on the break-down threshold of invariant tori in four dimensional maps
TRANSCRIPT
On the break–down threshold of invariant tori in
four dimensional maps
Alessandra Celletti
Dipartimento di Matematica
Universita di Roma Tor Vergata
Via della Ricerca Scientifica 1, I-00133 Roma (Italy)
Corrado Falcolini
Dipartimento di Matematica
Universita di Roma Tre
Largo S. L. Murialdo 1, I-00146 Roma (Italy)
Ugo Locatelli
Dipartimento di Matematica
Universita di Roma Tor Vergata
Via della Ricerca Scientifica 1, I-00133 Roma (Italy)
September 27, 2004
Abstract
We investigate the break–down of invariant tori in a four dimensional standard mappingfor different values of the coupling parameter. We select various two–dimensional frequencyvectors, having eventually one or both components close to a rational value. The dynamicsof this model is very reach and depends on two parameters, the perturbing and couplingparameters. Several techniques are introduced to determine the analyticity domain (in thecomplex perturbing parameter plane) and to compute the break–down threshold of theinvariant tori. In particular, the analyticity domain is recovered by means of a suitableimplementation of Pade approximants. The break–down threshold is computed through asuitable extension of Greene’s method to four dimensional systems. Frequency analysis isimplemented and compared with the previous techniques.
Keywords: Invariant tori, Greene’s method, Pade approximants, Frequency analysis
1 Introduction
The mechanism of break–down of invariant surfaces in nearly–integrable (continuous and dis-crete) systems has been widely studied through theoretical investigations and numerical ex-periments. In particular, the Kolmogorov–Arnold–Moser theory (see [17], [1] and [29]) allowsto prove the existence of an invariant torus provided the perturbing parameter is sufficientlysmall. Many different techniques have been developed to compute the experimental value of the
1
break–down threshold. However, most algorithms work undoubtely well for low–dimensionalsystems, but they are less effective when the dimension is increased. In this paper we concen-trate on 4–dimensional discrete systems, which may be viewed as surfaces of flows induced bysome Hamiltonians having 3 degrees of freedom. To be concrete, let us start by considering the2–dimensional symplectic standard mapping, described by the equations
{
y′ = y + ε sinxx′ = x+ y′
, (1.1)
where x ∈ T ≡ R/(2πZ), y ∈ R, ε ∈ R+ ∪ {0}. Let us consider an invariant curve withfrequency ω for the unperturbed system (i.e., taking ε = 0); KAM theorem ensures that suchcurve persists under perturbation, as far as the perturbing parameter is less than a criticalvalue, say ε = εc(ω), at which the curve breaks–down. A widespread method due to J. Greene([12]) allows to compute an accurate numerical estimate of the break–down threshold. Togive an example, consider the invariant curve with frequency equal to the golden ratio, i.e.
ω =√
5−12 , which is conjectured to be the most robust curve for the mapping (1.1). Greene’s
method provides an estimate of the transition value, amounting to about εc(ω) = 0.971635.However, we remark that a naive extension of Greene’s method to higher dimensional systems isnot straightforward and it does not provide accurate results as for the 2–dimensional standardmapping. Nevertheless, we believe that an effort in this direction is certainly worthwhile, sincehigher dimensional systems are definitely more interesting: beside the fact that they usuallydescribe more realistic physical systems, there is a huge variety of dynamical phenomena whichcan be viewed only in higher dimensions (Arnold’s diffusion, the coupling of rational andirrational behaviours, etc.).This paper will deal with a paradigmatic example, namely the so–called Froeschle map ([10]),which is a 4–dimensional symplectic mapping, denoted as Mε , b : T2 × R2 7→ T2 × R2 anddescribed by the equations
(~x′, ~y′) = Mε , b(~x, ~y) such that
y′1 = y1 + ε (sinx1 + b sin(x1 − x2))x′1 = x1 + y′1 mod 2πy′2 = y2 + ε (sinx2 − b sin(x1 − x2))x′2 = x2 + y′2 mod 2π
, (1.2)
where ε and b are positive parameters representing, respectively, the perturbing and couplingcoefficients. In the following, we will refer also to the so–called lift function Φε , b : R4 7→R4 , obtained through Mε , b by just removing the modulus operation. Let us define ωi =
limn→∞x(n)i −x
(0)i
2πn for i = 1, 2, where x(n)i denotes the n–th iterate of the variable xi with respect
to the lift Φε , b ; when the previous limits exist, we associate to an orbit the correspondingrotation vector ~ω = (ω1, ω2) . For instance, when ε = 0 the system reduces to two uncoupledcircle maps with frequencies identically equal to the initial values of the variables y1 and y2
divided by 2π . On the other hand, when b = 0 the system describes a set of two uncoupled2–dimensional standard mappings; therefore, the orbit is labelled by a frequency vector (ω1, ω2)whose components depend just on the two different dynamics induced by each standard map.In the following, we will focus on four different values of the frequency vector, taking bothcomponents strictly irrational, but considering also the case in which one component is closeto a rational number with a small integer value of the denominator (notice that we are onlyconcerned with maximal invariant tori).
2
In the present work, the different role of the parameters (ε, b) is investigated through the studyof the analyticity domain in the complex perturbing parameter ε as the coupling coefficientb is varied. Being the mapping 4–dimensional, a maximal invariant curve is parametrized bya 2–dimensional coordinate and it is represented by two Lindstedt’s series. The investigationof the domains of analyticity of such series leads to define the radius of convergence as theminimum distance ρP (ω) of the poles from the origin. The determination of the domains ofanalyticity and of the radius of convergence relies on the computation of the Pade approximantsin the style of [4], [3] (see also [5], [6]). As a byproduct, such technique provides informationabout the critical break–down threshold εc(ω), which can be evaluated as the intersection of thedomain of analyticity with the positive real axis. It is readily seen that whenever the domainof analyticity deviates from a circle, the two thresholds, ρP (ω) and εc(ω), might be different.Since Pade approximants present serious limitations in terms of computer implementations, weproceed to extend Greene’s method to the four dimensional mapping. Such technique relies onthe conjecture that the break–down of an invariant torus is strictly related to the transitionfrom stability to instability of the periodic orbits with frequency given by the rational approx-imants to the rotation vector of the invariant surface. In particular, one can define a quantityassociated to the periodic orbits, called the residue, whose behaviour, as the frequencies arevaried, provides a simple algorithm to determine the critical break–down value. Partial justifi-cations of such conjecture has been presented in [9], [25] for the 2–dimensional case; moreover,the approach used in [9] has been shown to apply to higher dimensional systems in [35] (seealso [36]). In this work, we revisit Greene’s method for mappings of the type (1.2), by provid-ing a heuristic justification. Our present approach is based on the estimate of the size of theresonant regions close to an invariant torus. In our opinion, this argument gives support to thewhole conjecture on which the method is based; moreover, our estimates of the resonant regionsare interesting in themselves and could be applied in future works on symplectic mappings.In order to check the reliability of Greene’s method, we compare the results with those obtainedby implementing frequency analysis (see, e.g., [21] and [22]). In the last two decades, thefrequency analysis has been mainly used to investigate the structure of stable and chaoticregions of complicated systems (like in [33]). Nevertheless, it can be used also to give ratheraccurate estimates of the transition value εc(ω), when dealing with simple mappings like (1.2).
This paper is organized as follows. The choice of the rotation vectors is discussed in section2. Section 3 is devoted to the introduction of Pade’s method and its implementation to themapping (1.2), while Greene’s method is presented in section 4, where a heuristic justificationis provided. The implementation of frequency analysis is discussed in section 5. A comparisonof the results is provided in section 6.
2 The 4–dimensional mapping and the choice of the rotation
vectors
We consider the mapping (1.2) and we select a frequency vector ~ω = (ω1, ω2). Let us supposeto fix a positive non–zero value of b, say b = b > 0 and suppose that the frequency vector ~ω =(ω1, ω2) is diophantine. Let us recall that a vector (ω1, . . . , ωn) ∈ Rn is said to be diophantine
3
if
|k1ω1+. . .+knωn+kn+1| ≥γ
(|k1| + . . . + |kn| + |kn+1|)τ, ∀ (k1 , . . . kn , kn+1) ∈ Zn+1\{~0}
(2.3)for suitable positive constants γ and τ ≥ n (of course, we are mainly interested to the casen = 2). We intend to follow the behaviour of the invariant torus associated to (ω1, ω2) as theperturbing parameter ε increases from zero. For ε sufficiently small, KAM theory ensures theexistence of a torus with the same frequency vector of the unperturbed case. As ε increases, thetorus is more and more displaced and distorted until one reaches a critical value, say ε = εc(b),at which the invariant torus breaks–down.
As it will be widely discussed in the rest of the paper, the choice of the rotation vector stronglyinfluences the dynamical scenario. We will limit ourselves to consider a very particular subset ofdiophantine vectors whose definition requires some notions of number theory (for more details,see Appendix 1 of [24] and references therein). Let α be an algebraic real number of degreen + 1 (i.e., α ∈ R is a root of a polynomial equation of degree n + 1 with integer coefficientsand it is not a root of any such equation of degree j with 0 < j ≤ n); consider a vector(ω1, . . . , ωn) ∈ Rn, such that
1ω1...ωn
=
1 0 . . . 0b1... Abn
1α...αn
, (2.4)
where both the vector (b1, . . . , bn) and the n×n dimensional matrix A have rational coefficientsand detA 6= 0 . Number theory ensures that the vector (ω1, . . . , ωn) is badly approximable, i.e.it satisfies the diophantine inequalities (2.3) with a fixed γ > 0 and τ = n . Let us introduce thePisot–Vijayaraghavan (PV) numbers: a PV number is the only root external to the unit circlein the complex plane of a monic polynomial, i.e. a polynomial with integer coefficients and withthe coefficient of the highest order equal to one. The smallest PV–number of degree 2 is thefamous golden mean (
√5 + 1)/2 , while at third degree the smallest one is the real solution of
the third order polynomial x3 − x− 1 = 0, amounting to s = 1.324717 . . . In the following, wedeal just with badly approximable vectors generated by a formula of type (2.4) with n = 2 andα = s . By the way, let us recall that the badly approximable vectors given by (2.4) with n = 1and α = (
√5+1)/2 correspond to the so called noble tori, which are expected to be the locally
most robust in the framework of the standard mapping (1.1) (see [26]). The frequency vectorsconsidered in the present work are interesting also because they are the natural extension ofthe noble tori for 4–dimensional mappings.Based on the previous definition of the PV–number s, we proceed to select four rotation vectors,defined as follows:
~ωu =
(
1
s, s− 1
)
= (0.754877..., 0.324717...)
~ωa =
(
1
s2,1
s
)
= (0.569840..., 0.754877...)
~ωc =
(
2(s − 1) − 1
24s, s− 1
)
= (0.617982..., 0.324717...)
4
~ωt =
(
2(s − 1) − 1
24s,4
3(s− 1) − 1
40s
)
= (0.617982..., 0.414085...) . (2.5)
Notice that s− 1 is close to 13 , so that 1
s is close to 34 and 1
s2 to 916 (as well as to 1
2); finally, we
select 2(s− 1)− 124s and 4
3(s− 1)− 140s as being respectively close to the golden ratio
√5−12 and
to√
2− 1. In view of these relations, we motivate our choice of the rotation vectors, remarkingthat both components of ~ωu, ~ωa and the second component of ~ωc are close to rational numberswith a small integer denominator. In summary, we have two vectors, ~ωu and ~ωa, which areclose to a rational vector; ~ωc has mixed components (irrational and quasi–rational) and onefrequency (~ωt) having both components irrational.
Classical results indicate that invariant KAM tori are labelled by a diophantine vector, whilerationally dependent frequencies correspond to periodic orbits. Periodic orbits with rotationvector given by the best rational approximants to a diophantine frequency vector are particu-larly interesting, since they approach closer and closer the corresponding invariant torus (if itexists). Let ~ω = (ω1, ω2) be a diophantine vector such that1 0 ≤ ω1 < 1 and 0 ≤ ω2 < 1; thesequence {pi
ri, qi
ri}i≥0 of its best approximants is defined so that
∀ (p′, q′, r′) such that 0 ≤ p′ ≤ r′ , 0 ≤ q′ ≤ r′ , 1 ≤ r′ ≤ ri , (p′, q′, r′) 6= (pi, qi, ri) :N~ω(pi, qi, ri) < N~ω(p′, q′, r′) ,
(2.6)where r0 = 1 and for any (p, q, r) ∈ Z3 the quantity N~ω is defined as
N~ω(p, q, r) ≡ Sup(|rω1 − p|, |rω2 − q|) .
The latter definitions can be easily used to determine numerically the first best approximantsof a vector ~ω; a few are listed in Table 1 in the case ~ω = ~ωu .Let us recall that for a badly approximable vector ~ω the following diophantine inequalities hold:
N~ω(p, q, r) ≥ Γ√r
∀ (p, q, r) ∈ Z3\{~0} , (2.7)
for a fixed value of Γ > 0 (in the general case of an n–dimensional badly approximable vector,the expression
√r must be replaced by r1/n). Let us stress that many nice properties (related
to the continued fraction expansions) of the best approximants of the 1–dimensional badlyapproximable vectors are no longer valid for n ≥ 2 ; nevertheless, (2.7) and Dirichlet theoremimply that
Γ√ri
≤ N~ω(pi, qi, ri) ≤1√ri+1
;
as a consequence, sudden jumps of the sequence {ri}i≥0 are not allowed when a badly approx-imable vector is considered. Our choice of the frequency vectors (generated by the PV–numbers) is mainly due to the fact that the numerical methods, which will be described in the followingsections, give sharper results when the sequence of best approximants behaves quite regularly.
1The assumption that 0 ≤ ω1 < 1 and 0 ≤ ω2 < 1 is not restrictive, especially for mappings like (1.2) dueto the well known periodicity properties in the variables (y1, y2). However, in the general case one just needsto modify the definition (2.6) so that the integers p′ and q′ vary between 0 and Mr′ , where M is the smallestinteger greater than both p′ and q′ .
5
Table 1: First few best approximants to ~ωu
i 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
pi 1 2 7 9 28 86 114 351 1081 1432 4410 13581 17991 41824 55405qi 0 1 3 4 12 37 49 151 465 616 1897 5842 7739 17991 23833pi 1 3 9 12 37 114 151 465 1432 1897 5842 17991 23833 55405 73396
3 Pade’s method
3.1 Lindstedt series expansions
Let us rewrite the mapping (1.2) as
y′1 = y1 + ε f1(x1, x2)x′1 = x1 + y′1y′2 = y2 + ε f2(x1, x2)x′2 = x2 + y′2
. (3.8)
A KAM torus with frequency vector ~ω = (ω1, ω2) can be parametrized by the equations
x1 = θ1 + u1(θ1, θ2; ε)x2 = θ2 + u2(θ1, θ2; ε)y1 = v1(θ1, θ2; ε)y2 = v2(θ1, θ2; ε)
, (3.9)
where (θ1, θ2) ∈ T2 and with the property that the flow in the parametric coordinates is linear:θ′1 = θ1 +ω1, θ
′2 = θ2 +ω2. Let us define the operator D acting on a function u ≡ u(θ1, θ2; ε) as
Du(θ1, θ2; ε) ≡ u(θ1 +ω1
2, θ2 +
ω2
2; ε) − u(θ1 −
ω1
2, θ2 −
ω2
2; ε) ;
consequently one has that
D2u(θ1, θ2; ε) ≡ u(θ1 + ω1, θ2 + ω2; ε) − 2u(θ1, θ2; ε) + u(θ1 − ω1, θ2 − ω2; ε) .
By developing u(θ1, θ2; ε) in Fourier series, it is immediately seen that the inversion of theoperator D2 causes the appearance of small divisors of the form 1/(cos(nω1 +mω2) − 1), forsome integers n and m. Using (3.8) and (3.9), we get that the functions u1 and u2 must satisfy
D2u1(θ1, θ2; ε) = εf1(θ1 + u1(θ1, θ2; ε), θ2 + u2(θ1, θ2; ε))
D2u2(θ1, θ2; ε) = εf2(θ1 + u1(θ1, θ2; ε), θ2 + u2(θ1, θ2; ε)) . (3.10)
Expanding u1 and u2 in Taylor series around ε = 0, i.e.
u1(θ1, θ2; ε) ≡∞∑
j=1
u(1)j (θ1, θ2)ε
j , u2(θ1, θ2; ε) ≡∞∑
j=1
u(2)j (θ1, θ2)ε
j , (3.11)
and inserting the series expansions in (3.10), one obtains recursive expressions for the coeffi-
cients u(1)j , u
(2)j in terms of the previous coefficients; this procedure is an extension of the 2–D
6
case presented in [6]. This procedure has been implemented on a computer by writing a For-tran program which simulates an algebraic manipulator, apt to compute as many coefficientsas possible. To give an example, we report the first–order coefficients, which are obtained bysolving the equations
D2u(1)1 (θ1, θ2) = f1(θ1, θ2) = sin θ1 + b sin(θ1 − θ2)
D2u(2)1 (θ1, θ2) = f2(θ1, θ2) = sin θ2 − b sin(θ1 − θ2) .
The solution of the previous equations gives the following coefficients:
u(1)1 (θ1, θ2) =
sin θ12(cosω1 − 1)
+ bsin(θ1 − θ2)
2(cos(ω1 − ω2) − 1)
u(2)1 (θ1, θ2) =
sin θ22(cosω2 − 1)
− bsin(θ1 − θ2)
2(cos(ω1 − ω2) − 1).
3.2 Pade’s implementation
For fixed values of θ1 and θ2 the series in (3.11) can be approximated using Pade approximants.More precisely, for a given pair of non–negative integers n,m, one defines the Pade approximantof order [n,m] of a series u(ε) as the unique rational function providing the maximal possibleorder of tangency at the origin in the class of functions r ∈ C(z) : r = (a0+a1z+...+anz
n)/(b0+b1z+...+bnz
n); such functions turn out to be locally the best rational approximations to a givenpower series. In practice, we look for a pair of polynomials P,Q ∈ C(z), such that deg(Q) ≤ n,deg(P ) ≤ m , Q 6= 0 and
(Qu− P )(z) = O(zn+m+1) , (3.12)
when z → 0, and such that the ratio P/Q is unique; the case n = m has a special role in thetheory as described in [34], [2].The accuracy in the calculation of the coefficients of P and Q is an important issue, since thesystem of equations defining the Pade approximants tends to be very ill–conditioned and sincea large accuracy is needed to find zeroes of high degree polynomials. In our calculations, due tothe 4–dimensional nature of the problem, we have the constraint of computer memory which hasprevented us to evaluate the coefficients of u1(ε), u2(ε), and therefore also of the correspondingPade polynomials, with many digits of precision. In order to control the accuracy of the results,we performed all calculations first in double precision and then in quadruple precision, checkingthe discrepancies: in all samples we have considered, the maximum difference in the coefficientsof u(ε) is of the order of 10−14 and the maximum difference in the position of the poles is about2.64 · 10−4. The order of the (diagonal) Pade approximants is not high (at most n = m = 52),but we used 12 values of θ1 = θ2 to get better approximations; such values are equally spacedin the interval (0, 2π). The evaluation of the coefficients of the polynomials P and Q has beenperformed through the implementation of a Fortran program working with a precision of 40digits; the zeroes of the denominator Q were computed through the Mathematica software. Wediscard as spurious results the zeroes of the denominator Q which are close to zeroes of thenumerator P within a tolerance of 10−4 (experiments show that 10−4 is a reasonable tolerancevalue). We remark that the results show that the domains of analyticity are well approximatedalong all their boundaries; a higher number of poles would not significantly change their shapesas well as the calculation of the critical thresholds.
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-0.25 -0.2 -0.15 -0.1 -0.05 0.05 0.1 0.15 0.2 0.25eps_re
-0.25
-0.2
-0.15
-0.1
-0.05
0.05
0.1
0.15
0.2
0.25
eps_Im
a)
-0.25 -0.2 -0.15 -0.1 -0.05 0.05 0.1 0.15 0.2 0.25eps_re
-0.25
-0.2
-0.15
-0.1
-0.05
0.05
0.1
0.15
0.2
0.25
eps_Im
b)
Figure 1: Pade results of order [52/52] for ~ωu = (1/s, s−1); a) first component (u1) with b = 0.8,
ρ(1)P = 0.19, ε
(1)P = 0.23; b) second component (u2) with b = 0.8, ρ
(2)P = 0.20, ε
(2)P = 0.23.
Pade approximants can be used to solve the problem of analytic continuation of power series orto localize the singular points. Among many interesting consequencies of Pade approximants([34], [2]), there is the numerical evidence that, even if n and m are not too large, the polesof P/Q (that is the zeroes of Q which are not also zeroes of P ) are close to the poles of thefunction u (see (3.12)). Therefore, the computation of Pade approximants provides the shape ofthe analyticity domain, from which one can compute an estimate of the radius of convergenceρ as the minimum distance of the poles from the origin of the complex ε–plane.When dealing with a 4-D mapping, one must approximate both functions u1 and u2 appearingin (3.11), so to obtain two different analyticity domains with associated radii of convergence
ρ(1)P and ρ
(2)P . The global radius of convergence ρP = ρP (~ω) will be determined as the minimum
between ρ(1)P and ρ
(2)P .
Another quantity which can be computed through the analysis of the domain of convergenceas provided by Pade approximants is the so–called break–down threshold of invariant tori.Indeed, for a fixed frequency vector ~ω, let the critical threshold εP = εP (~ω) be defined as thevalue of the perturbing parameter at which the analytic KAM torus with rotation vector ~ωbreaks–down. For 2–D mappings it is generally believed that such value coincides with theintersection of the analyticity domain with the positive real axis ([8], [4], [3], [6]). In the case ofa 4–D map, we compute εP as the minimum of the two values which are obtained by analyzing
the domains associated to u1 and u2, say ε(1)P and ε
(2)P . For low–order Pade approximants (i.e.,
when n and m are small), it is quite difficult to obtain a completely reliable estimate of suchintersections, since it could happen that no poles lie exactly on the positive real axis. In thiscase, we proceed to interpolate the poles which are closest to the positive real axis. If there arepoles exactly on the real axis, we avoid the interpolation and we calculate the closest one tothe origin.
8
We strongly remark that the two quantities ρP (~ω) and εP (~ω) are not necessarily coinciding,since the shape of the analyticity domain is in general not circular. As an example, we reportin Figure 1 the results of the computation of the poles of the functions u1 (Figure 1a) andu2 (Figure 1b) associated to (1.2) and to the rotation vector ~ωu. The coupling parameter isfixed to b = 0.8; the order of the Pade approximants is n = m = 52, which coincides with themaximum order we can determine within our computer limitations of memory and CPU time.Different colours are used to distinghuish the 12 equally spaced values of θ1 = θ2. The analysisof the graphs presented in Figure 1 suggests that the analyticity radius ρP (~ωu) is close to 0.19,while the break–down threshold εP (~ωu) amounts to about 0.23 (see also Figures 4a, 4b).
4 Greene’s method
In his celebrated article, Greene provided a criterion (see [12], assertion VI) to determine thebreak–down threshold in the perturbing parameter ε for the 1–D tori (for shortness 1–D meansone dimensional, ..., n–D stands for n–dimensional), which are invariant under the applicationof the 2–D standard mapping. Greene’s residue criterion is based on a conjecture which iseasy to explore numerically. The present section aims to describe the extension of the residuecriterion to 4–D standard mappings of the type (1.2); actually, our analysis applies to a generic2n–D standard mapping which generalizes (1.2) for all n ≥ 2, but we prefer to focus on thecase n = 2 in order to make the notation simpler.
Let us start by setting the notation. Let (x(0)1 , x
(0)2 , y
(0)1 , y
(0)2 ) denote the initial condition and
let (x(r)1 , x
(r)2 , y
(r)1 , y
(r)2 ) be the r–th iterate under the mapping (1.2). A periodic orbit of period
r must satisfy (x(r)1 , x
(r)2 , y
(r)1 , y
(r)2 ) = (x
(0)1 , x
(0)2 , y
(0)1 , y
(0)2 ). Denote by V the matrix associated
to the tangent map at the point (x(0)1 , x
(0)2 , y
(0)1 , y
(0)2 ), namely
V =
1 + c1,r−1 + dr−1 −dr−1 1 0−dr−1 1 + c2,r−1 + dr−1 0 1
c1,r−1 + dr−1 −dr−1 1 0−dr−1 c2,r−1 + dr−1 0 1
. . .
1 + c1,0 + d0 −d0 1 0−d0 1 + c2,0 + d0 0 1
c1,0 + d0 −d0 1 0−d0 c2,0 + d0 0 1
,
(4.13)
where cm,j ≡ ε cos x(j)m and dj ≡ εb cos(x
(j)1 − x
(j)2 ), for m = 1, 2 and j = 0 , . . . , r − 1. Since
every matrix appearing at the right hand side of (4.13) is symplectic, V will be symplectic aswell.Consider a frequency vector ~ω = (ω1, ω2) and its sequence of best approximants {(pi/ri , qi/ri)}i≥0,as defined in section 2. Suppose that we are able to determine the periodic orbit with rotationvector (pi/ri , qi/ri) for i ≥ 0. Let us denote by Vi the matrix related to the tangent mapassociated to the periodic orbit with frequency (pi/ri , qi/ri) (see (4.13)) and let λ1,i , . . . , λ4,i
be the eigenvalues of Vi. We will show (see Lemma 4.1) that for a symplectic matrix Vi, itis always possible to arrange the order of the eigenvalues in such a way that λ1,iλ3,i = 1and λ2,iλ4,i = 1. Consequently, we define the residue Ri associated to the periodic orbit withfrequency (pi/ri , qi/ri) as
Ri = maxm=1,2
|Rm,i| , where Rm,i =2 − λm,i − λ2+m,i
4for m = 1, 2 . (4.14)
The residue criterion is based on the following
9
Conjecture: Let us consider (a) a standard mapping of type (1.2), (b) a badly approximablefrequency vector ~ω = (ω1, ω2) (i.e. a diophantine vector ~ω satisfying the inequalities in (2.3)with τ = n = 2) and (c) the sequence of residues Ri associated to the periodic orbits withfrequencies (pi/ri , qi/ri), where {(pi/ri , qi/ri)}i≥0 is the sequence of the best approximantsrelated to ~ω. Define the quantity
µ(~ω) = lim supi→∞
logRi√ri
;
then, there exists an analytic (in the angle coordinates) invariant KAM torus with frequencyvector ~ω if and only if µ(~ω) < 0.
Remark 4.1 (i) We assumed that µ(~ω) does not substantially depend on the initial conditionson the periodic orbit (as shown in [9]) and on the character of the periodic orbit (i.e. ellipticor hyperbolic). (ii) We quote the existence of counterexamples to different formulations of theresidue criterion, like that provided by Herman (see [16] and the discussion in sections 9 and10 of [25]) in the context of smooth, non–analytic invariant tori. (iii) Let us stress that welimited ourselves to consider the case of badly approximable frequency vectors, otherwise it isnot easy to determine the exponent α such that logRi should be asymptotic to rα
i .
4.1 Heuristic justification of the conjecture
As a matter of fact, it is easier to support the part of the residue conjecture claiming that ifthe analytic KAM torus exists, then µ < 0. Indeed, this has been rigorously proved for 2–Dmappings (see [9] and [25]) and generalized in more dimensions in [35]. Our approach is basedon two arguments: the investigation of the dynamics nearby a KAM torus and the geometricalinterpretation of the residue. For simplicity we will describe separately these two arguments.
4.1.1 On the size of the resonant regions nearby a KAM torus
Consider a real analytic Hamiltonian H(~p, ~q, t), which is 2π–periodic in time and such thatits canonical flow after t = 2π induces exactly the symplectic map (1.2). It is known that ifε is small enough then the so–called interpolating Hamiltonian H exists2 (see [18] and [19]).We assume there exists an analytic KAM torus with frequencies (ω1, ω2, 1); then, there isan analytic canonical transformation (~p, ~q, t) = C( ~P , ~Q,Q0), which brings the Hamiltonian inKolmogorov’s normal form. More precisely, using the coordinates ( ~P , P0, ~Q,Q0) ∈ U ×R×T3
with U ⊆ R2 (P0 being conjugated to Q0), the Hamiltonian takes the (autonomous) form (inthe extended phase space):
H(~P , P0, ~Q,Q0) = ~ω · ~P + P0 + R(~P , ~Q,Q0) ,
where R = O(‖~P‖2).The residue criterion is essentially related to the fact that, in a small enough neighborhoodof a fixed analytic KAM torus, the size of the resonant regions shrinks exponentially to zero(with respect to the distance from the torus). This fact was pointed out by many authors,but for our purposes we find convenient to use normal forms as described in [27]. One can
2The introduction of the interpolating Hamiltonian could be bypassed after a careful extension of the tech-nique to construct the normal forms of symplectic maps as described in [14].
10
remove the angular dependence (up to an optimal order in the sense of the classical estimateson the Birkhoff normal form) in H through an analytic canonical transformation, such thatthe Hamiltonian takes the form
H( ~J, J0, ~ψ, ψ0) = ~ω · ~J + J0 +
M∑
m=2
Z(m)( ~J) + R(M)exp ( ~J, ~ψ, ψ0) , (4.15)
where (i) ( ~J, J0, ~ψ, ψ0) is a new set of action–angle coordinates; (ii) the functions Z (m) =
O(‖ ~J‖m) for m = 2 , . . . , M ; (iii) the remainder term R(M)exp = o(‖ ~J‖M ) for ‖ ~J‖ → 0; (iv)
when the distance from the torus with frequency (ω1, ω2, 1) (i.e., can be simply defined as‖ ~J‖) is small enough, there is a positive fixed parameter ∗ for which the estimate
R(M)exp = O
(
exp
[
(
−∗
)1/(τ+1)])
(4.16)
holds3.
Let us now consider the sequence {(pi/ri , qi/ri)}i≥0 of the best approximants related to ~ω andlet us focus on large values of the index i in order to be in a small neighborhood of the toruslabelled by (ω1, ω2, 1) (where (4.16) applies). We want to estimate the size of the resonant regioncorresponding to the frequency (pi/ri , qi/ri , 1). To this end, we perform another canonicaltransformation shifting the origin of the actions (i.e., ~J = ~I+ ~I∗i , J0 = I0, ~ψ = ~ϕ and ψ0 = ϕ0);
the vector ~I∗i is determined4 so that the Hamiltonian in the new coordinates becomes
H(~I, I0, ~ϕ, ϕ0) =piI1ri
+qiI2ri
+ I0 +M∑
m=2
Z(m)(~I) + R(M)exp (~I, ~ϕ, ϕ0) ,
where (i) an unessential additive constant was omitted; (ii) the functions Z (m) = O(‖~I‖m) for
any m = 2 , . . . , M ; (iii) the remainder term R(M)exp is obtained by (4.15) through a straight-
forward substitution of the coordinates (by abuse of notation, we keep the symbol R(M)exp ). Due
to the extremely small size of the remainder term as given by estimate (4.16), a new canonicaltransformation removing the non–resonant terms (i.e., the harmonics in the Fourier expansion
3The integer parameter M (ruling the order in up to which the elimination of the angular dependence is
performed) is determined so to keep the size of the remainder term R(M)exp as smaller as possible; in fact M is
actually linked to by a relation like M = O“
(1/)1/(τ+1)”
. This allows to get the estimate (4.16), which is
shown to be valid as an upper bound in [27], but it is also generically optimal as discussed in [28]. As a finalremark about the estimate (4.16), let us stress that the eventual occurrence of a power law prefactor in is notconsidered at all in this analysis.
4In a small enough neighborhood of the origin of the action variables ~J , the correspondence between (J1, J2)
and the frequencies“
ω1+PM
m=2∂ Z(m)
∂J1( ~J) , ω2+
PMm=2
∂ Z(m)
∂J2( ~J)
”
is invertible provided that the determinant of
the Hessian of Z(2) differs from zero. This new condition is always verified if the parameter ε appearing in (1.2)is small enough. In fact, due to the smallness of ε, we can ensure the non–degeneracy of the Hessian of thequadratic part of the Hamiltonian interpolating the mapping (1.2); a similar result follows for the Hamiltonian(4.15). As a consequence of this analysis, we remark that in a small enough neighborhood of the origin of theactions ~J , the following estimate holds: ‖~I∗
i ‖/Θ ≤ ‖(pi/ri − ω1 , qi/ri − ω2)‖ ≤ Θ‖~I∗
i ‖, where the value of theparameter Θ ≥ 1 depends essentially on the Hessian matrix associated to Z(2).
11
of R(M)exp such that k1pi + k2qi + k3ri 6= 0) will be surely effective up to very high order har-
monics; recall that such very high order resonant terms will be extremely small, due to theexponential decay of the Fourier coefficients. Therefore, after performing the reduction of theterms depending on the angles, the Hamiltonian can be approximated in a small neighborhoodof the origin as (for simplicity we keep the same notation for the coordinates)
H(~I, I0, ~ϕ, ϕ0) ≃piI1ri
+qiI2ri
+ I0 + hnorm(~I) + hres(~ϕ,ϕ0) , (4.17)
where hnorm(~I) contains only the quadratic terms of the integrable part of the approximationand hres can be expressed as
hres(~ϕ,ϕ0) =∑
~k∈Z3\{~0}
k1pi+k2qi+k3ri=0
[
c~k cos (k1ϕ1 + k2ϕ2 + k3ϕ0) + d~ksin (k1ϕ1 + k2ϕ2 + k3ϕ0)
]
,
(4.18)for suitable real coefficients c~k and d~k
. Let us stress that hres will satisfy an estimate similarto (4.16).
Consider the 4–D Poincare map (ϕ′1, ϕ
′2, I
′1, I
′2) = Φ(ϕ1, ϕ2, I1, I2) induced by the Hamiltonian
(4.17), when I0 is neglected and the value of the angle ϕ0 is an integer multiple of 2π. Denote byW s
i and W ui , respectively, the stable and unstable manifolds related to the hyperbolic periodic
points having a rotation vector equal to (pi/ri , qi/ri). Let us focus on the size of the resonantregion related to the frequency vector (pi/ri , qi/ri), expressed by
δm,i ≡ Sup(ϕ
(s)1 ,ϕ
(s)2 ,I
(s)1 ,I
(s)2 )∈W s
i
(ϕ(u)1 ,ϕ
(u)2 ,I
(u)1 ,I
(u)2 )∈W u
i
|I(s)m − I(u)
m | for m = 1, 2 ;
notice that we compute the size according to the two possible directions of the action coordi-nates, but hereafter the estimates on the width δm,i will be uniform with respect to the indexm.From (4.17) and (4.18), it follows that the size of the resonant region related to the frequencyvector (pi/ri , qi/ri) can be estimated as
maxm=1,2
δm,i ≤ α ·∑
~k∈Z3\{~0}
k1pi+k2qi+k3ri=0
√
c2~k+ d2
~k. (4.19)
In order to recover the latter estimate, let us start by considering a Hamiltonian of the typeH~k
= piI1ri
+ qiI2ri
+ I0 + hnorm(~I) + c~k cos (k1ϕ1 + k2ϕ2 + k3ϕ0) + d~ksin (k1ϕ1 + k2ϕ2 + k3ϕ0),
which is a pendulum–like integrable Hamiltonian, for which one can easily bound the size ofthe resonant region of the associated mapping. The estimate in (4.19) is obtained by addingall the bounds over the resonant regions associated to each H~k
. Let us recall that the effect ofthe splitting of the separatrices (due to the interaction between resonant terms associated todifferent harmonic vectors ~k) is expected to be negligible, since the norm of the integer vectors~k appearing in (4.18) increases when (pi/ri , qi/ri) → ~ω and since all coefficients c~k and d~k
are exponentially small with respect to the norm of ~k. As a final remark about the estimatein (4.19), we notice that α is a coefficient related to the hessian matrix of hnorm and it canbe determined so that it does not depend on the index i (indeed, the quadratic term hnorm
12
appearing in (4.17) will tend to Z (2) in (4.15) for i → ∞, i.e. when the best approximant(pi/ri , qi/ri) approaches ~ω).Recalling the estimate (4.16) and the expression (4.17) for the approximated Hamiltonian, theupper bound in (4.19) implies that
maxm=1,2
δm,i ≤ exp
(
− ′
‖~I∗i ‖
)1/(τ+1)
, (4.20)
where (i.e., the distance from the torus labelled by the frequency vector ~ω) has been replacedby ‖~I∗i ‖ and the coefficient ′ > 0 is given by the product of all the constants involved in the
previous estimates. From the discussion of footnote n. 4, it turns out that ‖~I∗i ‖ is proportional
to ‖(pi/ri−ω1 , qi/ri−ω2)‖, which is greater than Γ/r3/2i , whenever ~ω is badly approximable (so
that τ = 2), due to the inequalities in (2.7). Therefore, the inequality (4.20) can be rewrittenas
maxm=1,2
δm,i ≤ exp (−β√ri) , (4.21)
where the coefficient β > 0 is given in terms of the constants involved in the previous estimates.
4.1.2 About a geometrical meaning of the residue
A key point of the residue criterion is the assumption that close to a periodic point the localbehaviour of the mapping is sufficiently well approximated by the tangent map. Therefore,the following interpretation of the meaning of the residue takes advantage from the fact thatthe matrix V associated to the tangent map is symplectic. As a consequence, the proofs ofthe following statements are based on rather simple arguments of symplectic linear algebra.Since these arguments present the same degree of difficulty for a generic dimension, we selecta symplectic space of generic dimension 2n.
Let us start our investigation of the properties of V with the following
Lemma 4.1 Let V be a 2n × 2n–D symplectic real matrix such that all its eigenvalues aredistinct. There exists a symplectic basis {~v1, . . . , ~v2n} of eigenvectors in C2n, such that thecorresponding eigenvalues {λ1, . . . , λ2n} satisfy the relation
λjλj+n = 1 for j = 1, . . . , n .
The proof is presented in appendix A. We remark that if λj ∈ C \ R, then either there exist
θj ∈ (0, π) ∪ (π, 2π) , such that λj = exp(iθj) and λj+n = exp(−iθj) (4.22)
or there exist
{i, ζj , θj}, where i 6= j, ζj ∈ R+ \ {1}, θj ∈ (0, π) ∪ (π, 2π), such thatλj = ζj exp(iθj), λj+n = ζj exp(−iθj), λi = 1
ζjexp(iθj), λi+n = 1
ζjexp(−iθj).
(4.23)In the following, we will consider those matrices V whose corresponding eigenvalues are of thetype (4.22), so that it is possible to find a canonical basis, such that the action of the linearoperator associated to V can be decoupled over n 2–D subspaces. On the other hand, when
13
some eigenvalues of the type (4.23) are considered, the decoupling cannot be performed in somesubspaces more than 2–D. Let us stress that the case of eigenvalues of type (4.22) is often5 metin numerical applications of the residue criterion.Our interpretation of the geometrical meaning of the residue relies on a careful translation inhigher dimensions of that proposed by Greene at the end of section 2 of [12]. This interpretationis essentially based on a simple remark found in section 2 of [11] about the quadratic forms thatare invariant under the application of a 2 × 2–D symplectic matrix. The additional difficultydue to the fact that we are dealing with a 2n× 2n–D matrix V can be overcome, since there isa linear symplectic transformation that allows to decouple the application of V as the productof n 2 × 2–D linear area–preserving maps, acting separately. This statement is ensured by thefollowing Proposition, whose proof is provided in appendix A.
Proposition 4.1 Let V be a 2n × 2n–D symplectic real matrix such that all its eigenvalues{λ1, . . . , λ2n} are distinct; moreover, for j = 1, . . . , 2n assume that either λj ∈ R or |λj | = 1.
Therefore, there exists a basis {~e1, . . . , ~en, ~d1, . . . , ~dn} of vectors in R2n such that(i) the basis is symplectic;(ii) for j = 1, . . . , n, the pair {~ej , ~dj} is an orthonormal basis of the subspace span(~ej , ~dj);
(iii) for j = 1, . . . , n, the vectors ~ej and ~dj can be expressed as a linear combination (withcomplex coefficients) of the vectors ~vj and ~vj+n appearing in lemma 4.1 (and viceversa), so
that V~ej ∈ span(~ej , ~dj) and V ~dj ∈ span(~ej , ~dj);
(iv) for j = 1, . . . , n, the condition ~ej · V ~dj + ~dj · V~ej ≥ 0 is satisfied; the following equalities
hold: ~ej · V~ej = ~dj · V ~dj and
det
(
~ej · V~ej ~ej · V ~dj
~dj · V~ej ~dj · V ~dj
)
= 1 ;
(v) if there exists another basis {~e∗1, . . . , ~e∗n, ~d∗1, . . . , ~d∗n} of real vectors satisfying (i)–(iv) and
if the generic case ~ej · V ~dj + ~dj · V~ej > 0 applies, then, either ~e∗j = ~ej and ~d∗j = ~dj , or ~e∗j = −~ejand ~d∗j = −~dj.
The properties of a quadratic form, which is invariant with respect to the matrix V, are de-scribed by the following
Proposition 4.2 Let V be a 2n× 2n–D symplectic real matrix satisfying all the hypotheses ofProposition 4.1. There exists a quadratic form, which is invariant with respect to the changeof basis associated to the matrix V: if we denote by Q the 2n × 2n–D real symmetric matrixrelated to the quadratic form, then VTQV = Q.Let {~e1, . . . , ~en, ~d1, . . . , ~dn} be a basis of vectors in R2n satisfying (i)–(iv) of Proposition 4.1and denote by aj, bj and cj the (unique) real coefficients such that
(
~ej · V~ej ~ej · V ~dj
~dj · V~ej ~dj · V ~dj
)
=
(
aj cj + bjcj − bj aj
)
.
5Over all the periodic orbits computed to obtain the results in section 6, the situation with eigenvalues oftype (4.23) occurred only twice. In both cases the initial values of the angles were equal to zero. The first caseconcerns the orbit corresponding to the frequency vector (465/816 , 616/816), when ε = 0.1875 and b = 0.4; thesecond case concerns the vector (7959/12879 , 5333/12879), when ε = 0.875 and b = 0.001. See appendix B fordetails about the procedure to determine periodic orbits.
14
Then, the matrix Q satisfies(i) ~ei ·Q~ej = ~ei ·Q~dj = ~di ·Q~dj = 0, for any i, j = 1, . . . , n with i 6= j;(ii) for j = 1, . . . , n, the following matrix equation holds:
(
~ej ·Q~ej ~ej ·Q~dj
~dj ·Q~ej ~dj ·Q~dj
)
=
(
bj − cj 00 bj + cj
)
;
(iii) for j = 1, . . . , n, the following equation holds:
√
|bj − cj |√
|bj + cj |=
2√
|σj(σj − 1)||bj + cj |
, (4.24)
where σj = (2 − λj − λj+n)/4 is the residue associated to the pair of eigenvalues {λj , λj+n}appearing in Lemma 4.1.
Proposition 4.2 (whose proof is given in appendix A) ensures that there is a linear canonicaltransformation such that, in the new coordinates, the invariant quadratic form is associated tothe cartesian product of n different families of conic curves, each of them lying in the symplecticsubspace spanned by the vectors ~ej and ~dj for j = 1, . . . , n. In fact, let us remark that in thesesubspaces the conic invariant curves are of the form (bj −cj)x2 +(bj +cj)y
2 = ζ, where ζ ∈ R isa parameter ruling the size of the curve; therefore, these conics are either ellipses or hyperbolasaccording to6 bj−cj and bj+cj having either the same or opposite signs, respectively. Therefore,the quantity
√
|bj − cj |/|bj + cj| appearing in (4.24) can assume two geometrical meanings:if we are dealing with a family of ellipses, then (4.24) is equal to the ratio of their semimajoraxes; in the case of hyperbolas, (4.24) is equal to the tangent of half of the angle betweenthe asymptotes. Thus, the residue is related by (4.24) to the geometry of the invariant curvesassociated to the linear symplectic mapping induced by the matrix V.
4.1.3 Comparison between the dynamical and geometrical approaches to the
residue conjecture
We can summarize our investigation about the dynamics near a KAM torus with frequencyvector ~ω by saying that, in the proximity of the torus, the system looks more and more similarto an integrable one. Moreover, in a neighborhood of the torus, for each periodic orbit withfrequency (pi/ri , qi/ri), one can define a suitable canonical transformation showing that thesize of the resonant region becomes exponentially small according to the estimate (4.21). Letus remark that the sequence of canonical transformations will tend to a limit (provided by theKolmogorov’s normal form) for ri → ∞.Since the system looks more and more similar to an integrable one, we are led to provide asecond estimate of the size of a resonant region close to a KAM torus by using the linearizationof the mapping nearby a periodic point; this second estimate is a simple extension to higherdimensions of what has already done in section 4 (and numerically checked in section 5) of [23].Under very mild assumptions, we have seen that the invariant curves with respect to the tangentmap are 2 families (hereafter we limit again ourself to the case n = 2) of ellipses or hyperbolas.We know that the separatrices (in the limit of a pendulum–like integrable system) draw a sort
6Let us recall that in terms of the residues σj , if 0 < σj < 1 the curve is an ellipse and if σj < 0 or σj > 1we are dealing with an hyperbola (as remarked in [12]). Finally, the cases σj = 0 or σj = 1 are not compatiblewith the condition that the eigenvalues are distinct.
15
of “cat’s eye” connecting two consecutive hyperbolic points. Moreover, for continuity reasons,it is natural to imagine that dealing with ellipses the direction of the semimajor axes becomescloser to a parallel to the tangent plane to the KAM torus, whenever the periodic orbits areapproaching the torus. In the case of hyperbolas, the same should apply to the direction of thebisectors of the smallest angle between the asymptotes. Therefore, we are led to estimate thesize of a resonant region by multiplying the left hand side of (4.24) by the the distance betweentwo consecutive periodic points (which is proportional to the inverse of the period). Whenwe consider a periodic orbit of frequency (pi/ri , qi/ri) and we calculate the correspondingcoefficients aj , bj and cj (j = 1, 2) defined in Proposition 4.2, we find that all the coefficientsare usually O(ri). In conclusion, there is only one way to make the estimate of the size of aresonant region exponentially small with respect to
√ri: the residue appearing in the right
hand side of (4.24) must satisfy an inequality of the form (4.21), which is equivalent to say thatthe limit µ(~ω), appearing in the residue conjecture, must be negative.
Let us try to support the remaining part of the conjecture, i.e. if the sequence of the residuesdecreases exponentially then the torus with frequency vector ~ω exists and it is analytic. Consideragain the families of conic curves invariant with respect to the linearization of the mappingabout the sequence of periodic orbits of frequencies (pi/ri , qi/ri) (where {(pi/ri , qi/ri)}i≥0 isthe sequence of best approximants related to ~ω). Let us assume that µ(~ω) < 0 and that thedirections of both the semimajor axes of the ellipses and the bisectors of the smallest anglebetween the asymptotes of the hyperbolas are converging to a plane7. Under these conditions,the interpretation of the geometrical meaning of the residue leads to believe that the size ofthe resonant regions associated to the best approximants becomes exponentially small withrespect to
√ri. Since the resonant regions associated to the best approximants are expected
to be the largest ones and since the number of the resonant regions grows as a polynomialof ri, we conclude that the total fraction of volume occupied by high order resonances8 willdecrease exponentially with respect to their order. Therefore, in a neighborhood of the phasespace about the periodic orbits of frequency (pi/ri , qi/ri), we are led to infer that the majorityof this neighborhood will be filled by tori when ri → ∞; consequently, the torus correspondingto the frequency vector ~ω shall exist.
4.2 Implementation of Greene’s method
The residue conjecture is used as in the 2–D standard map, in order to determine numericallythe break–down threshold in the context of the 4–D mappings of type (1.2). Hereafter, we willdenote by εG the threshold provided by Greene’s method.
Let us consider the map Mε , b in (1.2) for fixed values of ε and b. First, we must find the
7This plane will be tangent to the KAM torus of frequency ~ω. The convergence of the directions associatedto the elliptic and hyperbolic families of invariant curves is usually not considered, because it is expected to bea consequence of the exponential decay of the residues.
8We are now referring to resonances satisfying a unique law of the form k1pi + k2qi + k3ri = 0 with a fixed~k ∈ Z
3\{~0}; notice that our discussion (as well as the numerical applications) only deals with the resonant regionsassociated to a periodic orbit of frequency (p/r , q/r), which satisfies two resonant laws. In fact, we are confidentthat the size of the resonances associated to a single law can be bounded by using those related to the periodicorbits, as suggested by the geography of the resonances described in the geometric part of the Nekoroshev’stheorem (see, e.g., the original papers [30] and [31], and the recent tutorial work [13]). The geography of theresonances is very well numerically investigated in [15].
16
Figure 2: Logarithm of the residue of some periodic orbits with frequencies tending to ~ωu =(1
s , s − 1), for a few values of the small parameter ε and for b = 0.4. The periodic orbits havefrequencies in the set {(pi/ri , qi/ri)}i≥0 of the best approximants to ~ωu, as listed in Table 1.
periodic orbits with frequencies (pi/ri , qi/ri), where {(pi/ri , qi/ri)}i≥0 is the sequence of bestapproximants to the rotation vector ~ω = (ω1, ω2). The numerical determination of the periodicorbits is definitely the most expensive operation in terms of CPU–time. A technique for thesearch of the periodic orbits, based on the particular symmetry properties of Mε , b, is describedin appendix B. Once we find a periodic orbit with frequency (pi/ri , qi/ri), we can compute thecorresponding residue Ri according to the definition (4.14). In principle, one should numericallydetermine µ(~ω) as the lim supi→∞ logRi/
√ri, but this task is practically impossible. Therefore,
by studying the behaviour of the first terms of the sequence {Ri}, the residue conjecture canbe used to discern between the existence or non–existence of the torus, according to whetherthe sequence shows a convergence to 0 or a divergence to +∞.To be more precise, let us refer to a concrete example. In Figure 2 we investigate for somevalues of ε the behaviour of the sequence of the residues Ri associated to Mε , 0.4 and to ~ω = ~ωu.Looking at the sequence related to ε = 0.2, we are led to conclude that the KAM torus withfrequency ~ωu does exist; on the other hand, for ε = 0.35, it clearly seems that such KAM torusdoes not exist.The procedure can be carried out in a completely automatic way, if one provides a safe andefficient criterion to decide whether the sequence of residues Ri converges to 0 or diverges to+∞. To this end, let us first choose i∗ so that for i ≥ i∗ the rate of convergence of (pi/ri , qi/ri)to ~ω is regular enough (to fix the ideas, for each vector ~ω considered in section 6 and inthe present one, we defined i∗ = 5). As a practical criterion, in our calculations we assumedthat limi→+∞Ri = 0, if for three consecutive integer values of i ≥ i∗ one has that either (i)
17
Ri < 2.5 × 10−5 or (ii) Ri < 2.5 × 10−2 and Ri−1/Ri > 1.5. Analogously, we assumed thatlimi→+∞Ri = +∞, if for three consecutive times it occurs that either (i) Ri > 2.5× 103 or (ii)Ri > 2.5 and Ri/Ri−1 > 1.5 or (iii) the corresponding periodic orbit has not been found9. Letus stress that the lower [upper] bounds on the values of Ri, under [over] which the trend goingto 0 [+∞] is considered to be significant, have been chosen after remarking that (typically)Ri ∼ 10−1 for the first few indexes i ≥ 0, whenever ε is not far from εG. Of course, our criteriaare highly questionable and, in general, one should keep in mind that they must be changedso to give reliable answers according to the regularity of the sequence N~ω(pi, qi, ri), as definedin section 2. As a final remark, we stress that for the 2–D standard mapping the behaviourof the residues presented in Figure 2 is definitely more regular. Moreover, Greene conjectured(see [12], assertion V) that if ε = εG, then limi→+∞Ri = 0.25. Passing to the 4–D case, the lossof regularity of the residues’ behaviour makes more delicate the detection of the asymptoticlimit.The automatic determination of the value εG can be completed by using a bisection procedureon the small parameter ε. In fact, one first defines two bounds εinf and εsup such that oneexpects that εG ∈ (εinf , εsup). Then, one investigates the convergence to 0 or the divergenceto +∞ of the sequence of the residues for ε = (εinf + εsup)/2; in case of convergence, onedefines εinf = ε, otherwise one sets εsup = ε. Finally, one iterates this procedure until thedesired approximation on the value of εG is reached. For instance, by starting with εinf = 0 andεsup = 0.4 and looking at Figure 2, one concludes that εG ∈ (0.3125, 0.325) (with ~ω = ~ωu andb = 0.4). Following this procedure, let us stress that we exploit the implicit assumption that(at least in the range of the parameter values we are considering) the residues Ri are increasingfunctions of ε, as supported also by Figure 2.
5 Frequency analysis
The method of frequency analysis is essentially based on a visual exploration; in this respect, itis certainly instructive to present this technique on a concrete example. Our approach basicallyfollows the suggestions presented in the tutorial work [22] and it is inspired by the investigationperformed in [33], especially for what concerns the frequencies’ diffusion.We report in Figures 3a and 3b the results of the application of the frequency analysis, aimedto study the existence of the KAM torus with frequency ~ωu, in the framework of the symplecticmapping (1.2) with ε = 0.3125 and b = 0.4. Figures 3c and 3d concern the case ε = 0.325.Each pair of plots (i.e., Figures 3a–3b and 3c–3d) contains a double check on the existence ofthe torus with frequency ~ωu: the first check deals with the regularity of the actions–frequenciesmap, while the second one is based on the frequencies diffusion. Let us focus on Figure 3a andlet us ignore, for a moment, the differencies between the symbols used to mark the dots. Every
point locates an initial condition (0, 0, y(0)1 , y
(0)2 ), since we set the initial angles equal to zero. For
each initial condition, we iterate the map 2T = 218 times and we calculate the corresponding
9By recalling the discussion in subsection 4.1.1, when the KAM torus is approached, the dynamics becomesmore and more similar to that of an integrable system, so that the determination of the nearby periodic orbitsis relatively easy. On the other hand, the existence of remarkable chaotic zones should prevent the existenceof nearby tori; as a consequence, the determination of periodic orbits is more difficult. These arguments aresupported by the remark that, usually, the sequence of the residues begins to significantly increase (with respectto the index i), thus obstructing the determination of the periodic orbits.
18
frequencies νj,m for all j,m = 1, 2 as the maximum points of the functions |Aj,m(ν)|2 with
Aj,m(ν) =1
T
mT∑
t=(m−1)T
y(t)j exp[ı(x
(t)j − 2πνt)]W(t − (m− 1)T ) , (5.25)
where W(t) = 1 + cos(
2π(t−T/2)T
)
is the so called Hanning filter adapted to the window [0, T ].
For each point (y(0)1 , y
(0)2 ) in the regular grid of Figure 3a, we plot a symbol in Figure 3b, which
corresponds to the average value of the frequencies ~ω = (ω1, ω2), with ωj = (νj,1 + νj,2)/2 forj = 1, 2. Let us remark that Figure 3b is centered around ~ω = ~ωu; on the contrary, Figure 3a iscentered about a vector ~yu, such that10 the region filled in Figure 3b by the points correspondingto those in the regular grid (see Figure 3a) surrounds a neighborhood of ~ωu.As pointed out in [22], in a neighborhood of a KAM torus the local correspondence betweenactions and frequencies must be invertible and regular (as can be explained by using theapproach to the KAM theorem described in [32]). Concerning Figure 3, we are led to concludethat the KAM torus with frequency ~ωu exists if and only if the dots lie on a regular gridclose to the origin of the plots on the right panel. Therefore, the study of the regularity of theactions–frequencies map in Figure 3 clearly shows that for ε = 0.3125 the torus with frequency~ωu still persists the perturbation, while for ε = 0.325 it seems to disappear. Let us remark thatthe frequencies are sometime located close to a straight line or they may form an empty strip;this phenomenon is due to the occurrence of some remarkable resonance.Let us now describe the check concerning the frequencies’ diffusion along the orbits. For eachpair of corresponding points in Figures 3a–b (i.e., the initial actions ~y on the left and thecorresponding average values of the frequencies ~ω), we calculate the frequencies’ diffusion as∆ω = |ν1,1 − ν1,2| + |ν2,1 − ν2,2|. We denote with the same symbol the corresponding pointsin Figures 3a–b according to the order of magnitude of ∆ω, by adopting the scale reported onthe top of the Figure; similarly for Figures 3c–d.As proved in Proposition 1 of [22], when we consider an orbit on a KAM torus, then ∆ω =O(1/T 4) (the value 4 of the exponent in this asymptotic law is due to the fact we have adoptedthe Hanning filter). On the other hand, when we study a chaotic orbit, ∆ω cannot be madearbitrarily small by increasing the number of iterations 2T . Therefore, for large enough valuesof T , ∆ω provides a sort of measure of the degree of chaoticity. Moreover, since in a smallneighborhood of a KAM torus the size of the resonant regions shrinks exponentially to zero(recall [27]), we are led to conclude that if we see around the origin many dots marked with�, then the torus with frequency ~ωu exists (compare with Figures 3b–d). On the contrary, ifmany ◦, • or + appear, then the torus does not exist. In practice, the study of the frequencies’diffusion presented in Figure 3 shows that the torus with frequency ~ωu exists for ε = 0.3125,while it disappears for ε = 0.325. This conclusion validates the study of the regularity of theactions–frequencies map and the results obtained with Greene’s method.
10The vector ~yu is a priori unknown and it is usually determined by trials, starting from the knowledge ofa periodic orbit with rotation frequency very close to ~ωu (see appendix B for a method to determine periodicorbits).
19
6 A comparison of the different methods and conclusions
We present the results obtained by implementing the different techniques discussed in theprevious sections, i.e. Greene’s method, Pade approximants and frequency analysis. We considerthe 4–D standard mapping (1.2) and the rotation vectors ~ωu, ~ωa, ~ωc, ~ωt introduced in (2.5).We know that for b = 0 the 4–D map decouples in two distinct 2–D standard maps; beinginterested to the transition from the 2-D to the 4–D case, we start by implementing Greene’smethod for the 2–D mapping (in its –nowadays– standard formulation provided in [12]) inorder to compute the break–down threshold associated to the components of the frequencyvectors (2.5). In particular, we are interested to investigate the invariant tori with (irrational)frequencies given by 1/s , s − 1 , 1/s2 , 2(s − 1) − 1
24s , 43(s − 1) − 1
40s . The results shown inTable 2 are presented in the form of a critical interval with accuracy parameter δ , say εG ± δ ,meaning that for ε ≤ εG − δ we have evidence that the residues of the approximating periodicorbits decrease to zero, while for ε ≥ εG + δ we have a numerical indication of the divergenceof the residues.We apply Greene’s method (as described in section 4) for different values of the couplingparameter, i.e. b = 10−4, 10−3, 10−2, 0.1, 0.4 and 0.8. The results are gathered in Table 3 andare again given in the form of an interval with accuracy parameter δ ≤ 0.01 .
Next, we apply the frequency analysis as explained in section 5, in order to determine thebreak–down threshold for all frequencies and values of b as listed in Table 3. We use a doublecheck provided by the regularity of the actions–frequencies map and by the measure of thefrequencies’ diffusion. Let us stress that it is extremely difficult to carry on the frequencyanalysis in a really automatic way: first of all, we did not find a criterion different from thevisual one, in order to discriminate whether an invariant torus exists or not; moreover, severaltrials are usually needed in order to select the center ~yu (as discussed in the footnote n. 10)and the size of the regular grid of initial conditions. For these reasons, we limit ourselves toapply the frequency analysis in order to confirm the results previously obtained by Greene’smethod: we do not aim to provide the break–down thresholds through an independent use ofthe frequency analysis. In all cases listed in Table 3, we find that the frequency analysis is inagreement with the results of Greene’s method.
Finally, we present the results of the application of Pade approximants related to the functionsu1(θ1, θ2; ε) and u2(θ1, θ2; ε) appearing in (3.11). We compute their Pade approximants of order[52/52] for 12 different values of θ1 = θ2, equally spaced in the interval (0, 2π). The couplingparameter b is again fixed as b = 10−4, 10−3, 10−2, 0.1, 0.4 and 0.8. Small squares of differentcolours (grey–scale) are used to represent poles in the complex ε–plane for different values ofθ1 = θ2. Spurious poles are discarded and the radius of convergence ρP = ρP (~ω) (computed as
the minimum between the radii ρ(1)P and ρ
(2)P associated to the two components u1 and u2) has
been computed according to section 3.2.In computing the value of εP , there is an intrinsic problem due to the spreading of Pade polesaround the boundary of the analyticity domain. More precisely, one has to take into accountthe following cases, which might occur when computing the intersection of the boundary withthe positive real axis.(a) There might be several poles which lie ”exactly” on the positive real axis. This is a typicalsituation which occurs when varying θ1, θ2 in the interval [0, 2π). In our calculations, thespreading of such points is within an error less than 5 · 10−2, though it may vary with b and ω.
20
For sake of simplicity, we reported in Tables 4,5 the distance of the closest pole to the origin.(b) If there are no real poles, we are led to interpolate on the cloud of points which lie inthe proximity of the real axis. Also in this case the main source of error is due to the widthof the spreading, which is again of the order of 5 · 10−2. We have bypassed this hindrance byconsidering a sufficiently large number of values of θ1, θ2 so that we always had at least onepole on the real axis (compare with Figure 4d)).We remark that the above errors might be decreased by improving the precision of Pade’scalculations and, in particular, by increasing the order of the approximating polynomials.Figure 4 provides the analyticity domains of the components u1 and u2 for the rotation vector~ωc. The coupling parameter is fixed to b = 0.0001 in Figures 4a, 4b and b = 0.01 in Figures4c, 4d. We recall that the first component of the frequency vector is close to the golden ratio√
5−12 , while the second component s − 1 = 0.3247 . . . is close to the rational number 1
3 . Asa consequence of this coupling between an irrational and a quasi–rational coordinate, for bsmall the analyticity domain assumes a flower–shape (see Figure 4a) with 6 petals. Indeed, theresults for the 2–D standard mapping provided in [5], [6], showed that the analyticity domainof a frequency close to a rational number p
q is flower–shaped with 2q petals. We remark thatthe effect of the quasi–rational component is weakened by taking a higher value of the couplingparameter; in fact, Figures 4c and 4d suggest that the domain of analyticity is approaching acircular shape.We present in Tables 4 and 5 the results for the rotation numbers (2.5) concerning the com-putation of the analyticity radius ρP and the break–down threshold εP . The results are givenin the form r1 − r2, where r1 refers to the first component u1 and r2 refers to the secondcomponent u2. A comparison between Tables 3 and 5 shows a remarkably good agreement ofthe results obtained applying Greene’s method and the computation of the break–down valueεP by means of Pade approximants.
Let us briefly compare the performances of the three methods described in the present work. Ifone is exclusively interested in the determination of the break–down threshold, Greene’s methodis definitely faster and more accurate. We think that the uncertainity interval of Greene’s resultsreported in Table 3 could be likely reduced of at least one order of magnitude, just with theresources available to us; instead, such precision is probably beyond the current possibilities ofthe frequency analysis and (even far beyond) of Pade’s method. Nevertheless, since Greene’smethod is based on a conjecture that requires the knowledge of an asymptotic limit, this methodmust be carefully used in order to avoid some pitfalls. Therefore, for a reliable determination ofthe break–down threshold, we propose the complementary use of the three methods. Moreover,let us recall that each of these techniques provides additional information to that of the criticalthreshold: the size of the resonant regions nearby a KAM torus can be estimated by theresidues, the geography of the ordered and chaotic regions can be reconstructed thanks tofrequency analysis, the shape of the analyticity domain is given by Pade approximants.
Let us focus on the consequences of the study of the analyticity domains. As remarked before,the domains should approach a circular shape when the coupling parameter b is increased. Bycomparing the values of ρP to those of εP (see Tables 4 and 5, respectively) for values of b notvery small, there are not the remarkable differences that one can appreciate when frequenciesas close to the resonance 1/3 as the number s− 1 are studied in the framework of the standardmapping (see Figure 3 in [5]). In particular, our study suggests that perturbative computer–assisted proofs on the existence of KAM tori (eventually limited by the radius of analyticity,
21
Table 2: Break-down threshold as given by the Greene’s method (εG) for b = 0 .
Frequency Critical value for b = 0
1s ≃ 0.7548 0.6932 ± 0.0001s− 1 ≃ 0.3247 0.7039 ± 0.00021s2 ≃ 0.5698 0.8325 ± 0.00012(s− 1) − 1
24s ≃ 0.6179 0.9563 ± 0.000343(s− 1) − 1
40s ≃ 0.4140 0.9531 ± 0.0001
Table 3: Break-down threshold as obtained with the Greene’s method (εG) and confirmed byusing the frequency analysis.
~ωu ~ωa ~ωc ~ωt
b = 10−4 0.62 ± 0.01 0.607 ± 0.008 0.593 ± 0.007 0.918 ± 0.007b = 10−3 0.55 ± 0.01 0.533 ± 0.007 0.518 ± 0.007 0.865 ± 0.01b = 10−2 0.468 ± 0.007 0.445 ± 0.005 0.368 ± 0.007 0.7 ± 0.01b = 0.1 0.393 ± 0.007 0.33 ± 0.008 0.254 ± 0.01 0.362 ± 0.008b = 0.4 0.318 ± 0.007 0.18 ± 0.008 0.193 ± 0.007 0.137 ± 0.008b = 0.8 0.23 ± 0.01 0.13 ± 0.008 0.167 ± 0.008 0.083 ± 0.006
see, e.g., [7]) might nearly reach the critical threshold, when coupled mappings are consideredwith a non negligible interaction.We remark that the values of the critical threshold εP and of the radius of convergence ρP
decrease when the coupling coefficient b increases. In particular, there is also an abrupt decreaseof the critical threshold, whenever an irrational number is coupled with an almost rational value;see, for example, the number close to the golden ratio (0.6179...) in Table 2 and compare theresults of ~ωc for b = 10−4 in Tables 3, 4, 5. Moreover, the decrease rate seems to dependstrongly on the vector frequency (see Figure 5); for instance, for small values of b the invarianttorus corresponding to the vector ~ωt looks largely the most robust, while it is the weakestfor b = 0.8. Other changes in the robustness hierarchy of the tori has been already remarked(see [20]); in the present case, we can give a heuristic explanation. Let us remark that the vector((√
5 − 1)/2 ,√
2 − 1) (which is very near to ~ωt) is diophantine and it satisfies the inequalitiesin (2.3) with τ = 3 ; indeed, it can be completed with ω3 =
√10 in such a way that it is given
by the formula (2.4) with α =√
5 +√
2 , while the same cannot apply when n = 2 . Therefore,by recalling how the size of the resonant regions nearby a KAM torus depends on τ (see theestimate (4.20)), one expects that the torus corresponding to ((
√5 − 1)/2 ,
√2 − 1) (and its
neighbors as ~ωt) is weaker than those related to (genuine) badly approximable vectors, at leastwhen the coupling is effective. For small values of b , looking at Table 2, one expects that forcontinuity reasons ~ωt is more robust than the others.
Acknowledgements. We than R. Schoof for very useful discussions on number theory.
22
Table 4: Pade’s results: radius of convergence (ρP ) for the first and second components, u1 andu2.
~ωu ~ωa ~ωc ~ωt
b = 10−4 0.56 − 0.56 0.58 − 0.59 0.56 − 0.56 0.89 − 0.88b = 10−3 0.53 − 0.51 0.54 − 0.51 0.51 − 0.49 0.81 − 0.82b = 10−2 0.45 − 0.45 0.46 − 0.43 0.36 − 0.35 0.67 − 0.67b = 0.1 0.38 − 0.39 0.32 − 0.32 0.24 − 0.23 0.31 − 0.31b = 0.4 0.28 − 0.27 0.16 − 0.17 0.19 − 0.18 0.13 − 0.13b = 0.8 0.19 − 0.20 0.11 − 0.11 0.15 − 0.15 0.08 − 0.08
Table 5: Pade’s results: critical threshold (εP ) of the first and second components, u1 and u2.
~ωu ~ωa ~ωc ~ωt
b = 10−4 0.59 − 0.61 0.60 − 0.64 0.59 − 0.56 0.89 − 0.91b = 10−3 0.53 − 0.53 0.55 − 0.51 0.56 − 0.50 0.84 − 0.84b = 10−2 0.50 − 0.46 0.47 − 0.45 0.36 − 0.35 0.72 − 0.72b = 0.1 0.39 − 0.40 0.33 − 0.34 0.25 − 0.24 0.41 − 0.39b = 0.4 0.31 − 0.30 0.18 − 0.17 0.20 − 0.18 0.13 − 0.14b = 0.8 0.23 − 0.23 0.12 − 0.13 0.17 − 0.17 0.09 − 0.09
A Proofs of Lemma 4.1, Proposition 4.1 and Proposition 4.2
Let ~u ≡ (α1, . . . , α2n) ∈ C2n, ~v ≡ (β1, . . . , β2n) ∈ C2n; let ~u · ~v ≡ ∑2nj=1 αjβj (this is an abuse
of notation, since the standard inner product in C2n requires that the coefficients αj must besubstituted by their complex conjugates). Let J be the 2n× 2n–D skew symmetric matrix, i.e.
J =
(
0 In
−In 0
)
,
where In is the n× n–D identity matrix.
Proof of Lemma 4.1. Since all the eigenvalues are distinct, there is a set of eigenvectors{~u1, . . . , ~u2n} which is a basis in C2n. Therefore, since V is symplectic, we can write the followingequalities:
~ui · J~uj = ~ui · VTJV~uj = (V~ui) · JV~uj = λiλj (~ui · J~uj) , i , j = 1 , . . . , 2n ,
where λj is the eigenvalue corresponding to ~uj. It follows that, for any fixed pair of indexes(i, j), either ~ui · J~uj = 0 or λiλj = 1.We claim that for j = 1, . . . , 2n, there exists only one index j∗ such that λj∗λj = 1. In fact, ifthere exists another j ′ 6= j such that λj′λj = 1, then it would contradict that all the eigenvaluesare distinct; on the other hand, if λiλj 6= 1, for i = 1, . . . , 2n, then it follows that ~ui · J~uj = 0so that J should be degenerate, which is clearly false.
23
By simply reordering the indexes, we can ensure that λjλj+n = 1 for j = 1, . . . , n. Finally, letus define
~vj = ~uj and ~vj+n =~uj+n
~uj · J~uj+nfor j = 1 , . . . , n ;
it is easy to verify that
~vi · J~vj =
{
1 if j = i+ n0 if j 6= i+ n
for 1 ≤ i ≤ j ≤ 2n . (A.1)
Q.E.D.The following Lemma proves part of Proposition 4.1.
Lemma A.1 Let V be a 2n × 2n–D symplectic real matrix satisfying all the hypotheses ofProposition 4.1. There exists a basis {~e1, . . . , ~en, ~d1, . . . , ~dn} of vectors in R2n, such that itsatisfies the points (i), (ii) and (iii) of Proposition 4.1.
Proof . Let the basis {~v1, . . . , ~v2n} ∈ C2n and the corresponding eigenvalues {λ1, . . . , λ2n} beas in Lemma 4.1. Knowing only the eigenvalues, one can easily determine11 n real linear oper-ators Πj , such that Πj(C
2n) = span(~vj, ~vj+n). First, let us introduce the real linear operatorsrepresented by the matrices
Pj = V V − (λj + λj+n)V + I2n , j = 1, . . . , n ;
by recalling that all the eigenvalues are supposed to be distinct, one can easily verify that thekernel of Pj coincides with span(~vj , ~vj+n). Thus, the operators Πj are given by the matrices
Πj = P1 . . .Pj−1Pj+1 . . .Pn , j = 1, . . . , n .
Let us now proceed to the definition of the basis {~e1, . . . , ~en, ~d1, . . . , ~dn} we are looking for.First, choose n vectors ~e1, . . . , ~en such that ~ej ∈ span(~vj , ~vj+n) ∩ R2n and ~ej · ~ej = 1; then,
determine ~dj ∈ span(~vj , ~vj+n) ∩ R2n, j = 1, . . . , n, so that ~ej · ~dj = 0 (one can use the Gram–
Schmidt procedure) and ~ej · J ~dj = 1. Let us check that the selected basis satisfies point (i),i.e.
~ei · J~ej = ~di · J ~dj = 0 and ~ei · J ~dj =
{
1 if j = i0 if j 6= i
for i , j = 1 , . . . , n . (A.2)
The latter equations are true by construction when i = j. If i 6= j, let us remark that, forinstance, ~ei · J ~dj = α~vi · J~vj + β~vi · J~vj+n + γ~vi+n · J~vj + δ~vi+n · J~vj+n = 0 (α, β, γ and δ aresuitable complex coefficients), since all the symplectic products involving vectors of the basis{~v1, . . . , ~v2n} are equal to zero in view of (A.1). The same remark applies also to ~ei · J~ej and~di · J ~dj .Let us now prove point (ii) of the thesis, i.e.
~ej · ~ej = 1 , ~ej · ~dj = 0 and ~di · ~dj = 1 for j = 1 , . . . , n .
11The operators Πj can be used to construct the basis {~e1, . . . , ~en, ~d1, . . . , ~dn} we are looking for. Moreover,let us remark that the determination of Πj as real operators is the only point of the proof which is not possibleto obtain, whenever the eigenvalues of type (4.23) are considered.
24
The first two equalities are true by construction. Let us remark that the vector J ~dj is parallel
to ~ej, because both belong to span(~vj , ~vj+n) and ~ej · ~dj = ~dj · J ~dj = 0; therefore, the equations
~ej · ~ej = 1 and ~ej · J ~dj = 1 imply that ~dj · ~dj = 1.Point (iii) of the thesis is trivially true by construction. Q.E.D.
Recalling the procedure to construct the basis used in the proof of Lemma A.1 and recallingpoints (i) and (ii) of Proposition 4.1, one immediately obtains the following
Corollary A.1 Let the basis {~e∗1, . . . , ~e∗n, ~d∗1, . . . , ~d∗n} be such that it satisfies (i), (ii) and (iii)
of Proposition 4.1. Any other basis {~e1, . . . , ~en, ~d1, . . . , ~dn} satisfies (i), (ii) and (iii) if and onlyif there exist θ1 ∈ [0, 2π] , . . . , θn ∈ [0, 2π], such that
~ej = cos θj~e∗j + sin θj
~d∗j and ~dj = − sin θj~e∗j + cos θj
~d∗j for j = 1 , . . . , n . (A.3)
In other words, such bases are uniquely determined up to n rotations in the subspaces span(~e∗j , ~d∗j ).
We are now ready to complete the
Proof of Proposition 4.1. Let {~e∗1, . . . , ~e∗n, ~d∗1, . . . , ~d∗n} be a basis constructed as describedin the proof of Lemma A.1. We will find another basis which satisfies points (i)–(iii) by usingCorollary A.1 and by determining the values of the rotation angles so to fit with point (iv).Let the basis {~e1, . . . , ~en, ~d1, . . . , ~dn} and the angles θ1 , . . . , θn be as in Corollary A.1. Let usintroduce the 2 × 2–D matrix A∗
j and the values of the real coefficients α∗j , β
∗j , γ∗j and δ∗j , as
follows:
A∗j =
(
α∗j + δ∗j γ∗j + β∗j
γ∗j − β∗j α∗j − δ∗j
)
=
(
~e∗j · V~e∗j ~e∗j · V ~d∗j~d∗j · V~e∗j ~d∗j · V ~d∗j
)
, j = 1 , . . . , n ;
similarly, for what concerns Aj, αj , βj , γj and δj we have:
Aj =
(
αj + δj γj + βj
γj − βj αj − δj
)
=
(
~ej · V~ej ~ej · V ~dj
~dj · V~ej ~dj · V ~dj
)
, j = 1 , . . . , n .
If we denote by R(θ) the 2× 2–D rotation matrix of angle θ, the equations in (A.3) imply thatAj = R(−θj)A
∗jR(θj); by straightforward calculations, one immediately gets the relations
αj = α∗j , βj = β∗j , γj = γ∗j cos(2θj) + δ∗j sin(2θj) , δj = δ∗j cos(2θj)− γ∗j sin(2θj) . (A.4)
Finally, if δj 6= 0 or γj 6= 0 , there are just two values θj ∈ [0, π] and θj + π such that δj = 0
and γj ≥ 0 , j = 1, . . . , n, so that ~ej · V ~dj + ~dj · V~ej ≥ 0 and ~ej · V~ej = ~dj · V ~dj .In order to complete the proof of point (iv), let us denote by B the 2n × 2n–D matrix whosecolumns are the vectors ~e1, . . . , ~en, ~d1, . . . , ~dn. One immediately gets the relations BTJB =BJBT = VTJV = J . Starting from (A.2) when i = j, we can write(
0 1−1 0
)
=
(
~ej · J~ej ~ej · J ~dj
~dj · J~ej ~dj · J ~dj
)
=
(
(BTV~ej) · JBTV~ej (BTV~ej) · JBTV ~dj
(BTV ~dj) · JBTV~ej (BTV ~dj) · JBTV ~dj
)
. (A.5)
Let us now recall that V~ej ∈ span(~ej , ~dj) and V ~dj ∈ span(~ej , ~dj); then, the vectors BTV~ej and
BTV ~dj have just the j–th and the j+n–th component different from zero. Thus, we can rewrite(A.5) as
(
0 1−1 0
)
=
(
~ej · V~ej ~dj · V~ej~ej · V ~dj
~dj · V ~dj
)(
0 1−1 0
)(
~ej · V~ej ~ej · V ~dj
~dj · V~ej ~dj · V ~dj
)
,
25
which is satisfied if and only if the determinant of the last matrix is equal to 1. This concludesthe proof of point (iv).Point (v) is a trivial consequence of Corollary A.1 and of the fact that there are only two valuesθj ∈ [0, π] and θj + π such that in (A.4) δj = 0 and γj ≥ 0 , ∀ j = 1, . . . , n. Q.E.D.
Let us now proceed to the
Proof of Proposition 4.2. Since {~e1, . . . , ~en, ~d1, . . . , ~dn} is a basis, the matrix Q is uniquelydetermined by (i) and (ii). Let us prove that this implies VTQV = Q. To this end, it is enoughto prove that for i, j = 1, . . . , n,
~ei ·Q~ej = ~ei · VTQV~ej , ~ei ·Q~dj = ~ei · VTQV ~dj , ~di ·Q~dj = ~di · VTQV ~dj .
After having recalled that V~ej ∈ span(~ej , ~dj) and V ~dj ∈ span(~ej , ~dj), the property (i) allowsto deduce that both the left and right hand sides of the previous equations are equal to zero,when i 6= j. For i = j we have just to show that
(
aj cj − bjcj + bj aj
)(
bj − cj 00 bj + cj
)(
aj cj + bjcj − bj aj
)
=
(
bj − cj 00 bj + cj
)
;
this can be directly verified, after recalling that a2j + b2j − c2j = 1 in view of point (iv) of
Proposition 4.1. As a consequence of points (i)–(iii) of Proposition 4.1 and of the definition ofthe coefficients aj, bj and cj , the eigenvalues are given by the expressions
λj = aj −√
a2j − 1 and λj+n = aj +
√
a2j − 1 .
Using the latter formula and the relation a2j + b2j − c2j = 1, one can easily verify (iii). Q.E.D.
B A method to determine periodic orbits
The search for the periodic orbits associated to the symplectic mapping Mε , b in (1.2) can takeadvantage of the symmetry properties of the map. The essential remark is that Mε , b can beexpressed as the product of two involutions, analogously to what occurs for the 2–D standardmap. Adapting the technique used in appendix A of [12] to the mapping in (1.2), one can easilyprove the following
Lemma B.1 Let us consider an orbit {(x(j)1 , x
(j)2 , y
(j)1 , y
(j)2 )}j≥0 induced by the mapping Mε , b.
Assuming that x(0)1 = x
(0)2 = x
(r)1 = x
(r)2 = 0, the orbit is periodic of period 2r, i.e.
M2rε , b(x
(0)1 , x
(0)2 , y
(0)1 , y
(0)2 ) = (x
(0)1 , x
(0)2 , y
(0)1 , y
(0)2 ) .
When looking for a periodic orbit with rotation vector (p/r , q/r), Lemma B.1 implies that one
must solve the system of two equations in two unknowns (precisely, y(0)1 and y
(0)2 ), given by
{
x(r)1 = 2pπ
x(r)2 = 2qπ
with (x(r)1 , x
(r)2 , y
(r)1 , y
(r)2 ) = Φr
ε , b(0, 0, y(0)1 , y
(0)2 ) , (B.1)
where Φrε , b : R4 7→ R4 is the lift of the mapping Mε , b. Let us remark that the solutions of the
equations (B.1) depend on ε (b is kept fixed in the following discussion); by abuse of notation,
26
the solution will be denoted as y(0)1 (ε) and y
(0)2 (ε). These functions are, in general, not well
defined, due to bifurcation phenomena of periodic orbits12; nevertheless there are usually somecontinuous branches of these functions for those values of ε, such that there exists a KAM torus
close to the periodic orbit. Hereafter the functions y(0)1 (ε) and y
(0)2 (ε) will denote one of such
continuous branches.
Numerical experiments suggest to use Newton’s method together with a sort of continuationprocedure in order to solve (B.1). More precisely, let us suppose that we want to solve (B.1)for a fixed value ε = ε∗; we summarize our numerical algorithm as follows.
(i) Equations (B.1) are solved by (y(0)1 , y
(0)2 ) = (2pπ/r, 2qπ/r) for ε = 0.
(ii) Define initially N = 1 and ∆ε = ε∗.
(iii) Consider the set {(y(0)1 (εj), y
(0)2 (εj))}N
j=1 of solutions of the equations in (B.1), corre-
sponding to the set of values {εj}Nj=1 of the parameter ε; then, extrapolate an initial guess
(υ1(εN + ∆ε), υ2(εN + ∆ε)) for the solution of (B.1) with ε = εN + ∆ε.(iv) Try to solve (B.1) with ε = εN + ∆ε, by using the Newton’s method with approximatesolution (υ1(εN + ∆ε), υ2(εN + ∆ε)).
(v) If the solution (y(0)1 (εN + ∆ε), y
(0)2 (εN + ∆ε)) can be found as in (iv) within an acceptable
numerical error, check whether the equation
(4pπ, 4qπ, y(0)1 (εN + ∆ε), y
(0)2 (εN + ∆ε)) = Φ2r
εN+∆ε, b(0, 0, y(0)1 (εN + ∆ε), y
(0)2 (εN + ∆ε)) (B.2)
holds true (within the numerical error). If one of the previous tests is not verified, then reduce∆ε, setting ∆ε = ∆ε/2 and return to point (iii).(vi) If ε∗ = εN +∆ε, we succeeded to find the periodic orbit, otherwise increase by 1 the valueof N and go back to point (iii).
Remark B.1 1) We found that the extrapolation at point (iii) allowed us to keep the steps∆ε large enough, whenever it was based on the two parabolas passing through the last three
pairs of points of the set {(y(0)1 (εj), y
(0)2 (εj))}N
j=1 for N ≥ 3; for numerical purposes, we usedtwo lines instead of the parabolas for N = 2, and two lines with constant ordinates for N = 1.
2) When the Newton method is applied as in (iv), the jacobian of the map giving (x(r)1 , x
(r)2 ) as
a function of (y(0)1 , y
(0)2 ) can be approximated by the 2× 2–D submatrix in the top-right corner
of V as given by (4.13).3) If the value of εN + ∆ε is not so large that the bifurcation phenomena occur, then equa-tion (B.2) is usually verified by replacing 4pπ, 4qπ and 2r by 2pπ, 2qπ and r , respectively.4) In order to prevent that our algorithm undergoes a loop, we stop the program at point (v),whenever the step ∆ε becomes lower than a fixed (small) threshold. Indeed, the branches of
y(0)1 (ε) and y
(0)2 (ε) cannot be infinitely prolonged by using this method: at some upper threshold
on ε the periodic orbit gets lost in a chaotic region.
12A periodic orbit may bifurcates when, by increasing parameter ε, the solutions of (B.1) do not satisfy both
y(0)1 = y
(r)1 and y
(0)2 = y
(r)2 . Thanks to lemma B.1, one can easily verify that there are at least two different
solutions of (B.1), which originate two periodic orbits with the same frequencies. The orbits described in footnoten. 4 provide two examples of bifurcating periodic orbits.
27
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30
Figure 3: Frequency analysis of the symplectic mapping Mε , 0.4 in a neighborhood of the fre-quency vector ~ωu. a) a regular grid of initial conditions. b) Values of the frequencies maximizingthe norm of the function (5.25) for the orbits induced by Mε , 0.4 with ε = 0.3125 and startingfrom the initial conditions reported in Figure a. c) Another regular grid of initial conditions.d) Same as Figure b with ε = 0.325 for the initial conditions reported in Figure c. Every pairof corresponding points in Figures a–b (c–d, respectively) is marked with the same symbol,according to the size of the frequency’s diffusion along the orbit. The scale of the frequency’sdiffusion is reported on the top. See the text for more details.
31
-0.7-0.6-0.5-0.4-0.3-0.2-0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7eps_Re
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
eps_Im
a)
-0.7-0.6-0.5-0.4-0.3-0.2-0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7eps_Re
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
eps_Im
b)
-0.4 -0.3 -0.2 -0.1 0.1 0.2 0.3 0.4eps_Re
-0.4
-0.3
-0.2
-0.1
0.1
0.2
0.3
0.4
eps_Im
c)
-0.4 -0.3 -0.2 -0.1 0.1 0.2 0.3 0.4eps_Re
-0.4
-0.3
-0.2
-0.1
0.1
0.2
0.3
0.4
eps_Im
d)
Figure 4: Pade results of order [52/52] for ~ωc = [2(s − 1) − 1/(24s), s]; a) first component
(u1) with b = 0.0001, ρ(1)P = 0.56, ε
(1)P = 0.59; b) second component (u2) with b = 0.0001,
ρ(2)P = 0.56, ε
(2)P = 0.56; c) first component (u1) with b = 0.01, ρ
(1)P = 0.36, ε
(1)P = 0.36; d)
second component (u2) with b = 0.01, ρ(2)P = 0.35, ε
(2)P = 0.35.
32
Figure 5: Breakdown threshold εc (given by Greene’s method) as a function of the couplingparameter b . The four different plots refer to the frequency vectors ~ωu , ~ωa , ~ωc and ~ωt , asdefined in (2.5).
33