invariant manifolds for nonsmooth systems
TRANSCRIPT
Physica D 241 (2012) 1895–1902
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Physica D
journal homepage: www.elsevier.com/locate/physd
Invariant manifolds for nonsmooth systemsD. Weiss a, T. Küpper b,∗, H.A. Hosham b
a Mathematical Institute, University of Tübingen, Auf der Morgenstelle 10, D-72076, Germanyb Mathematical Institute, University of Cologne, Weyertal 86-90, D-50931, Germany
a r t i c l e i n f o
Article history:Available online 29 July 2011
Keywords:Invariant manifoldNonlinear piecewise dynamical systemsInvariant conesPeriodic orbitsGeneralized Hopf bifurcation
a b s t r a c t
For piecewise smooth systems we describe mechanisms to obtain a similar reduction to a lowerdimensional system as has been achieved for smooth systems via the center manifold approach. Itturns out that for nonsmooth systems there are invariant quantities as well which can be used for abifurcation analysis but the form of the quantities is more complicated. The approximation by piecewiselinear systems (PWLS) provides a useful concept. In the case of PWLS, the invariant sets are givenas invariant cones. For nonlinear perturbations of PWLS the invariant sets are deformations of thosecones. The generation of invariant manifolds and a bifurcation analysis establishing periodic orbits aredemonstrated; also an example for which multiple cones exist is provided.
© 2011 Elsevier B.V. All rights reserved.
1. Introduction
Within the frame of the standard theory of smooth dynamicalsystems, it is well-known that qualitative changes (bifurcations)are typically due to changes in lower dimensional subsystems.The classical approach to analyze the qualitative behavior takesadvantage of a reduction of the original high dimensional systemto a lower dimensional system carrying the essential dynamics ofthe full system via the center manifold theorem. This approach isbased on information provided by the linearization and requiressmoothness of the functions determining the dynamical process.In that way a bifurcation and stability analysis of complex systemscan substantially be reduced to the study of low dimensionalsystems by retaining the relevant interactions.
The investigation of nonsmooth dynamical systems and theirbifurcations is a research topic of present interest. Following theclassical approach it is a natural question to explore if similarreduction techniques are at hand without a requirement ofsmoothness.
The standard center manifold approach is related to propertiesof a linearized system, especially involving the number ofeigenvalues with vanishing real part determining the center spacespanned by the corresponding eigenvectors. Due to a lack ofdifferentiability the notion of a linearization is not at hand;likewise there is no way to define eigenvalues. Instead of theanalytical description of eigenvalues crossing the imaginary axis
∗ Corresponding author.E-mail addresses:[email protected] (D. Weiss),
[email protected] (T. Küpper), [email protected](H.A. Hosham).
0167-2789/$ – see front matter© 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.physd.2011.07.012
a rather geometric view turned out to be of help which forexample can be explained in the case of Hopf bifurcation [1–4].The analytical criterion giving rise to Hopf bifurcation is given bythe transversal crossing of a pair of complex eigenvalues throughthe imaginary axis. Since eigenvalues are not defined in a properway, the corresponding geometric feature of a change in the phasespace from a stable focus to an unstable focus through a centerturns out to be the appropriate mean to describe a process whichcan be carried over to nonsmooth systems as well. For planarsystems this view has been explored in [5–9] in connection withgeneralizedHopf bifurcation. For the special case of planar systemsthere was of course no need for any reduction. Based on thisexperience an extension to higher dimensional systems has beendeveloped in [10,11] for piecewise linear systems. Here the notionof an invariant cone appeared generalizing the focus to an objecton a cone consisting of periodic orbits or orbits spiraling ‘‘in’’and ‘‘out’’ of zero, respectively. In the case of smooth systems,the cone reduces to an object which can be regarded as a flat,degenerated cone. It is the key observation to view that cone as ageneralized invariant ‘‘manifold’’ determining the dynamics. In factit is themain result of this paper to establish that even for nonlinearperturbations of piecewise linear systems there is a cone-likeinvariant ‘‘manifold’’ carrying the essential dynamics of the fullsystem under appropriate conditions. In that way a reductionprocedure to a two-dimensional surface has been established fornonsmooth systems allowing a bifurcation and stability analysis ofa reduced system.
The similarity with smooth systems already suggests tounderstand this procedure as a generalization of a center manifoldreduction. The connections will be supported by the mechanismsemployed to proof this result which make use of a Poincaré mapwith eigenvalue close to 1. We emphasize that the role of the cone
1896 D. Weiss et al. / Physica D 241 (2012) 1895–1902
as a key element in the reduction procedure has been developedwithin this approach; the existence of invariant cones consistingof periodic orbits, though, has already been observed in [1,12,2] inconnection with piecewise linear system.
The first results in the construction of invariant cones havebeen obtained for piecewise linear systems PWLS [11]. In thatcase the Poincaré map is (at least formally) explicitly given. Duedo piecewise linearity it pertains linear behavior in one way,on the other hand a strong nonlinear component enters via theintersection times. Different scaling properties of those timesin various directions offer a key to understand an extension tononlinear perturbations of PWLS.
As main result we will show that for nonlinear perturbations ofhomogeneous PWLS (PWNS) there is an invariant cone-like surfacetangent at the origin to a cone of a basic PWLS. Although this resultseems obvious at a first glance the proof requires subtle estimatesinvolving different directions. The appropriate set to construct thePoincaré map for the PWNS is given by a small sector definedaround the generating vector of the cone of the PWLS. The Poincarémap is then defined as sum of the Poincaré map of the PWLS anda nonlinearity. Using the Hadamard graph transform the existenceof an invariant generating curve is established. Once existence ofthe invariant and attractive surface has been established it can beused for bifurcation analysis.
We illustrate this process by an example for which the Poincarémap can explicitly be worked out.
Consequently it is possible to determine the leading coefficientsof the function generating the invariant surface.
For simplicity we restrict our attention to piecewise smoothsystems consisting of 2 components separated by a hyperplaneM := {ξ ∈ Rn
|eT1ξ = ξ1 = 0}, where ei denotes the ith unitvector:
ξ =
f+(ξ), ξ1 > 0,f−(ξ), ξ1 < 0 (1)
with smooth functions f+, f−:Rn→ Rn; we further assume that
f+(ξ) = A+ξ + g+(ξ) (2a)
f−(ξ) = A−ξ + g−(ξ) (2b)
with constantmatricesA± and nonlinear Ck-parts g±(ξ) = o(∥ξ∥),k ≥ 1. Here we restrict our attention to trajectories with immedi-ate transition between the halfspaces Rn
−:= {ξ ∈ Rn
| ξ1 < 0}and Rn
+:= {ξ ∈ Rn
| ξ1 > 0}. In the following we refer to sys-tem (1) as ⊖-system and ⊕-system in the half-spaces Rn
−and Rn
+
respectively. Define
W±
< := {ξ ∈ M | eT1A±ξ < 0},
W±
> := {ξ ∈ M | eT1A±ξ > 0}.
Then for any initial value ξ ∈ W−< ∩ W+
< the trajectory ϕ(t, ξ) =
etA−
ξ of the homogeneous PWLS
ξ =
A+ξ, ξ1 > 0,A−ξ, ξ1 < 0, (3)
enters Rn−immediately in forward time due do the fact that both
quantities eT1A±ξ have negative sign. Assume that ϕ(t, ξ) reaches
M again for the first time t−(ξ) at η := ϕ(t−(ξ), ξ) ∈ M; hencethere exists the intersection time
t−(ξ) = inf{t > 0 | eT1eA−tξ = 0}.
Let W< ⊂ W−< ∩ W+
< be the set on which t− is defined. We knowt− is smooth in ξ ∈ W< with η ∈ W−
> and constant on half-rays(see [11]), i.e.
t−(λξ) = t−(ξ), 0 < λ < ∞. (4)
Fig. 1. Different dynamics on cones, µ < 1, µ = 1 and µ > 1, respectively.
Differentiating this identity with respect to ξ gives
t(j)− (λξ)λj= t(j)− (ξ), 0 < λ < ∞ (5)
for j ≥ 1, indicating possible difficulties for λ → 0. On the otherhand we find that all derivatives of t− when applied in direction ofthe ray vanish.
Lemma 1. For j ≥ 1we get t(j)− (λξ)ξ = 0 for ξ ∈ W< and η ∈ W−> .
Proof. Differentiating equation (4) with respect to λ gives theresult for j = 1. The statement for j > 1 then follows by inductionand by differentiating 0 = eT1e
A−t−(ξ)ξ with respect to ξ , becauseof the transversality guaranteed by η ∈ W−
> . �
In the same way t+(η) and W> can be defined, where similarresults hold for t+.
For initial data ξ ∈ W< and η ∈ W> we define P−(ξ) :=
et−(ξ)A−
ξ , P+(η) := et+(η)A+
η and the Poincaré map for PWLS (3)with P−(ξ) ∈ W> by P(ξ) := P+(P−(ξ)) [13,14].
An invariant cone C is then generated by an ‘‘eigenvector’’ ξ ∈
W< of the nonlinear eigenvalue problem
P(ξ ) = µξ (6)
with some real positive ‘‘eigenvalue’’ µ. The orbits with initialvalue λξ , λ > 0, form the invariant cone C in the phase spaceRn anchored in 0 ∈ Rn. The surface of the cone is smoothexcept possibly at the intersection of C and M. We recall that forcontinuous PWLS (A+ξ = A−ξ, ξ ∈ M) the cone is C1 even at M.The value µ > 0 determines the dynamic on the cone. For µ > 1and µ < 1 solutions spiral out and in, respectively. If µ = 1, thenthe cone consists of periodic orbits, Fig. 1.
In order to study the attractivity of the cone C we consider theeigenvalues of the Jacobian P ′(λξ ), which are independent ofλ > 0due to the homogeneity of the system: since the intersection timest± are constant on half-rays, we get the identity
P(λξ) = λP(ξ), 0 < λ < ∞.
Differentiating this identity with respect to ξ yields
P ′(λξ) = P ′(ξ),
which confirms independency of λ. By differentiating with respectto λ we obtain
P ′(λξ )ξ = P(ξ ) = µξ .
Hence µ is an eigenvalue of P ′(ξ ) with eigenvector ξ . For theremaining n − 2 eigenvalues µi, i = 1, . . . , n − 2, we assume
|µi| ≤ α < min{1, µ}. (7)
Remark 1. The attractivity condition (7) guarantees that allsolutions with initial values close to C are attracted to the cone. Incase of contracting spiraling on C itself these solutions convergefaster to the cone than to the origin. These statements will bediscussed in more detail in Section 2.
D. Weiss et al. / Physica D 241 (2012) 1895–1902 1897
In the following we assume that the corresponding PWLS pos-sesses an attractive invariant cone generated by ξ ∈ W< (see (6)),which is transversal to M, i.e.
eT1A+ξ · eT1A
−ξ > 0,
eT1A+η · eT1A
−η > 0,(8)
with η := P−(ξ ) ∈ W>.The main result is the following:
Theorem 1. Assume that the conditions (2), (3) and (6)–(8) holdfor the corresponding PWLS and g±. Then, there exists a sufficientlysmall δ and a C1-function h: [0, δ) → M satisfying h(0) = 0 and∂∂uh(0) = ξ such that
{h(u) | 0 ≤ u < δ}
is locally invariant and attractive under the Poincaré map ofsystem (1). For k = 2 the function h is Ck in case of µ ≥ 1 andCmin(k,j) in case of µ < 1 and α < µj.
We decompose the Poincaré maps of PWLS and PWNS into a linearpart and a nonlinearity. The Poincaré map P for PWNS will bewritten using properties of the Poincaré map of PWLS, P .
2. The piecewise linear system
We first decompose P using the derivative at ξ and anappropriate nonlinear term Q as
P(ξ) = P ′(ξ )ξ + Q (ξ).
Using the properties of P we immediately obtain Q (ξ ) = 0,Q ′(ξ ) = 0 and
Q (λξ) = λQ (ξ),
Q ′(λξ) = Q ′(ξ), 0 < λ < ∞.
Hence, the function Q ′ is constant on half-rays. Differentiating thesecond equation with respect to ξ gives
Q (j+1)(λξ)λj= Q (j+1)(ξ), 0 < λ < ∞,
for j ≥ 1, again indicating possible difficulties for λ → 0. Onthe other hand we find vanishing derivatives of the return timet− applied in direction of the ray lead to corresponding results forderivatives of Q .
Lemma 2. For j ≥ 1 we get Q (j+1)(λξ)ξ = 0 for ξ ∈ W< andP−(ξ) ∈ W>.
Proof. The statement follows due toQ (j+1)(ξ) = P (j+1)(ξ), P ′−(ξ)ξ
= P−(ξ) and
P (j+1)− (λξ)ξ = 0,
P (j+1)+ (P−(λξ))P−(ξ) = 0,
which is guaranteed by Lemma 1 and the corresponding propertiesof t+. �
To simplify matters we linearly transform the coordinates ofsystem (1) by a constant matrix1 00 T
, T ∈ R(n−1)×(n−1)
to get ξ = e2 ∈ Rn and
P ′(ξ ) =
µ 00 As
, (9)
Fig. 2. ε-sector of the cone in M.
where the eigenvalues of thematrix As are exactly theµi. Note thaton the one hand M remains the separating plane, on the other thetransformation of the last (n−1) components is independent of ξ1.
We decompose
P =
PcPs
, ξ =
yz
according to the blocks in (9), so that y is a scalar and z ∈ Rn−2.Due to assumption (7) we can choose a norm on Rn−2 such that
∥As∥ =: α < min{1, µ}.
On Rn−1 we define a norm by
∥ξ∥ = max{|y|, ∥z∥}.
Remark 2. Due to the properties of the derivative of Q we are ableto obtain an estimate for Q ′ in a neighborhood of the vector ξ :
∥Q ′(ξ)∥ ≤ Lε, ξ Sε(ξ ),
Sε(ξ ) := {(y, z)T ∈ M | y > 0, ∥z/y∥ ≤ ε}, for some constant Lε
with Lε → 0 for ε → 0 (see Fig. 2).
We now use the property of the sector Sε(ξ ) to obtain anestimate relating relations of Pc and Ps, hence the approximationproperty mentioned in Remark 1. For ξ ∈ Sε(ξ ) we know ∥z∥/y ≤
ε and hence ∥ξ∥ = y for ε < 1. According to Remark 2 andQ (y, 0) = 0 we know
∥Ps(ξ)∥ ≤ ∥Asz∥ + ∥Q (ξ)∥≤ (α + Lε)εy, (10)
Pc(ξ) ≥ (µ − Lεε)y.
Combining these two estimates we see
∥Ps(λξ)∥
Pc(λξ)≤
α+Lεµ−Lεε
ε < ε (11)
for sufficiently small ε. Additionally we get α + Lε < 1 for smallvalues of ε. Hence (10) shows the local attractivity of the cone C,whereas (11) guarantees that in case of contracting spiraling on Csolutions close to the cone converge faster to the cone than to theorigin.
3. The piecewise nonlinear system
As we are interested to prove the existence of a local manifoldwe use the usual techniques of ‘‘cut-off and scale’’ and withoutrestriction we end up with a piecewise nonlinear system of theform
ξ =
A+ξ + g+(ξ), ξ1 > 0,A−ξ + g−(ξ), ξ1 < 0, (12)
where the nonlinear perturbations g± are Ck-maps, k ≥ 1, definedon the whole phase space Rn with supp g± ⊂ {ξ ∈ Rn
| ∥ξ∥ ≤
δ} and g± = o(∥ξ∥), ∥ξ∥ → 0. Obviously we find a constanto(1) depending on the scaling parameter δ with ∥g±∥ + ∥g ′
±∥ ≤
o(1), δ → 0.A global invariantmanifold of system (12) gives a local invariant
manifold of system (1).
1898 D. Weiss et al. / Physica D 241 (2012) 1895–1902
3.1. The Poincaré map
The PoincarémapsP−,P+ andP = P+(P−(ξ))will be definedon sectors Sϵ(ξ ) resp. Sϵη (η) as long as ε, εη and δ are sufficientlysmall.
We decompose the Poincaré map of system (12) using thePoincaré map of the PWLS:
P (ξ) = P(ξ) + R(ξ), R(ξ) := P (ξ) − P(ξ).
Due to the compact support of the perturbations g± depending onδ, we know R(ξ) = 0 for ∥ξ∥ ≥ const · δ, const sufficiently large.In this section we will study further properties of the remainingterm R.
A crucial step in the definition ofP−,P+, relies on the definitionand properties of the intersection times τ± of PWNS (12), which incase of the ⊖-system is given by (we omit the (−)-indices)
τ−(ξ) = inf{τ > 0 | F(τ , ξ) = 0},
F(t, ξ) = eT1
eAtξ +
t
0eA(t−s)g(y(s, ξ))ds
,
where y(t, ξ) is the solution of y = Ay+ g(y), y(0) = ξ . ApplyingGronwall’s Lemma it is obvious, that we have y(t, ξ) = eAtξ +
o(∥ξ∥) for t ∈ [0, T ]. The existence of Ck-functions τ± for initialvalues close to the cone is guaranteed by the Implicit FunctionTheoremdue to the transversality condition (8) and the hypothesison the perturbations g± (see proof of Lemma 3). Furthermore weknow that τ±(ξ) is ‘‘close’’ to t±(ξ) for small perturbations g±:
eT1A−et
∗A−
ξt−(ξ) − τ−(ξ)
= eT1
τ−(ξ)
0eA
−(τ−(ξ)−s)g−(y−(s, ξ))ds
with intermediate time t∗. Due to the transversality condition (8)and the hypothesis on g− we find
τ−(ξ) − t−(ξ) = o(1) (13)
for δ → 0or∥ξ∥ → 0. Differentiating the equations defining t−(ξ)
and τ−(ξ) with respect to ξ and using (8) and the properties of g−
we find τ ′(ξ) = O(∥ξ∥−1) and with (13) additionally
τ ′
−(ξ) − t ′
−(ξ) = o(∥ξ∥
−1) (14)
for δ → 0 or ∥ξ∥ → 0.
Remark 3. In case of k ≥ 2 we assume without loss of generalityg± = O(∥ξ∥
2). Thus we can replace the o-terms in (13) and (14)by O(∥ξ∥) and O(1) respectively. Similarly we conclude
τ ′′
−(ξ) − t ′′
−(ξ) = O(∥ξ∥
−1).
All o- and O-terms are independent of ε.
Corresponding to Lemma 1 and (5) we get
Lemma 3. The intersection time τ− is Ck in Sε(ξ ), ε suitably small.Let ξ ∈ Sε(ξ ) with ∥ξ∥ = 1. For 0 ≤ j ≤ k we get
τ(j)− (λξ) = O(λ−j), 0 < λ.
In case of k ≥ 2 we gain one power of λ in direction of the ray ξ :
τ(j+1)− (λξ)ξ = O(λ−j), 0 < λ.
Similar results hold for τ+.
Proof. Let ε and δ be sufficiently small, so that transversalityof the perturbed vector fields is still given. Application of theImplicit Function Theorem implies the existence of τ−(ξ) for ξ =
(1, z)T , ∥z∥ ≤ ε. The existence on Sε(ξ ) can be concluded applyingthe Contraction-Mapping Theorem to the operator t → t −1
λβF(t, λξ) for λ > 0, where eT1A
−eτ−(ξ)A−
ξ = β > 0.The statements about τ− and τ ′
−are given by (13), (14) and (5) in
case of k = 1. Let k ≥ 2. By (14), Remark 3 and Lemma 1 we findτ ′(λξ)ξ = O(1).
The statement for higher derivatives follows inductively by dif-ferentiating F(τ (ξ), ξ) = 0 with respect to ξ , by the transversalitycondition (8) and by the observation ∂
∂t F(τ (λξ), λξ) = O(λ). �
Lemma 4. The remaining term R is Ck in Sε(ξ ). Furthermore there isa constant Kδ independent of ε with Kδ → 0 for δ → 0 and
∥R(ξ)∥ + ∥R′(ξ)∥ ≤ Kδ, ξ ∈ Sε(ξ ).
Let ξ ∈ Sε(ξ ) with ∥ξ∥ = 1. In case of k = 2 we get
R′′(λξ) = O(1), 0 < λ < ∞.
Proof. Since the intersection times τ± are Ck the same holds forthe remaining term R. Using P±(ξ) = eA
±τ±ξ + τ±0 eA
±(τ±−s)g±
(y±(s, ξ))dswe get
R(ξ) =eA
+τ+eA−τ− − eA
+t+eA−t−ξ
+ eA+τ+
τ−
0eA
−(τ−−s)g−(y−(s, ξ))ds
+
τ+
0eA
+(τ+−s)g+(y+(s, P−(ξ)))ds,
τ− = τ−(ξ), τ+ = τ+(P−(ξ)), t− = t−(ξ), t+ = t+(P−(ξ)).Obviously the Lemma is true for the last two terms. For the firstterm we write equivalentlyeA
+τ+eA−τ− − eA
+t+eA−t−ξ = eA
+τ+eA
−τ− − eA−t−ξ
+eA
+τ+ − eA+t+eA
−t−ξ .
By differentiating [eA−τ− −eA
−t− ]ξ with respect to ξ and using (13)and (14) and Remark 3 it is easy to conclude that the Lemma holdsfor this term and thus for R. �
3.2. Hadamard’s graph transform
Using the explicit form of P leads to the composition ofP whichwe can use to define Hadamard’s graph transformation:
P (ξ) =
µ 00 As
ξ + R(ξ),
R(ξ) := Q (ξ) + R(ξ). Obviously the remaining term R is Ck inSε(ξ ) and we get
∥R′(ξ)∥ ≤ Lε,δ := Lε + Kδ, (15)
so that Lε,δ can be made as small as necessary by setting ε and δsuitably small.
We will prove the existence of a smooth function H: [0, ∞) →
Rn−2 with H(0) = 0, which satisfies the invariance condition
H(Pc(y,H(y))) = Ps(y,H(y)) (16)
for y ≥ 0 using Hadamard’s Graph Transform T :D → D definedby
[TH](ζ ) := Ps(y,H(y)), ζ ≥ 0, (17)
and ζ = Pc(y,H(y)). Obviously, a fixed point of the operator Tvanishes at y = 0 and fulfills the invariance condition (16).
D. Weiss et al. / Physica D 241 (2012) 1895–1902 1899
In case of k = 1 we define D as a set of maps H: [0, ∞) →
Rn−2, satisfying H(0) = 0, ∥H∥∞ ≤ ε, graph(H) ⊂ Sε(ξ ), i.e.(y,H(y)) ∈ Sε(ξ ) for all y ≥ 0 and ∥H(y1)−H(y2)∥ ≤ ε|y1−y2| fory1, y2 ≥ 0. Due to Remark 2, Lemma 4 and the following Lemma 5the existence of a fix-point of the operator T can be proved quitesimilar to [15]. We only have to make use of Q (ξ ) = 0 and toguarantee graph(H) ⊂ Sε(ξ ) for H = TH , which can easily be seen:We will showP (Sε(ξ )) ⊂ Sε(ξ )
which holds for ε and δ sufficiently small. For ξ ∈ Sε(ξ ) we getsimilar to (10)∥Ps(ξ)∥ ≤ ∥Asz∥ + ∥Q (ξ)∥ + ∥R(ξ)∥
≤ (α + Lε + Kδε−1)εy,
Pc(ξ) ≥ (µ − Lεε − Kδ)y.Combining these two estimates we get for sufficiently small ε andδ as in (11)
∥Ps(ξ)∥
Pc(ξ)≤
α + Lε + Kδε−1
µ − o(ε) − Kδ
ε ≤ ε.
Lemma 5. For each ζ ≥ 0 and each H ∈ D there is a unique y =
ω(ζ ,H) with
Pc(y,H(y)) = ζ .
Furthermore the function ω( . ,H) is Lipschitz-continuous withconstant 1/(µ − Lε,δ).
Proof. Since
|Rc(y1,H(y1)) − Rc(y2,H(y2))|≤ Lε,δ max{|y1 − y2|, ∥H(y1) − H(y2)∥}
≤ Lε,δ|y1 − y2|, (18)
the functionPc( . ,H( . )) given byPc(y,H(y)) = µy+Rc(y,H(y))≥ 0 is strictly monotonically increasing as long as Lε,δ < µ. Hencethere exists such a function ω( . ,H). Using (18) a second time wefind |ζ1 − ζ2| ≥ µ|y1 − y2| − Lε,δ|y1 − y2|. �
The differentiability of the fix-point H = TH can be shown asin [16]. To prove H ′(0) = 0 we use Lemma 5 and the invariancecondition (16) to conclude
lim supζ→0
∥H(ζ )∥
|ζ |≤
1µ − Lε,δ
lim supy→0
∥AsH(y) + Rs(y,H(y))∥|y|
≤α + Lε,δ
µ − Lε,δ
lim supy→0
∥H(y)∥|y|
.
For k = 2 we define D as a set of C1-maps H: [0, ∞) → Rn−2,satisfying the additional conditions• ∥H∥1,∞ := max{∥H∥∞, ∥H ′
∥∞} ≤ ε,• ∥H ′(y1) − H ′(y2)∥ ≤ L′
|y1 − y2|
for all y1, y2 ≥ 0, where the Lipschitz-constant L′ will be deter-mined later. D is a Banach space with respect to the norm ∥ · ∥1,∞.
Additionally to Lemma 5 we need.
Lemma 6. The function y = ω( . ,H) is continuously differentiablewith
|y′
1 − y′
2| ≤ K |ζ1 − ζ2| (19)
for y′
i :=∂∂ζ
ω(ζi,H), where the constant K is independent of L′
for (ε + Lε,δ)L′≤ const. Furthermore for yi := ω(ζ ,Hi) and
y′
i :=∂∂ζ
ω(ζ ,Hi) we find
|y1 − y2| + |y′
1 − y′
2| ≤ Kε,δ∥H1 − H2∥1,∞
for some constant Kε,δ with Kε,δ → 0 for ε, δ → 0.
Proof. Obviously the function ω( . ,H) is C1 together with P andH . Differentiating ζ = Pc(y,H(y)) gives
1 = µy′+ R′
c(y,H(y))
1H ′(y)
y′.
Setting y′
i := y′(ζi), i = 1, 2, and using the abbreviations R′
j :=
R′(yj,H(yj)), H ′
j := H ′(yj), j = 1, 2, we estimate
µ|y′
1 − y′
2| ≤
R′
1
1H ′
1
y′
1 − R′
2
1H ′
2
y′
2
≤
R′
1
1H ′
1
(y′
1 − y′
2)
+
R′
1
0
H ′
1 − H ′
2
y′
2
+
R′
1 − R′
2
1H ′
2
y′
2
,
where the first term can be estimated by Lε,δ|y′
1 − y′
2|, the secondterm by Lε,δL′
|y1 − y2 ∥ y′
2| and the third using Lemmas 2 and 4:[R′
1 − R′
2]
1H ′
2
≤ K |y1 − y2|.
More precisely we get[R′
i,1 − R′
i,2]
1H ′
2
=
R′′
i,∗
1H ′
∗
,
1H ′
2
|y1 − y2|
with intermediate value y∗, where Ri is the ith component of Rand R′′
i,∗ := R′′
i (y∗,H(y∗)), H ′
∗:= H ′(y∗). FinallyQ ′′
i,∗
1H ′
∗
,
1H ′
2
= O(ε)L′,R′′
i,∗
1H ′
∗
,
1H ′
2
= O(1),
prove the first statement.We now set yi := ω(ζ ,Hi), i.e.
ζ = Pc(yi,Hi(yi)) = µyi + Rc(yi,Hi(yi)).
Hence
µ|y1 − y2| ≤ ∥R(y1,H1(y1)) − R(y2,H2(y2))∥≤ Lε,δ
|y1 − y2| + ∥H1 − H2∥∞
and therefore
|y1 − y2| ≤Lε,δ
µ − Lε,δ
∥H1 − H2∥∞.
For y′
i :=∂∂ζ
ω(ζ ,Hi) and Rk,i := R(yi,Hk(yi)), Hk,i = Hk(yi) weget
µ|y′
1 − y′
2| ≤
R′
1,1
1
H ′
1,1
y′
1 − R′
2,2
1
H ′
2,2
y′
2
≤
R′
1,1
1
H ′
2,1
y′
1 − R′
2,2
1
H ′
2,2
y′
2
+
Lε,δ
µ − Lε,δ
∥H ′
1 − H ′
2∥∞
≤
R′
2,1
1
H2,1
y′
1 − R′
2,2
1
H2,2
y′
2
+
R′
1,1 − R′
2,1
1H2,1
|y′
1|
+Lε,δ
µ − Lε,δ
∥H ′
1 − H ′
2∥∞,
1900 D. Weiss et al. / Physica D 241 (2012) 1895–1902
where the first term is already estimated above. For the secondterm we findR′
i,1,1 − R′
i,2,1
1H2,1
=
R′′
i (y1, z∗)
0
H1,1 − H2,1
,
1
H2,1
≤ [O(ε) + O(δ)]∥H ′
1 − H ′
2∥∞
with intermediate value z∗. More precisely:Q ′′
i (y1, z∗)
0
H1,1 − H2,1
,
1
H2,1
≤ O(ε)∥H ′
1 − H ′
2∥∞R′′
i (y1, z∗)
0
H1,1 − H2,1
,
1
H2,1
≤ O(δ)∥H ′
1 − H ′
2∥∞
for y1 = O(δ). �
In the following we will prove that T :D → D is a contraction.Defining H := TH for H ∈ D gives
H(ζ ) = Ps(y,H(y)), y = ω(ζ ,H).
Clearly, H is a continuously differentiable function with H(0) = 0.Further we get with Q (y, 0) = 0, Remark 2 and Lemma 4
∥H(ζ )∥ ≤ ∥As∥∥H(y)∥ + ∥R(y,H(y))∥
≤ (α + Lε + Kδε−1)ε.
Using Q ′(y, 0) = 0, Lemmas 2 and 4 we find
∥H ′(ζ )∥ ≤ ∥As∥∥H ′(y)∥|y′
| +
R′(y,H(y))
1H ′(y)
|y′|
≤α + O(ε) + Kδε
−1
µ − Lε,δ
ε.
Eventually we can guarantee
∥H∥1,∞ ≤ ε
for ε and δ sufficiently small.Indeed, H ′ is Lipschitz continuous: For ζ1, ζ2 ≥ 0 we define
y1 := ω(ζ1,H) and y2 := ω(ζ2,H). Then
∥H ′(ζ1) − H ′(ζ2)∥ ≤ ∥As∥∥H ′
1y′
1 − H ′
2y′
2∥
+
R′
1
1H ′
1
y′
1 − R′
2
1H ′
2
y′
2
.
The first term of the right-hand side can be estimated by
∥H ′
1y′
1 − H ′
2y′
2∥ ≤ ∥H ′
1(y′
1 − y′
2)∥ + ∥H ′
1 − H ′
2∥|y′
2|
≤ ε|y′
1 − y′
2| + L′|y1 − y2 ∥ y′
2|
whereas the second term is already estimated in the proof ofLemma 6. We then arrive at
∥H ′(ζ1) − H ′(ζ2)∥ ≤ (αε + Lε,δ)|y′
1 − y′
2|
+ (αL′+ Lε,δL′
+ O(1))|y1 − y2 ∥ y′
2|,
where the O-term is independent of L′. Using Lemma 6 weeventually get
∥H ′(ζ1) − H ′(ζ2)∥ ≤αL′
+ Lε,δL′+ O(1)
(µ − Lε,δ)2|ζ1 − ζ2|,
where theO-term is still independent of L′. Choosing L′ sufficientlylarge and ε, δ small we end up with
∥H ′(ζ1) − H ′(ζ2)∥ ≤ L′|ζ1 − ζ2|
which proves T (D) ⊂ D.
For any H1,H2 ∈ Dwe define Hi = THi and yi, y′
i as in Lemma 6.Then by definition (17) and Lemma 6 we find
∥H1(ζ ) − H2(ζ )∥ ≤ ∥As∥∥H1,1 − H2,2∥ + ∥R1,1 − R2,2∥
≤ α(ε|y1 − y2| + ∥H1 − H2∥∞)
+ Lε,δ
|y1 − y2| + ∥H1 − H2∥∞
≤ [α + o(1)]∥H1 − H2∥1,∞
for ε, δ → 0.Furthermore we get
∥H ′
1(ζ ) − H ′
2(ζ )∥ ≤ ∥As∥∥H ′
1,1y′
1 − H ′
2,2y′
2∥
+
R′
1,1
1
H ′
1,1
y′
1 − R′
2,2
1
H ′
2,2
y′
2
,
where the first term can be estimated by
∥H ′
1,1y′
1 − H ′
2,2y′
2∥ ≤ ∥H ′
1,1∥|y′
1 − y′
2|
+∥H ′
1,1 − H ′
1,2∥ + ∥H ′
1,2 − H ′
2,2∥|y′
2|
≤ ε|y′
1 − y′
2| +1
µ − Lε,δ
L′|y1 − y2| + ∥H ′
1 − H ′
2∥∞
and the second term is already estimated in the proof of Lemma 6:R′
1,1
1
H ′
1,1
y′
1 − R′
2,2
1
H ′
2,2
y′
2
≤
Lε,δ
µ − Lε,δ
∥H ′
1 − H ′
2∥∞ + o(1)∥H1 − H2∥1,∞
for ε, δ → 0. Finally we end up with
∥H ′
1(ζ ) − H ′
2(ζ )∥ ≤α + Lε,δ + o(1)
µ − Lε,δ
∥H1 − H2∥1,∞
for ε, δ → 0.Since α < min{1, µ} we can guarantee that T is a contraction
for ε, δ sufficiently small, hence by the Contraction-MappingTheorem there is a fixed point defining the invariant graph.
Remark 4. Theorem 1 holds even in case of k = 3. The proofdepends crucially on Lemma 3, which guarantees
R′′′(λξ)ξ = O(λ−1),
R′′′(λξ)ξ 2= O(1), 0 < λ < ∞.
Remark 5. Once the existence of H(y) = a1y + a2y2 + · · ·, hasbeen established, we can use H to determine the dynamics on{h(y) = (y,H(y)) | 0 ≤ y < δ}.
For example to determine periodic solutions we consider thefixed point equation P (ξ) = ξ , reduced to the first component
µy + Rc(y,H(y)) = y, (y ≥ 0).
Dividing by y we obtain
µ − 1 = Rc(y,H(y))/y. (20)
Solutions y > 0 of (20) then lead to periodic orbits.
4. Example
We illustrate the result by an example where the piecewiselinear system has been designed according to the choice providedin [11]. Further we have chosen a nonlinearity such that thesolution is explicitly known for comparison.
The ⊕-system is taken in normalized form; for the ⊖-systemwe have chosen the situation represented by δ = 0 in [11]. Hencewe consider the system
ξ = A±ξ + g±(ξ), ±eT1ξ > 0, (21)
D. Weiss et al. / Physica D 241 (2012) 1895–1902 1901
where
A+=
λ+−ω+ 0
ω+ λ+ 00 0 µ+
,
A−=
λ−− ω−
−2ω− ακ
ω− λ−+ ω− αω−
0 0 µ−
with κ = λ−
− µ−− ω− and
g+(ξ) = ρ+
00
ξ 21 + ξ 2
2
, g−(ξ) = ρ−
ξ 2300
.
Note that the eigenvalues of A± are given by λ±± iω±, µ±, and
PWNS (21) has transversal crossing if ξ2[−2ω−ξ2 + α(λ−− µ−
−
ω−)ξ3 + ρ−ξ 23 ] < 0.
For ξ = (y, z)T ∈ W−< the intersection time τ−(ξ) = τ− is
determined as smallest positive root of equation
0 = −2syeλ−τ− + α[(c − s)eλ−τ− − eµ−τ− ]z + G1(τ−)z2 (22)
with
G1(τ−) = ρ−
(µ−− κ)(s − c) + 2ω−c
(λ− − 2µ−)2 + ω−2eλ−τ−
+ ρ−
2µ−− λ−
− ω−
(λ− − 2µ−)2 + ω−2e2µ
−τ− ,
where we have used the abbreviations s := sin(ω−τ−) and c :=
cos(ω−τ−). Further the map P− is given by
P−(y, z) =
(c + s)eλ−τ− αseλ−τ−
0 eµ−τ−
yz
+
G2(τ−)z2
0
,
where
G2(τ−) = ρ−
(λ−− 2µ−)s − ω−c
(λ− − 2µ−)2 + ω−2eλ−τ−
+ ρ−
ω−
(λ− − 2µ−)2 + ω−2e2µ
−τ− .
For η = (y, z)T ∈ W+> , we have t+ := τ+(η) = π/ω+ independent
of the nonlinearity g+ and therefore
P+(y, z) =
−eλ+t+y
eµ+t+z + G3y2
,
where G3 = ρ+
e2λ
+t+ − eµ+t+/(2λ+
− µ+).For P (y, z) = P+(P−(y, z)) we obtain
P (y, z) =
µ1(τ−) d(τ−)
0 µ(τ−)
yz
+
−G2(τ−)eλ+t+z2
G3((c + s)y + αsz)eλ−τ− + G2(τ−)z2
2
with
µ1(τ−) = −(c + s)eλ−τ−+λ+t+ ,
d(τ−) = −αseλ−τ−+λ+t+ ,
µ(τ−) = eµ−τ−+µ+t+ .
We write P (y, z) as
P (y, z) =
µ1 d0 µ
yz
+ R(y, z),
with µ1 = µ1(t−(ξ )), d = d(t−(ξ )), µ = µ(t−(ξ )); further t−(ξ )is determined as smallest positive root of
0 = −2s + α[c − s − e(µ−−λ−)t−
m, (23)
where we have used the abbreviations s = sin(ω−t−(ξ )), c =
cos(ω−t−(ξ )), and the invariant ‘‘eigenvector’’ ξ satisfying P(ξ ) =
µξ is chosen as ξ = (y, z)T = (1,m)T with m = (µ − µ1)/d.Note that we want to consider the situation that µ ≈ 1; at-
tractivity of the cone is then guaranteed if |µ1| < min{1, µ}. ByTheorem 1, we know there exists a local invariant set tangent tothe cone at 0 which is generated by a graph of the form H(y) =
my+a2y2+O(y3). UsingQ (ξ ) = 0, Q ′(ξ ) = 0 and g± = O(∥ξ∥2),
we obtain
Q (y,H(y)) = O(y3),
R(y,H(y)) =
b1b2
y2 + O(y3).
After lengthy computations we obtain
b1 = −G2 −
eT2A−η
eT1A−ηG1eλ+t+m2,
b2 = −eT3A
−η
eT1A−ηG1eµ+t+m2
+ G3η22,
where η = P−(ξ ) and Gi = Gi(t−(ξ )).To determine the coefficient a2 we substitute H into the
equation representing the invariance condition; hence
H(P1(y,H(y))) = P2(y,H(y)),
which leads to
mµy + m(da2 + b1)y2 + a2µ2y2
= mµy + µa2y2 + b2y2 + O(y3),
and thus
a2 =b2 − mb1µ2 − µ1
.
Finally we can use the expression for H(y) to study bifurcation ofperiodic orbits by determining fixed points of the reduced system
P1(y,H(y)) := µy +db2 − µ(1 − µ)b1
µ2 − µ1y2 + O(y3).
Thus the fixed point is approximately given by
y∗≈
1 − µ
db2 − µ(1 − µ)b1(µ2
− µ1).
Using this general form various situations can be explicatedby a special choice of parameters. The simulation is done withparameters set at λ+
= −0.5, λ−= 0.5, µ+
= 0.2, α = 0.5,t+ = π, w+
= w−= 1.0, ρ− = −0.01, ρ+ = 0.1 and bifurcation
parameter µ− close to µ−
0 := −µ+t+/t−(ξ ) ≈ −1.0604, wheret−(ξ ) ≈ 0.59253 is determined by
0 = −2s + α[c − s − e−(µ+t++λ−t−(ξ ))]m,
wherem =1−µ1
d . We get µ(µ−
0 ) = 1 and ∂∂µ− µ(µ−
0 ) > 0. For thisset of values in PWLS (i.e. ρ− = ρ+ = 0), the phase space con-tains two attractive invariant cones (Fig. 3), likewise there are twogeneralized center manifolds of PWNS (21), Fig. 4. These attractivecones are given by ξ = (0, 1, 0)T and ξ = (0, 1,m) with eigen-values µ1 = 1, µ ≈ 0.067 and µ = 1, µ1 ≈ −0.388 respectivelyand are separated by a nonattractive cone.Recall, that the existence
1902 D. Weiss et al. / Physica D 241 (2012) 1895–1902
0
–1
–1.5
–2
–2.5
–30.5
0
–0.5–1 –1
–0.50
0.5
xy
z
–0.5
Fig. 3. Two attractive invariant cones of PWLS for µ−= µ−
0 .
0.5
0
–0.5
–1
–1.5
–2
–2.50.5
0– 0.5
–1 –1–0.5
00.5
x
z
y
H(y)
H(y)
Fig. 4. Two generalized center manifolds of PWNS (ρ−= −0.01, ρ+
= 0.1) forµ−
= µ−
0 .
–0.01
–0.015
–0.02
–0.025–2
–1
0
1
2
3
4 –8–6
–4–2
02
yx
×10-3
×10-3
z
Fig. 5. Stable periodic orbit of PWNS for µ−= −1.06 > µ−
0 .
of two attractive cones is not possible in continuous PWLS (see [1]Theorem 2). A periodic orbit on the manifold generated by Hopfbifurcation is shown in Fig. 5.
Acknowledgment
The research of the author (H.A. Hosham) was supported bythe Department of Mathematics, Faculty of Science, Al-AzharUniversity of Assiut, Egypt.
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