whereas the other two are purely imaginary, as is shown
in the top right panel in Figure 9. Thus, it is not expected that localized solutions that are asymptotic to u´unifexist. Secondly, although the Turing bifurcation curve for reciprocal oscillons Γd has a different condition for
ω ă ωβ (we require that αω ą αωα) as for ω ą ωβ (we require that zω ą zωz) these conditions are actually
3We remark that [8] defines ´z instead of z. Hence, all inequalities involving z are reversed in [8] from the inequalities discussed
in this section.
29
consistent at the transition point ω “ ωβ since zω ą zωz can be rewritten
αp1` β2qpω ´ ωαq ´ 2βp1` αβqpω ´ ωβq ą 0.
Comparison between theoretical results in 1D and 2D: The only theoretical 2D result that we are aware
of is our Theorem 1 that describes the bifurcation of standard planar oscillons with monotone tails: its hypotheses
and conclusions are the same as for the 1D case studied in [8]. As discussed in more detail in section 7, we believe
that our techniques can be used to obtain theoretical results for Turing bifurcations to standard planar oscillons
as well as for the bifurcations to reciprocal planar oscillons along Γ0 and Γb.
We recall that 1D Turing bifurcations require a subcriticality condition on the nonlinearity: for the 2D Swift–
Hohenberg equation, it was shown in [32] that the same condition is needed for cubic nonlinearities (i.e., for
systems with a u ÞÑ ´u symmetry), whilst no such condition is needed for quadratic-cubic nonlinearities, which
break this symmetry. The normal form for Turing bifurcations from u “ 0 in the CGL obeys the u ÞÑ ´u
symmetry, which suggests that a subcriticality condition might be needed for planar standard oscillons: we tried
to find oscillons numerically for the planar CGL near Turing bifurcations in regions where the bifurcation is
supercritical and were indeed not successful. On the other hand, Turing bifurcations from u “ u`unif will likely
not obey the u ÞÑ ´u symmetry, thus suggesting that this condition may not be needed for planar reciprocal
oscillons: we did not attempt to find reciprocal oscillons in the supercritical region.
6.2 Overview of numerical results for planar standard oscillons
Our numerical results for standard oscillons of (6.2) are summarized schematically in Figure 12 for four different
parameter sets for pβ, α, µq, which are listed in Table 2. The schematic diagrams in Figure 12 are not meant to
represent all possible behaviors of oscillons far from onset; a more detailed study is needed. Standard oscillons
with monotone tails bifurcate from Γ0, while standard oscillons with oscillatory tails bifurcate from Γa. Solutions
terminate at Γ˚; we discuss this curve in section 6.3 below. For reference, we also plot Γb and Γd, the bifurcation
curves for reciprocal oscillons. Each bifurcation curve Γj is highlighted whenever the spatial eigenvalues are as
shown in Figure 9; otherwise (i.e. when the spatial eigenvalues are rotated by 90 degrees from the orientation in
Figure 9), we represent Γj with the same line style as in the legend but thin and black. Our numerical results
are plotted either as graphs of ||u||L2 versus γ or in the pω, γq plane. In either case, it is understood that the
other parameters are held constant.
We remark that our numerical findings for standard as well as for reciprocal planar oscillons showed qualita-
tive differences to the 1D computations reported in [8], which we comment on in more detail in the following
sections.
6.3 Planar standard oscillons bifurcating from Γ0
To simplify notation, we define the critical parameter values ωβ :“ µ{β and ωα :“ µ{α. The hypotheses in
Theorem 2 are then equivalent to
(i) ω ą ωβ : equivalent to the hypothesis c03 ă 0, and
(ii) αωα ą αω: necessary so that all four spatial eigenvalues immediately below Γ0 are real (see Figure 9, and
also Figure 4 which contains additional information about spatial eigenvalues in the pµ, γq plane).
Alternatively, the existence conditions (i) and (ii) are equivalent to ωα ą ω ą ωβ for α ą 0, and ω ą ωα, ωβ for
α ă 0. Since the existence region for ω rescales with the magnitude of the parameters α, β, and µ, we define
parameter regions based solely on the signs of α and µ, which we label counterclockwise in the pα, µq plane
30
!
�
Q1
!� !↵!!� !↵
�
Q2
↵ > 0, µ < 0↵ < 0, µ < 0
�0 �0
�⇤
�d
�⇤
!
�
Q3
!↵ !�
↵ < 0, µ > 0
�0
�⇤
!
�
Q4
!� !↵
↵ > 0, µ > 0
�0
�⇤
�d
�0
�d
�0
�d
�0
�0
!z
!z
↵ > �
↵ < �
�d
�d
!z
�d
�d
!z
�b
�b
�b
�b
�b
�b
�a
�a
�a
�a
z < 0
z > 0
z > 0
z > 0
�a
�a�a
Figure 12: Expected existence regions of planar standard oscillons in each of the parameter quadrants. We refer
to Figure 9 for the interpretation of the curves shown here: the segments of the curves Γj that coincide with
bifurcations are highlighted; otherwise, we use the same line style as in Figure 9, but the curve is thin and black.
Small-amplitude oscillons with monotone tails bifurcate into the parameter region below Γ0 provided that also
ω ą ωβ :“ µ{β; small-amplitude oscillons with oscillatory tails bifurcate into the parameter region below Γa.
The dark salmon shaded regions indicate the numerically observed existence region for oscillons with monotone
tails; the light green shaded regions indicate the existence region for oscillons with oscillatory tails. Solutions
bifurcating from Γ0 are observed to develop oscillatory tails for γ ă γa (with the notation of (6.5)). In parameter
regions Q1, Q2, and Q4, all localized solutions terminate in a stationary one-dimensional front at Γ˚. In Q3,
Γ˚ ends at the curve Γd; the termination of solutions for parameter values to the left of Γd in Q3 is not well
understood. 2D stable planar oscillons were found between two saddle nodes in a small subset of Q2; see also
Figure 13. In Figures 13-16, we show the results of numerical continuation along the vertical blue lines indicated
above.
31
β α µ z :“ αp1´ β2q ´ 2β
Q1 10.0 0.5 10.0 ´69.5
Q2 2.5 ´2.0 0.5 5.5
Q3 2.5 ´2.0 ´0.5 5.5
Q4 0.5 10.0 ´5.0 6.5
Table 2: Values of the fixed parameters for our numerical computations in each region.
starting with α ą 0, µ ą 0
Q1 :“ α ą 0, µ ą 0, Q2 :“ α ă 0, µ ą 0, Q3 :“ α ă 0, µ ă 0, Q4 :“ α ą 0, µ ă 0.
We also set α ă β in region Q1 and α ą β in region Q4 so that both conditions (i) and (ii) are satisfied
simultaneously. We performed a few selected example computations in each parameter region: in particular, we
did not carry out an exhaustive study, and further study is required to determine whether the solution behavior
indicated in Figure 12 is representative. In particular, large amplitude solutions may interact with Γb and Γd
in complicated ways, based on the sign of z. Below, we report on one computation for a single parameter set,
listed in Table 2, in each of the quadrants.
An example bifurcation diagram for localized solutions emerging from Γ0 in Q2 is shown in Figure 13 along with
some representative solution profiles. Near Γ0, localized solutions have small amplitude. As γ is decreased away
from γ0, the profile grows and narrows; as a result, the solution L2 norm increases. For γ ă γa, solutions develop
oscillatory tails. We observe that the solutions resemble one-dimensional stationary fronts for γ far enough below
γ0. At this point, a small decrease in the value of γ causes the front interface to shift to the right, which, in turn,
increases the solution L2 norm; in fact, we observe that the continuation curve asymptotes in the pγ, ||u||L2q-
plane at some value of γ, which we call γ˚ :“ γ˚pωq. In two-parameter pω, γq-space we denote the boundary of
the oscillon existence region corresponding to pω, γ˚pωqq by Γ˚. We observe that solutions along Γ˚ resemble
one-dimensional stationary fronts. To verify this observation, we computed the curve in pω, γq space along which
one-dimensional stationary fronts connect u “ u`unif and u “ 0 and compared it with Γ˚: they coincide, and the
resulting solution profiles look identical. Since u`unif exists in the region γ ă γ0 only for ω ą ωβ , this observation
provides insight into why Γ˚ bifurcates from the point ω “ ωβ . Similarly, since standard oscillons terminate at
Γ˚, this provides another rationale for the standard oscillon existence condition ω ą ωβ , which was necessary in
the proof of Theorem 2 and which we observed numerically, as indicated in Figure 12. Finally, these observations
explain why we observe γ˚ ą γb, provided Γ˚ exists, as is also indicated in Figure 12.
Next, we comment on 2D stability. First, planar oscillons are always unstable near onset. In regions Q3 and
Q4, the essential spectrum is in the right half-plane due to the instability of the background state u “ 0; hence,
standard oscillons can never stabilize in these regions. In the parameter region Q2, standard oscillons were found
to be 2D stable in between two saddle nodes, whenever such saddle nodes exist; see Figure 13. A two-parameter
continuation of the saddle nodes in region Q2 with z ą 0 indicates that they collide in a cusp near Γ˚ in the
parameter region where both zω ą zωz and γ˚ ą γd. The numerically computed existence region of stable
standard oscillons is indicated by the spotted area in Figure 12. In region Q1, we do not observe saddle nodes
of standard oscillons, and these never stabilize.
Our numerical results on standard oscillons in the planar CGL are consistent with the results for 1D oscillons
in [8] with the following difference: As indicated in Figure 13, planar standard oscillons are unstable at onset
and stabilize at a saddle node at some value of γ above, but near, γ˚; the oscillons then destabilize at a second
saddle node, and the branch of unstable 2D oscillons asymptotically approaches γ˚ from above. In 1D, we again
find stable oscillons in only a small subset of Q2; however, the stability region extends below γ˚ and several
32
(a)(b)
(c)
(d)
�⇤ �0
||U|| L
2
�
(a) � = 3.59
ReU
s
(c) � = 2.69 (d) � = 2.53
(b) � = 2.88
Figure 13: Planar standard oscillons with monotone tails in region Q2 (bifurcation diagram [left] and sample
solution profiles [right]) with β “ 2.5, α “ ´2.0, and µ “ 0.5. We choose ω “ 3.65 so that two saddle-node
bifurcations occur along the solution branch: Solutions are 2D stable along the solid curve, and unstable along
the dashed curves. Note that solution profiles are plotted as functions of s “ r{L P r0, 1s with L “ 26.1. The
location of the continuation curve in pω, γq-space is indicated by the blue vertical line in Figure 12.
�
||U|| L
2
�⇤ �0
Figure 14: 1D standard oscillons in Q2. The parameter values are the same as those for the planar oscillons shown
in Figure 13. Solutions are 1D stable along the solid curve segments and unstable along the dashed segments.
Solution profiles are plotted as functions of s “ r{L P r0, 1s with L “ 50.0.
33
(a)
(b)
(c)
(d)
�⇤
||U|| L
2
�
(b) � = 37.89
(d) � = 24.75(c) � = 25.77
(a) � = 57.17
ReU
s
�a
Figure 15: Planar standard oscillons with oscillatory tails in region Q1 (bifurcation diagram [left] and sample
solution profiles [right]) with β “ 10.0, α “ 0.5, and µ “ 10.0. We choose ω “ 59.0 so that ω ą µ{α “ 20.0.
Solution profiles are plotted as functions of s “ r{L P r0, 1s with L “ 16.0. The location of the continuation curve
in pω, γq-space is indicated by the blue vertical line in Figure 12.
(a)(b)
(c)
(d)
||U|| L
2
0 �
(d) � = 2.4 ⇥ 10�2(c) � = 2.0 ⇥ 10�2
(b) � = 7.4 ⇥ 10�2(a) � = 2.8 ⇥ 10�1
ReU
s
�a
Figure 16: Planar standard oscillons with oscillatory tails in region Q3 (bifurcation diagram [left] and sample
solution profiles [right]) with β “ 2.5, α “ ´2.0, and µ “ ´0.5. We choose ω “ 0.13 so that ω is in the parameter
region to the left of Γd. Solutions do not stabilize at the saddle nodes due to the instability of the background state.
Solution profiles are plotted as functions of s “ r{L P r0, 1s with L “ 100.0. The location of the continuation
curve in pω, γq-space is indicated by the blue vertical line in Figure 12.
34
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
-1 -0.5 0 0.5 1 1.5 2
a
t
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
-1 -0.5 0 0.5 1 1.5 2
a
t
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
-1 -0.5 0 0.5 1 1.5 2
a
t
0.15
0.2
0.25
0.3
0.35
0.4
0.5 0.6 0.7 0.8 0.9 1
a
t
0.15
0.2
0.25
0.3
0.35
0.4
0.5 0.6 0.7 0.8 0.9 1
a
t
0.15
0.2
0.25
0.3
0.35
0.4
0.5 0.6 0.7 0.8 0.9 1
a
t!
��d
�⇤
�d
�⇤�0
!↵
!�
�↵µ�a
�a
�b
�b
�b
�0
Figure 17: Bifurcation diagram for planar standard oscillons in pω, γq-space in Q3 with β “ 2.5, α “ ´2.0, µ “
´0.5 and interval length L “ 75.0. The thin solid lines represent the continuation curves of the saddle nodes.
The dynamics appears to be organized by Γd in the region where Γ˚ ceases to exist. In the region shown, we have
zω ą zωz: hence, spatial eigenvalues exist in pairs on the imaginary axis along Γd, whilst they are rotated 90
degrees from how they are shown in Figure 9 along Γb.
stable solutions coexist, as indicated in Figure 14. In particular, both the existence and stability regions for 1D
standard oscillons are larger in this parameter region compared with 2D oscillons.
6.4 Planar standard oscillons near Turing bifurcations: collapsed snaking
As indicated in Figure 12, we find that standard oscillons with oscillatory tails bifurcate from Γa provided
αω ą αωα, as expected. An example bifurcation diagram and representative solution profiles are shown in
Figure 15. In Q1 and Q2 the behavior of these solutions is similar to that of the monotone tail solutions:
solutions have small amplitude near the bifurcation curve Γa; the amplitude becomes larger and the envelope
narrows as γ decreases; the continuation curve again terminates at Γ˚—the curve of one-dimensional stationary
front solutions. None of these solutions were found to be stable. Our numerical results on standard oscillons with
oscillatory tails in the planar CGL in Q1 and Q2 are consistent with the one-dimensional results in [8].
In Q3, Γ˚ cannot bifurcate from Γ0; see Figure 12: instead, we observe that Γ˚ terminates at Γd, where u`unifundergoes a bifurcation to spatially periodic waves; hence, one-dimensional stationary fronts in this regime have
oscillatory core. In parameter regions to the left of Γd, localized solutions do not terminate in one-dimensional
stationary fronts. An example bifurcation diagram and representative solution profiles in the parameter region
to the left of Γd are shown in Figure 16: Near Γa, oscillons have small amplitude with oscillatory tails. As γ is
decreased, oscillations are added to the core; as a result, the bifurcation curve demonstrates collapsed snaking.
We also performed a two-parameter continuation of the saddle nodes in the collapsed-snaking region: the result is
shown in Figure 17. We believe that these saddle-node curves, and therefore collapsed snaking, exist all the way
to the left of Γd until ω ą ´αµ, which is the intersection point of Γa with the γ “ 0 axis: First, we observe that
solution profiles exhibit similar behavior to that shown in Figure 16 along the bifurcation curves everywhere to
the left of Γd. In particular, as the parameter γ is decreased, we find that the profiles contain more and more core
oscillations. We believe that each new oscillation is introduced at a saddle node, but that the difference between
successive saddle nodes becomes too small to resolve numerically. As is shown in Figure 17, the dynamics of the
saddle nodes in parameter space near Γd is complicated and not well understood. We expect the same behavior
of solutions in Q3 regardless of the sign of z, with the difference that Γ˚ may bifurcate from Γb rather than Γd;
this case, we expect that the front solution will have a monotone rather than oscillatory core.
35
(a)(b)
(c)
(d)||U
|| L2
�
ReU
s
�a�⇤
(d) � = 2.200(c) � = 2.298
(b) � = 3.213(a) � = 4.875
Figure 18: Planar standard oscillons with oscillatory tails in Q4 (bifurcation diagram [left] and sample solution
profiles [right]) with β “ 0.5, α “ 10.0, and µ “ ´5.0. We choose ω “ 0.93 so that ω ą ωα but to the left of the
bifurcation curve Γb. Solution profiles are plotted as functions of s “ r{L P r0, 1s with L “ 150.0. The location
of the continuation curve in pω, γq-space is indicated by the blue vertical line in Figure 12.
I : z ą 0 α ą β α ą 0
II : z ă 0 α ą β α ą 0
III : z ă 0 α ă β α ą 0
IV : z ă 0 α ă β α ă 0
V : z ą 0 α ă β α ă 0
Table 3: Notation for the relevant parameter regions, given in [8]. The superscript ˘ refers to the sign of ´µ.
In Q4, the curve Γ˚ similarly terminates at Γb, the curve along which u˘unif bifurcates. Hence, for ω to the right
of this intersection point, solution profiles do not approach one-dimensional fronts, as indicated in Figure 12.
Preliminary computations indicate that in Q4 to the right of Γb the one-parameter bifurcation diagram and
solution profiles resemble those for Q3 to the left of Γd: we observe saddle nodes as oscillations are added to the
core. More detailed numerical computations are needed in order to characterize the similarities and difference
between solutions in Q3 and in Q4.
6.5 Planar reciprocal oscillons
As with standard oscillons, we expect that the behavior of reciprocal oscillons far from onset will depend on the
values of the fixed parameters α, β and µ. While it suffices to consider the four parameter regions Q1 to Q4 to
describe the typical behavior of standard oscillons, this is no longer true for reciprocal oscillons as the sign of
z and the location of ωz relative to the other critical points ωα and ωβ become relevant. In what follows, we
therefore use the parameter region notation from [8], which we recall in Table 3. The superscript ˘ refers to
the sign of ´µ.4 We now report on preliminary findings in regions III˘ and I´ and note that a more exhaustive
study is needed to cover the other regions.
4Recall that [8] uses the parameter `µ whereas we have used ´µ. Therefore, consistency with the notation of [8] requires that
the superscript be given by the sign of ´µ.
36
(a)
(b)
(c)
(d)
||U|| L
2
�
s
ReU
(a) � = 12.8423 (b) � = 12.9392
(c) � = 12.9306 (d) � = 12.9303
�b
Figure 19: Planar reciprocal oscillons with monotone tails in III´ (bifurcation diagram [left] and sample solution
profiles [right]) with β “ 0.5, α “ 0.33, and µ “ 1.0. We choose ω “ 13.86 so that several saddle-node bifurcations
occur. Solutions are 2D stable along the solid curve and unstable along the dashed curve. Solution profiles are
plotted as functions of s “ r{L P r0, 1s with L “ 33.7. Γ˚ represents again the curve of stationary one-dimensional
fronts that connect u “ 0 and u`unif .
0
0.02
0.04
0.06
0.08
0.1
4 6 8 10 12
`
0
0.02
0.04
0.06
0.08
0.1
4 6 8 10 12
`
0
0.02
0.04
0.06
0.08
0.1
4 6 8 10 12
`
0
0.02
0.04
0.06
0.08
0.1
4 6 8 10 12
`
(a) Parameter region III´ as in Figure 19 with
µ “ 1.0.
-0.1
0
0.1
0.2
0.3
0.4
0 2 4 6 8 10 12 14
`
-0.1
0
0.1
0.2
0.3
0.4
0 2 4 6 8 10 12 14
`
-0.1
0
0.1
0.2
0.3
0.4
0 2 4 6 8 10 12 14
`
-0.1
0
0.1
0.2
0.3
0.4
0 2 4 6 8 10 12 14
`
(b) Parameter region III` with µ “ ´1.0.
Figure 20: 2D stability regions of planar reciprocal oscillons for β “ 0.5 and α “ 0.33. Note that the bifurcation
curves Γj are plotted relative to Γβ.
37
(a)
||U|| L
2
�
ReU
s
(b)
Figure 21: Planar reciprocal oscillons with oscillatory tails in I´ (bifurcation diagram [left] and sample solution
profiles [right]) with β “ 0.0, α “ 4.0, µ “ 1.0, and ω “ 1.0; then γd “ 1.48, computed numerically. These
parameter values are taken from [6], where a stable planar reciprocal oscillon was found through direct simulations
at γ “ 1.8. Solutions are 2D stable along the two solid branches of the bifurcation curve that are delineated by
successive saddle nodes. Solution profiles are plotted as functions of s “ r{L P r0, 1s with L “ 70.0. The dotted
line represents the nontrivial uniform solution u`unif . The parameter value γ˚ corresponds to the existence of a
1D stationary Ising front that connects u`unif to itself.
�
||U|| L
2
s
ReU
(a)
(b)
Figure 22: Bifurcation diagram for 1D reciprocal oscillons in I´. The parameter values are the same as for the
planar oscillons shown in Figure 21. All solutions are 1D unstable. Solution profiles are plotted as functions of
s “ r{L P r0, 1s with L “ 70.0.
38
First, we focus on the regions III˘. An example bifurcation diagram and representative solution profiles in region
III´ is shown in Figure 19. We found that planar reciprocal oscillons bifurcate from Γb and terminate in the one-
dimensional stationary fronts that connect u “ 0 and u`unif . The reciprocal oscillons are unstable near onset but
can stabilize at saddle-node bifurcations. Continuing the locations of these saddle-node bifurcations delineates
the stability regions for reciprocal oscillons: Figure 20 summarizes our stability findings separately for regions
III´ and III`. Note that stable planar reciprocal oscillons seem to exist only in narrow wedges close to the curve
Γ˚. We remark that stable planar reciprocal oscillons were previously found in [54] for isolated parameter values
in regions III˘ using direct numerical simulations: Figure 20 traces out their stability regions.
Our second case study is for region I´. A stable planar reciprocal oscillon was found previously in [6] using direct
numerical simulations. We continued this solution in γ and found that it bifurcates from Γd and ends at a curve
Γ˚ that corresponds to stationary 1D Ising fronts, that is, to fronts that connect u`unif to ´u`unif . As indicated in
Figure 21, this reciprocal planar oscillon is 2D stable over a narrow range of γ values: both the narrow existence
region and the proximity of oscillons to the Ising fronts agree with the direct numerical simulations carried out
in [6]. We also computed the 1D reciprocal oscillons that bifurcate from Γd in this region and found that they
never stabilize in 1D; see Figure 22: this agrees with [8, Table 1], where it is noted that stable 1D reciprocal
oscillons should not exist in I´.
7 Discussion
Localized solutions to the one-dimensional 2:1 forced complex Ginzburg–Landau equation (CGL) were previously
studied in [8, 54]. Numerical results for planar oscillons in the 1:1 and 2:1 forced CGL were carried out, for
instance, in [25] and [6, 54], respectively. In this paper, we carried out numerical computations and a theoretical
analysis of oscillons in the planar CGL with 2:1 forcing.
Numerics: Using the numerical continuation software package AUTO07, we observed the existence of four
distinct localized radial steady-state solutions to the planar CGL:
(i) standard oscillons with monotone tails;
(ii) standard oscillons with oscillatory tails;
(iii) reciprocal versions of (i) and (ii); and
(iv) standard and reciprocal front-like solutions.
We used a far-field spatial eigenvalue analysis to predict parameter-space curves along which each solution type
is expected to bifurcate. Sketches of the various bifurcation curves and associated spatial eigenvalues were shown
in Figure 9. We observed numerically that solution type (i) bifurcates from the curve Γ0 along which two spatial
eigenvalues collide at the origin, provided that the other two spatial eigenvalues are real and bounded away from
the origin; we also observed that solution type (ii) bifurcates from the curve Γa along which pairs of eigenvalues
collide on the imaginary axis away from the origin. Preliminary computations indicate that the analogous
statements hold for the reciprocal version of each solution type. Away from onset, standard and reciprocal
oscillons often terminate in 1D stationary fronts: however, we also found far more complicated bifurcation
scenarios that involved, for instance, collapsed snaking; we also found reciprocal oscillons that terminate in Ising
fronts, a phenomenon not reported on previously in 1D. It is interesting to note that the situation for the 1:1
forced CGL seems to be simpler: the results presented in [25] indicate that the existence regions of oscillons are
organized by a Takens–Bogdanov point; features there included Hopf bifurcations of oscillons as well as their
termination along curves at which their period approaches infinity. The stability regions for both standard and
reciprocal oscillons in the 2:1 forced CGL were found to be narrow, in agreement with previous results in [6, 54]
39
where direct numerical simulations were utilized. We emphasize that a more detailed numerical study is needed
to completely characterize the behavior of solutions far from onset.
Theory: We proved rigorously that standard oscillons bifurcate along Γ0, provided the leading-order nonlinear
term has the correct sign. Our method consisted of matching solutions that stay bounded at the core with
those that decay in the far field. To find the latter, we utilized the geometric blow-up strategy of [32]. A major
difference from previous work is that the matching analysis required a center-manifold reduction and expansions
of invariant manifolds and foliations.
Connections with experiments: The narrowness of the stability regions for both standard and reciprocal
oscillons raises the question of how accessible these solutions are in experiments: it is difficult to answer this
question by extrapolating from the 2:1 forced Ginzburg–Landau equation. In several experiments, see for instance
[2, 6], the regions where oscillons are observed are also very narrow. What aids their experimental observability
is that the main bifurcation parameters are the amplitude and frequency of the forcing, which can usually be
adjusted during an experiment.
Open problems: In future work, we plan to apply the blow-up coordinates used in both this paper and [32] to
prove the existence of solutions of types (ii)-(iv) near onset. We expect that the general framework and strategy
will be the same, with the following modifications. For standard and reciprocal oscillons with oscillatory tails,
the dimension of the center manifold near onset will be four, and the dynamics will be different from the one
we encountered in this paper. The analysis on the four-dimensional center manifold should be similar to the
one carried out in [31], and it would be interesting to see whether additional subcriticality conditions will, or
will not, be needed for bifurcations to standard or reciprocal oscillons (see section 6.1 for a discussion). For
bifurcations to reciprocal oscillons with monotone tails, we expect to find a quadratic, rather than cubic, leading
term in the nonlinearity on the center manifold: this alters the dynamics, but the methods used here should
apply without major change. Finally, the front-like structures at which standard oscillons terminate should
emerge from codimension-two points at which the cubic coefficient changes sign: though the analysis will depend
on the quintic terms, we expect that the general approach taken here should apply to this case as well.
We also plan to justify the CGL as a normal form for periodically forced reaction-diffusion equations near a
supercritical Hopf bifurcation. While formal multiple scales analyses indicate clearly that the CGL describes,
to leading-order, the long-scale modulation of the amplitude of solutions to such systems, we believe it would
be useful to make this connection rigorous using the following two complementary approaches: First, we plan
to extend the spatial dynamics for time-periodic patterns that was developed in [38, Chapter 4] for autonomous
reaction-diffusion systems near Hopf bifurcations to nonautonomous systems with periodic forcing. We expect
to be able to prove that the stationary forced CGL is the normal-form equation in this setting on an appropriate
spatial center manifold in a space of time-periodic functions. Second, we plan to show that the time-dependent
forced CGL governs the dynamics of the envelope ApX,T q of solutions upx, tq of the form
upx, tq “ εApεx, ε2tqeiω0t ` ε2Rpx, tq ` c.c.
of the forced reaction-diffusion system over time scales of order 1{ε2 by showing that the remainder Rpx, tqremains bounded over that time scale; this is the approach taken to justify the complex Ginzburg–Landau
equation for unforced reaction-diffusion equations in [39]. One possible strategy for proving boundedness of the
remainder is to use the space-time normal forms described recently in [19–22].
Acknowledgments McQuighan was supported by the NSF through grants DMS-0907904 and DMS-1148284.
Sandstede was partially supported by the NSF through grant DMS-0907904. We would like to thank Victor
Brena-Medina for helpful comments on this manuscript.
40
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