Physica D 224 (2006) 90–101www.elsevier.com/locate/physd

From synchronization to Lyapunov exponents and back

Antonio Politia,∗, Francesco Ginellib, Serhiy Yanchukc,d,e, Yuri Maistrenkoc,f

a CNR - Istituto dei Sistemi Complessi, Via Madonna del Piano, 10, I-50019 Sesto Fiorentino, Italyb CEA – Service de Physique de l’Etat Condense, Centre d’Etudes de Saclay, 91191 Gif-sur-Yvette, France

c Institute of Mathematics, National Academy of Sciences of Ukraine, 01601 Kyiv, Ukrained Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germanye Institute of Mathematics, Humboldt University of Berlin, Unter den Linden 6, 10099 Berlin, Germany

f Institute of Medicine and Virtual Institute of Neuromodulation, Research Centre Julich, 52425 Julich, Germany

Available online 7 November 2006

Abstract

The goal of this paper is twofold. In the first part we discuss a general approach to determine Lyapunov exponents from ensemble rather thantime averages. The approach passes through the identification of locally stable and unstable manifolds (the Lyapunov vectors), thereby revealingan analogy with generalized synchronization. The method is then applied to a periodically forced chaotic oscillator to show that the modulus ofthe Lyapunov exponent associated to the phase dynamics increases quadratically with the coupling strength and it is therefore different from zeroalready below the onset of phase synchronization. The analytical calculations are carried out for a model, the generalized special flow, that weconstruct as a simplified version of the periodically forced Rossler oscillator.c© 2006 Elsevier B.V. All rights reserved.

Keywords: Synchronization; Lyapunov exponents

1. Introduction

As soon as synchronization phenomena in chaotic systemshave been discovered [1,2], the standard tools of nonlineardynamics have been implemented in order to clarify thisphenomenon. This is particularly true for the Lyapunovexponents (LEs), [3] because they measure the degree ofstability along different directions and are thus the naturalcandidates to quantify the degree of synchronization of differentregimes. However, several subtleties have been immediatelydiscovered. For instance, the negativity of the “transversal”LE is only a necessary condition for the stability of completesynchronization: (i) in low-dimensional systems, fluctuationsof the finite-time LEs may render the synchronized regimeunstable even when the “average” exponent is negative [4];(ii) in high dimensional systems, it has been ascertained thatthe propagation of finite-amplitude perturbations can sustain anunsynchronized regime, in spite of its linear stability [5,6]. A

∗ Corresponding author.E-mail address: antonio.politi@isc.cnr.it (A. Politi).

0167-2789/$ - see front matter c© 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.physd.2006.09.032

still open problem concerns the behaviour of the LEs in thecontext of the transition to a phase synchronized regime [7,9]even in the simple case of coherent chaotic oscillators. In somecases, the LE associated with the phase dynamics seems to bea proper order parameter, since it is numerically found to bezero below the transition, while it becomes negative beyond thetransition point [7]. However, negative values of the LE havealso been found both slightly before the transition [8,10] and infinite windows of the coupling parameter below the transitionitself [11]. Altogether, there is anyway the common beliefthat there exist a critical coupling strength beyond which theLE becomes strictly negative[10]. One exception is Ref. [12],which however refers to a simplified model (special flow — seebelow).

It is therefore important to clarify analytically the stabilityof the dynamics along the “phase” direction: in the absenceof coupling, this is a marginally stable direction and it is thusnatural to expect some difficulties. Here, we develop a methodthat allows concluding that the LE corresponding to the phasedynamics is different from zero (and possibly positive) as soonas the coupling is switched on. Therefore, the switching of a

A. Politi et al. / Physica D 224 (2006) 90–101 91

zero LE to negative values cannot be related with the onset ofphase synchronization.

One of the main problems is the lack of analytical methodsfor determining even perturbatively the LEs. Some ideashave been put forward for the maximum exponent [13,14],because almost any initial condition eventually grows withthe maximum rate and no special care is required to tune thedirection of the perturbation. However, very little is knownfor the other exponents, starting already from the second one.This is precisely what is needed to determine the stabilityof phase dynamics in the simplest system exhibiting phasesynchronization, i.e. in a periodically forced chaotic attractor,where the first LE accounts for the overall instability of thechaotic dynamics. Here, we attack and solve the problem bydeveloping a formalism to determine LEs as ensemble ratherthan time averages. Similar ideas have been already discussedby Ershov and Potapov [15], although they have not gonemuch beyond the level of formal statements. In fact, theirmethod relies on the determination of the growth rates ofhypervolumes of increasing dimension. While this idea provedvery effective for the development of a powerful algorithm tocompute the LEs [16], its ensemble-average extension has somelimitations due to the difficulty of disentangling the variousexponents. The advantage of our approach is that we are able toassociate each non-degenerate LE to a field of local directions,the Lyapunov vectors (LVs). Roughly speaking, the i th LV isdetermined into two steps: the first one consists in iteratingforward in time a hypervolume of dimension i in tangentspace to identify the local orientation of the most expandingi directions (this is also considered in [15]); the second stepconsists in iterating backward a vector lying within such ahypervolume. As a result, a coordinate-independent LV canbe determined: the LE is finally obtained by averaging thecorresponding instantaneous expansion/contraction rate overthe entire phase space, according to the invariant measure.

An objective identification of LVs is particularly interestingin the study of the hydrodynamic behaviour of extendedsystems. In the last years, mostly as a consequence of thepioneering work of Posch and collaborators [17,18], it hasbeen discovered that in models of fluids (more in generalin Hamiltonian systems) the directions corresponding to thesmallest (in absolute value) LEs almost coincide with long-wavelength Fourier modes. This observation has in turnsuggested that the Lyapunov analysis naturally leads to ahydrodynamic description without the need of introducing asuitable coarse graining. However, progress has been hinderedby the lack of an absolute definition of the Lyapunov “modes”that have been mostly identified with the vectors arising fromthe implementation of the Gram–Schmidt orthogonalizationprocedure during a standard computation of the LEs. The onlyexamples of a philosophy similar to that one outlined in thepresent paper concern the chronotopic Lyapunov approach [19]and the characterization of space–time chaos [20].

It is also interesting to notice that the problem of identifyingthe LVs is itself equivalent to a problem of (generalized)synchronization. In fact, the Lyapunov vectors are determinedby integrating a skew-product system composed of the original

nonlinear dynamics plus the “forced” evolution in tangentspace. As a result, the direction of the LVs varies in a possiblysingular way with the position in real space. However, thisdifficulty does not hinder the LE determination, which resultsfrom an average that is substantially insensitive to the presenceof local singularities.

An analytic investigation of the stability of phase dynamicsin a generic setup is an extremely difficult task because ofthe lack of structural stability of low-dimensional chaos. Forthis reason, it is convenient to consider suitable simplifiedmodels. The simplest system where phase synchronization hasbeen investigated is the so-called special flow [21]. This isbasically a skew-product system, where the phase dynamics isforced by the chaotic amplitude dynamics. In this system, it ispossible to establish analytically a certain number of results,because the phase evolution is basically unidimensional andthere is no need to deal with the problem of identifying thedirection of perturbations. In order to perform a more realisticanalysis of phase synchronization, a suitable coupling betweenphase and amplitude dynamics has been added to the specialflow [22]. Here, setting up a perturbative approach for theweakly forced Rossler oscillator, we show that the structure ofthe model proposed in Ref. [22] is quite similar to that oneexpected in generic chaotic systems, whenever the presenceof strong dissipations allows eliminating the stable directions.Furthermore, in order to simplify the analytic treatment ofthe LEs, we focus our attention on a model that we callthe generalized special flow (GSF), very similar to that oneanalyzed in Ref. [22] but characterized by a finite Markovpartition. As a result, we find that the modulus of the second LEexponent increases quadratically with the coupling strength andits corresponding smallness justifies the claims often found inthe literature that the second LE is equal to zero below the onsetof phase synchronization. In other words, we conclude that theLE is not the right order parameter to describe this transition.

More precisely, this paper is organized as follows. Inthe next section we introduce a general approach for thedetermination of Lyapunov exponents through an average overthe invariant measure. In Section 3 we present our case-studymodel, the GSF, deriving it as a discrete-time approximationof a periodically forced Rossler system. In Sections 4 and5 we illustrate the perturbation expansion for the secondLE, the corresponding LV and the invariant measure. Finallysome numerical results are presented in Section 6, along withconclusions.

2. A general approach for the determination of Lyapunovvectors and Lyapunov exponents

In this section we discuss a method to determine Lyapunovexponents from suitable ensemble averages. It is easy to writedown a formal meaningful definition, but the problem lies intranslating it into a workable procedure. With reference to anN -dimensional discrete-time system, described by the mappingrule

xt+1 = fd(xt ) x ∈ RN , (1)

92 A. Politi et al. / Physica D 224 (2006) 90–101

one can express the i th LE (as usual, LE are supposed to beordered from the largest to the smallest one) as

λ(i) =12

∫dxP(x) ln

[‖∂x fdV(i)(x)‖2

‖V(i)(x)‖2

](2)

where P(x) is the corresponding invariant measure, ∂x fd isthe Jacobian of the transformation, and the Lyapunov vectorV(i)(x) identifies the i th most expanding direction in x.

With reference to a continuous-time system, ruled by theordinary differential equation

x = fc(x) x ∈ RN . (3)

The i th LE is written as

λ(i) =

∫dxP(x)

[∂x fcV(i)(x)] • V(i)(x)

‖V(i)(x)‖2(4)

where • denotes the scalar product.Unless a clear procedure to determine the LV is given, Eqs.

(2) and (4) are nothing but formal statements. As anticipatedin the introduction, V(i)(x) can be obtained by following atwo-step procedure. We start with a generic set of i linearlyindependent vectors lying in the tangent space and let themevolve in time. This is the standard procedure to determineLEs, and it is well known that the hypervolume Y(i) identifiedby such vectors contains, for large enough times, the imost expanding directions. Furthermore, with reference to theset of orthogonal coordinates obtained by implementing theGram–Schmidt procedure, the component vk of a generic vectorv evolves according to the following differential equation (forthe sake of simplicity, we refer to continuous-time systems),

vk =

i∑j=k

σk, j (x)v j 1 ≤ k ≤ i (5)

where, as shown in Ref. [15], σk, j does not explicitly dependon time, but only through the position x in the phase space. Asa result, the i th Lyapunov exponent can be formally expressedas the ensemble average of the local expansion rate σi,i , i.e.,

λ(i) =

∫dxP(x)σi,i (x). (6)

By comparing with Eq. (4), one finds the obvious equality

σi,i =

[∂x fcV(i)(x)

]• V(i)(x)

‖V(i)(x)‖2. (7)

In Section 4, where this formalism is applied to a phase-synchronization problem, we find that the only workable wayto obtain an analytic expression for σi,i passes through thedetermination of the direction of the corresponding LV vectorV(i)(x).

Let us now consider the backward evolution of a genericvector V(i) ∈ Y(i). Its direction is identified by the (i − 1)-dimensional vector

u ≡ (u1, u2, . . . , ui−1) (8)

where uk = vk/vi . From the definition of u and from Eq.(5), one easily finds that the backward evolution follows theequation

uk = (σi,i − σk,k)uk −

i−1∑j=k+1

σk, j (t)u j − σk,i 1 ≤ k < i. (9)

This is a cascade of skew-product linear stable equations (theyare stable because the Lyapunov exponents are organized indescending order). The overall stability is basically determinedby the smallest (σk,k − σi,i ) that is obtained for k = i − 1. Itis, therefore, sufficient to turn our attention to the last (i − 1)component of the vector V. Its equation has the followingstructure

u(t) = γ u + σ(t) (10)

where γ = λi − λi−1 < 0 and we have dropped the subscripti for simplicity. The value of the direction u is obtainedby integrating this equation. By neglecting the temporalfluctuations of γ (it is not difficult to include them, but this isnot important for our final goal), the formal solution of Eq. (10)reads

u(x(t)) =

∫ t

−∞

eγ (t−τ)σ(x) dτ. (11)

This equation does not simply tell us the value of u at time t ,but the value of u when the trajectory sits in x(t). It is in factimportant to investigate the dependence of u on x. We proceedby determining the deviation δ j u induced by a perturbation δx jof x along the j th direction,

δ j u =

∫ t

−∞

eγ (t−τ)δ jσ(τ) dτ (12)

where, assuming a smooth dependence of σ on x (see below fora further discussion of this point),

δ jσ(τ) ≈ σx (τ )δx j (τ ) = σx (τ )δx j (t)eλ j (t−τ) (13)

(notice that the dynamics is flowing backward). If the Lyapunovexponent λ j is negative, δ jσ(τ) decreases for τ → −∞ andthe integral over τ in Eq. (12) converges. As a result, δ j uis proportional to δx j , indicating that the direction of the LVis smooth along the j th direction. If λ j is positive, δ jσ(τ)

diverges, and below time t0 where

δx j (t)eλ j (t−t0) = 1 (14)

linearization breaks down. In this case, δσ (τ) for τ < t0 isbasically uncorrelated with its “initial value” δ jσ(t) and onecan estimate δ j u, by limiting the integral to the range [t0, t]

δ j u(t) = δx j (t)∫ t

t0dτe(λ j +γ )(t−τ)σx (τ ) (15)

where t0 is given by Eq. (14). By bounding σx with constantfunctions and thereby performing the integral in Eq. (15), wefinally obtain

δ j u(t) ≈ δx j (t)+ δx j (t)−γ /λ j . (16)

A. Politi et al. / Physica D 224 (2006) 90–101 93

The scaling behaviour is finally obtained as the smallest numberbetween 1 and −γ /λ j . If we now introduce the exponent η jto identify the scaling behaviour of the deviation of the LVdirection when the point of reference is moved along the j thdirection in phase space, the results are summarized in thefollowing way

η j =

1 for λ j ≤ −γ

−γ /λ j for λ j > −γ.(17)

The former case corresponds to a smooth behavior (thederivative is finite), while the latter one reveals a singularbehaviour that is the signature of a generalized synchronization.

Although most of the assumptions made to derive theabove equation are quite plausible (even though not rigorouslyproved), there is one point that needs to be more carefullychecked: the smoothness of σ(x). In the absence of a morecareful analysis of this point, we can only claim that theabove equation provides an upper bound to the true range ofsmoothness for the LV direction.

3. From the periodically forced Rossler system to thegeneralized special flow

The first model where phase synchronization has beenexplored is the forced Rossler oscillator [7]. In this sectionwe derive a discrete-time mapping describing a forced Rosslersystem in the limit of weak coupling. We obtain what wecall the Generalized Special Flow (GSF), because it extendsa mapping previously introduced to characterize the onset ofphase synchronization [21].

The starting set of ordinary differential equations is

x = −y − z + εy cos(Ω t + ψ0)

y = x + a0 y − εx sin(Ω t + ψ0)

z = a1 + z(x − a2)

(18)

where ψ0 fixes the phase of the forcing term at time 0.The parameters a0, a1 and a2 will be chosen to keep theRossler system in the phase coherent regime, where the chaoticdynamics is rather simple and is characterized by weaklynonisochronous rotations in the (x, y) subspace. FollowingRef. [8], it is therefore convenient to introduce cylindricalcoordinates, namely u = (φ, r, z), (x = r cosφ, y =

r sinφ). For the future sake of clarity, let us denote with Scthe 3-dimensional space parametrized by such coordinates. Thedifferential equation (18) writes as

u = F(u)+ εG(u,Ω t + ψ0) (19)

where

F =

[1 +

z

rsinφ +

a0

2sin 2φ, a0r sin2 φ − z cosφ,

a1 + z(r cosφ − a2)]

(20)

G =

[− sin2 φ cos(Ω t + ψ0)− cos2 φ sin(Ω t + ψ0),

r√

2sin 2φ cos(Ω t + ψ0 + π/4), 0

].

Note that system (19) can be written in the equivalentautonomous form

u = F(u)+ εG(u, ψ), ψ = Ω ,

where ψ denotes the phase of the forcing term.We pass to a discrete-time description by monitoring the

system each time the phase φ is a multiple of 2π . In the newframework, the relevant variables are r , z, and ψ , all measuredwhen the Poincare section is crossed. The task is to determinethe transformation mapping the state (r, z, ψ) onto (r ′, z′, ψ ′).

In order to obtain the expression of the map, it is necessary toformally integrate the equations of motion from one section tothe next. This can be done by expanding around the unperturbedsolution for ε = 0 (which must nevertheless be obtainednumerically). The task is anyhow worth it, because it allowsdetermining the structure of the resulting map, which turns outto be (see Appendix A)

ψ ′= ψ + 〈T (0)〉Ω + A1 + ε(Bc

1 cosψ + Bs1 sinψ)

r ′= A2 + ε

(Bc

2 cosψ + Bs2 sinψ

)z′

= A3 + ε(Bc3 cosψ + Bs

3 sinψ)

(21)

where 〈T (0)〉 is the average period of the unperturbed Rossleroscillator and Am’s and Bm’s are functions of z and r . As isshown in Appendix A, they can be numerically determinedby integrating the appropriate set of equations. Up to firstorder in ε, the structure of the model is fairly general as it isobtained for a generic periodically forced oscillator representedin cylindrical coordinates (as long the phase of the attractor canbe unambiguously identified).

For the usual parameter values, the Rossler attractor ischaracterized by a strong contraction along one direction [23].As a result, one can neglect the z dependence since this variableis basically a function of r , and thus write

ψ ′= ψ + 〈T (0)〉Ω + A1(r)+ ε(Bc

1(r) cosψ + Bs1(r) sinψ)

r ′= A2(r)+ ε(Bc

2(r) cosψ + Bs2(r) sinψ) (22)

where all the functions can be obtained by integratingnumerically the equations of motion of the single Rossleroscillator.1

To simplify further manipulations, we finally recast Eq. (22)in the form

ψ ′= ψ + K + A1(r)+ εg1(r) cos(ψ + β1(r))

r ′= A2(r)+ ε g2(r) cos(ψ + β2(r)) (23)

where

Bci (r) = gi (r) cosβi (r)

Bsi (r) = −gi (r) sinβi (r) (24)

for i = 1, 2. The parameter K = 〈T (0)〉Ω − 2π representsthe detuning between the original Rossler-system averagefrequency and the forcing frequency Ω .

1 Strictly speaking, A and B functions in (21) and (22) are different (seeAppendix A). We use the same notations here to simplify the presentation.

94 A. Politi et al. / Physica D 224 (2006) 90–101

Fig. 1. Numerically computed functions Ai , gi and βi (i = 1, 2) for the Rossleroscillator. Rossler parameters have been chosen as in Ref. [21]: a0 = 0.2,a1 = 1 and a2 = 9.

The correctness of the scheme is confirmed in Fig. 1, whereall the functions defining the model have been numericallyobtained. The very fact that they all look as one-dimensionalcurves, confirms the conjecture that z-dependence can beneglected.

The GSF (23) generalizes the model introduced in Ref. [21],where the effect of the phase on the r dynamics was notincluded. This implies that the GSF loses the skew-productstructure. This has important consequences on the orientationof the second Lyapunov vector that we determine in the nextsections. Notice also that the GSF (23) generalizes and justifiesthe model invoked in Ref. [22].

In spite of the simplification introduced by removing the zvariable, a rigorous treatment of Eq. (23) for generic functionsg and β is still very difficult. A first obstacle may be the lack of afinite Markov partition for the unperturbed system, which doesnot allow us expressing the second order correction to the LV ina closed form (see Appendix B for details). A second obstacleis that the perturbation itself may and will in general destroythe Markov partition, making the invariant measure hardlyaccessible to a perturbative expansion. For both reasons, werestrict ourselves to considering specific A, g, and β functionswhich guarantee the existence of a finite Markov partitions in afinite range of the coupling constant. In the last section we shallcomment on the possibility to extend our formalism to a moregeneral setup.

For the sake of simplicity, we have decided to analyse thefollowing model,

r ′= f (r)+ 2εcg(r) cos(ψ + α)

ψ ′= ψ + K + ∆r + εb cosψ (25)

where

f (r) = 1 − 2|r |, g(r) = r2− |r | (26)

with r ∈ [−1, 1]. The tent-map choice for r ensures that [−1, 0]

and [0, 1] are the two atoms of a Markov partition. Moreover,since g(r) is equal to 0 for r = 0 and r = ±1, this remainstrue also when the perturbation is switched on. This is a keyproperty that is necessary to perform a completely analyticaltreatment in the following sections.

In this two-dimensional setup, the formal expression of thei th LE (2) writes

λ(i) =12

∫ 1

−1dr

∫ 2π

0dψP(r, ψ) ln

[‖J(r, ψ)V(i)(r, ψ)‖2

‖V(i)(r, ψ)‖2

](27)

and the Jacobian is

J(r, ψ)

=

(fr (r)+ 2εcgr (r) cos(ψ + α) −2εcg(r) sin(ψ + α)

∆ 1 − εb sinψ

)(28)

where the subscript r denotes the derivative with respect to r .The computation of the Lyapunov exponent therefore, requiresdetermining both the invariant measure P(r, ψ) and the localdirection of the Lyapunov vector V(i).

4. A perturbative calculation of the second Lyapunovexponent

In this section we derive a perturbative expression for thesecond LE of the GSF (25), by expanding Eq. (27). One ofthe key ingredients is the second LV, whose direction can beidentified by writing V = (V, 1) (for the sake of clarity, fromnow on, we omit the superscript i = 2 in V and λ, as we shallrefer only to the second direction). Due to the skew-productstructure of the unperturbed map (25), the second LV is, forε = 0, aligned along the ψ direction (i.e. V = 0). It is thereforenatural to expand V in powers of ε

V ≈ εv1(r, ψ)+ ε2v2(r, ψ). (29)

Accordingly, the logarithm of the norm of V is

ln ‖V‖2

= ln(1 + ε2v21) = ε2v2

1 (30)

while its forward iterate writes as (including only those termsthat contribute up to second order in the norm),

JV =

(ε fr (r)v1 − 2cεg(r) sin(ψ + α)

1 + ε(∆v1 − b sinψ)+ ε2∆v2

). (31)

Notice that we have omitted the (r, ψ) dependence of v1 andv2 to keep the notation compact.

A. Politi et al. / Physica D 224 (2006) 90–101 95

The Euclidean norm of the forward iterate is

‖JV‖2

= 1 + 2ε(∆v1 − b sinψ)+ ε2(∆v1 − b sinψ)2

+ 2∆v2 + [ fr (r)v1 − 2cg(r) sin(ψ + α)]2 (32)

and its logarithm is

ln ‖JV‖2

= 2ε(∆v1 − b sinψ)− ε2(∆v1 − b sinψ)2

− 2∆v2 − [ fr (r)v1 − 2cg(r) sin(ψ + α)]2. (33)

We now proceed by formally expanding the invariant measurein powers of ε

P(r, ψ) ≈ p0(r)+ εp1(r, ψ)+ ε2 p2(r, ψ). (34)

The determination of the pi coefficients is presented in thenext section, but here we anticipate that, as a consequenceof the skew-product structure for ε = 0, the zeroth-ordercomponent of the invariant measure does not depend on thephase ψ . Moreover, because of the structure of the tent-map, p0

is also independent of r , i.e. p0 = 1/4π . The second Lyapunovexponent can thus be written as

λ =

∫ 1

−1dr

∫ 2π

0dψ

(1

4π+ εp1(r, ψ)

)×2ε(∆v1(r, ψ)− b sinψ)− ε2

[(∆v1(r, ψ)− b sinψ)2

− 2∆v2(r, ψ)+ [ fr (r)v1(r, ψ)+ 2cg(r) sin(ψ + α)]2

+ v21(r, ψ)] + o(ε2). (35)

As the variable ψ is a phase, it is not a surprise thatsome simplifications can be found by expanding the relevantfunctions into Fourier components. We start writing the firstcomponent of the invariant measure as

p1(r, ψ) =1

2π

∑n

qn(r)einψ . (36)

We then turn our attention to the first order component v1(r, ψ)of the second LV (29). Due to the sinusoidal character of theforcing term in the GSF (25), it is easy to verify (see the nextsection) that v1(r, ψ) contains just the first Fourier component,

v1(r, ψ) = c[L(r) sin(ψ + α)+ R(r) cos(ψ + α)]. (37)

By now, inserting Eqs. (36) and (37) into Eq. (35) andperforming the integration over ψ , we obtain

λ = ε2∫ 1

−1dr

∆c[qr

1[L(r) sinα + R(r) cosα]

− q i1[L(r) cosα − R(r) sinα]]

+ bq i1 −

b2

8+ ∆

bc

4[L(r) cosα − R(r) sinα]

+c2

8(3 − ∆2)[L2(r)+ R2(r)]

+c2

2g2(r)+ c2 |r |

rg(r)L(r)

+

∆I2

4π(38)

where we have further decomposed q1(r) in its real andimaginary parts

q1(r) = qr1(r)+ iq i

1(r) (39)

and we have defined

I2 :=

∫ 1

−1dr

∫ 2π

0dψ v2(r, ψ), (40)

which accounts for the contribution arising from the secondorder correction to the LV. This expansion shows that thelowest-order contribution to the second Lyapunov exponent ofthe GSF scales quadratically with the perturbation amplitude.This is indeed a general result that does not depend on theparticular choice of the functions used to define the GSF, butonly on the skew-product structure of the unperturbed timeevolution and on the validity of the expansion assumed in(34) (we shall comment on this last issue in the next section).Moreover, tedious but straightforward calculations show thatthe first order contribution to the second LE vanishes for anygeneric periodic forcing function, containing higher harmonicsin ψ but no zeroth order terms.

By inserting the expression for I2 obtained in Appendix B(see Eq. (B.11)) in Eq. (38), we finally obtain the perturbativeexpression for the second LE,

λ = ε2

c2

30−

b2

4+

∫ 1

−1dr

[bq i

1(r)

+c2

16(6 − ∆2)[L2(r)+ R2(r)]

+∆cqr1(r)[L(r) sinα + R(r) cosα]

+∆c

(b

4− q i

1(r)

)[L(r) cosα − R(r) sinα]

+ c2 |r |

rg(r)L(r)+

∆c2

4r sin

(∆(1 − r)

2

)

× [L(r) cos K − R(r) sin K ]

] . (41)

Accordingly, the numerical value of the second LE canbe obtained by performing integrals which involve the fourfunctions qr

1(r), q i1(r), L(r), and R(r), that are determined in

the next section.

5. Determining the coefficients of the power expansion

After having more or less formally expanded the expressionof the second LE in powers of the coupling strength ε in theprevious section, now we show how the coefficients can bedetermined for both the invariant measure and the directionof the LV. Notice that the second part of the project passesthrough the implementation of the general ideas put forwardin Section 2.

5.1. The invariant measure

We start focusing our attention on the invariant measureP(r, ψ) which can be computed as a fixed point of the

96 A. Politi et al. / Physica D 224 (2006) 90–101

Frobenius–Perron equation

P ′(r ′, ψ ′) =P(r−, ψ−)

|det J(r−, ψ−)|+

P(r+, ψ+)

|det J(r+, ψ+)|(42)

where (r−, ψ−) and (r+, ψ+) are the two preimages of(r ′, ψ ′). It is important to notice that our choice of the mapguarantees that two solutions do exist in the whole rectangle[−1, 1] × [0, 2π ] in a finite range of ε-values. This will becrucial for obtained exact expressions. As has been shown inthe previous section, we are interested in solving the aboveequation up to first order. Accordingly, we write

P(r ′, ψ ′) = p0(r′, ψ ′)+

ε

2π

∑n

qn(r)einψ (43)

where we have expanded the first order contribution as inEq. (36). It is also necessary to expand the preimages

r±= r±

0 + εr±

1 (44)

ψ±= ψ±

0 + εψ±

1 (45)

where

r±

0 = ±1 − r ′

2(46)

ψ±

0 = ψ ′− K ∓ ∆

1 − r ′

2(47)

r±

1 = ∓c1 − r ′2

4cos(ψ±

0 + α) (48)

ψ±

1 = ±c∆1 − r ′2

4cos(ψ±

0 + α)− b cosψ±

0 . (49)

At zeroth order in ε, it is easy to see that the Frobenius–Perronequation (42) reduces to

p0(r′, ψ ′) =

12(p0(r

+

0 , ψ+

0 )+ p0(r−

0 , ψ−

0 )) (50)

whose solution is everywhere constant, as anticipated inSection 4. By imposing the normalization condition, oneobtains

p0 =1

4π. (51)

By then considering that

|det J(r, ψ)|−1=

12[1 + εb sinψ

+ εc sign r(∆g(r) sin(ψ + α)+ gr (r) cos(ψ + α))] (52)

and projecting Eq. (42) over its first Fourier component, wefinally obtain a closed equation for q1(r)

q1(r′) =

e−iK

2

[e−i∆r+

0

(q1(r

+

0 )− ib

4

+c

4eiαgr (r

+

0 )− ic

4eiα∆g(r+

0 )

)(53)

+ e−i∆r−

0

(q1(r

−

0 )− ib

4−

c

4eiαgr (r

−

0 )

Fig. 2. The first order contribution q1 to the probability density for theparameter values: ∆ = −0.18, α = −π/4, K = 0.04π . Solid and dashedlines refer to real and imaginary parts.

+ ic

4eiα∆g(r−

0 )

)]. (54)

The structure of this equation is very similar to aFrobenius–Perron equation for a one-dimensional system.The dimensionality reduction has been made possible byexploiting the skew-product structure of the unperturbedsystem. Considering also the simple expression of thepreimages of r ′ (they have to be determined at zeroth order),the above equation can be accurately solved by implementingthe standard method to solve a Frobenius–Perron equation (theonly limit being imposed by the numerical round-off).

In Fig. 2 we have plotted the real and imaginary parts of q1for three different choices of the parameters b and c. In all cases,one can see a very smooth dependence, which thus suggeststhe possibility to obtain accurate fully analytic expressions byexpanding polynomially q1(r). However, being more interestedin testing the overall validity of the perturbative approach, wedo not explore this possibility.

In fact, in order to test the general validity of the powerexpansion, we have numerically investigated three differentGSFs, corresponding to the following choices of the functionsf and g: (i) f (r) = 1 − 2|r |, g(r) = r2

− |r | as considered in(26); (ii) f (r) = 0.8 − 1.8|r | and g(r) = 1/2, for which thereis no finite Markov partition; (iii) f (r) = 2(1 − 2εc)(1 − |r |)

and g(r) = 1/2, for which the finite Markov partition existingin the unperturbed case is destroyed as soon as the perturbationis switched on.

In order to compare such models, we have computed thedeviation of the zero Fourier component of the invariantmeasure of the map (25) induced by a small finite coupling ε,

〈P(r, ψ)〉 =

∫ 1

−1dr

∫ 2π

0dψ[P(r, ψ)|ε − P(r, ψ)|ε=0]. (55)

As can be clearly seen in Fig. 3, in the first two cases thelinear term is even equal to zero, while relevant multiplicativelogarithmic corrections are present in the third case. This“pathological” behaviour is induced by the fact that as soon as

A. Politi et al. / Physica D 224 (2006) 90–101 97

Fig. 3. The deviation of the zero Fourier component of the invariant measure(see Eq. (55)) of map (25) as a function of ε. Three different choices of f and ghave been tested: (i) f (r) = 1 − 2|r | and g(r) = r2

− |r | (crosses), (ii) f (r) =

0.8 − 1.8|r | and g(r) = 1/2 (squares), and (iii) f (r) = 2(1 − 2εc)(1 − |r |)

and g(r) = 1/2 (diamonds). Parameter values have been fixed to ∆ = −0.18,α = −π/4, K = 0.04π , b = −0.6 and c = 2.8. Abscissas are divided by εto better emphasize deviations from linear scaling. The logarithmic deviationsdisplayed by the diamonds are emphasized in the inset.

the coupling is switched on, an infinite series of discontinuitiesin the invariant measure suddenly arises in the vicinity ofthe former fixed point r = −1. Although this latter specialcase, characterized by a non-vanishing zeroth harmonics inthe forcing term, bears no direct relationship with the Rosslersystem (18), it illustrates well the potential pitfalls of the powerexpansion (34) if the finite Markov partition for the unperturbeddynamics is destroyed by the forcing terms. It is also importantto notice that no peculiarity is found in the most genericsecond case, where no finite Markov partition characterizes thedynamics.

5.2. The direction of the second Lyapunov vector

In this subsection we derive a self-consistent equation for thesecond LV. We start from Eq. (31), retaining only the relevantperturbation terms

JV

=

([ fr (r)+ 2εcgr (r) cos(ψ + α)]V − 2εcg(r) sin(ψ + α)

∆V + 1 − εb sinψ

).

(56)

By computing the ratio between the components of JV weobtain the new value of the slope V ′

= εv′

1 + ε2v′

2 + · · ·,

v′1 + εv′2

=[ fr (r)+ 2εcgr (r) cos(ψ + α)] (v1 + εv2)− 2cg(r) sin(ψ + α)

∆εv1 + (1 − εb sinψ),

(57)

where we have again kept only the relevant terms (up to firstorder after dividing both sides by ε) and where v′

1 and v′

2 areboth functions of the iterates r ′ and ψ ′,

r ′= r ′

0 + 2εcg(r) cos(ψ + α) (58)

ψ ′= ψ ′

0 + εb cosψ (59)

Fig. 4. The functions L(r) and R(r) computed for ∆ = −0.18 and K = 0(solid curve) and K = −1 (dashed line).

where r ′

0 = f (r) and ψ ′

0 = ψ + K + ∆r are the unperturbediterates. By replacing the expressions for r ′ and ψ ′ in Eq. (57),at leading order, we obtain

v′

1( f (r), ψ + K + ∆r) = fr (r)v1(r, ψ)

− 2cg(r) sin(ψ + α). (60)

As anticipated in Section 2, this recursive relation can besolved by following backwards the dynamics of (r, ψ). It isworth stressing that the term 2cg(r) sin(ψ+α), i.e. the effect ofthe phase on the amplitude dynamics, acts as a source term inEq. (60). In its absence, the latter equation would yield a trivialvanishing solution for v1 (which in turn also implies v2 = 0, ascan be appreciated in Appendix B). It is therefore the feedbackof the phase on the amplitude dynamics that generates anontrivial orientation of the perturbed second Lyapunov vector.Furthermore, the structure of the source term 2cg(r) sin(ψ+α)

naturally suggests the Ansatz (37). By inserting it in Eq. (60) weobtain two recursive equations,

L(r) = sign(r)[

12(R( f (r)) sin(K + ∆r)

− L( f (r)) cos(K + ∆r))− g(r)

]R(r) = −

12

sign(r) [R( f (r)) cos(K + ∆r)

+ L( f (r)) sin(K + ∆r)] . (61)

This equation can be solved numerically, by considering it as arecursive relation to be iterated backward in time until the fixedpoint solution is eventually attained and the functions L and R,computed with the desired precision. In Fig. 4 we can see someexamples of how they look like.

From the analysis carried on in Section 2 and in particularfrom Eq. (17), we see that the condition for a smooth behaviourof the direction V along the phase direction is (noticing thathere, γ = λ2 −λ1) is λ1 > 2λ2 that is certainly verified and thisfully justifies the expansion in Fourier modes along such a di-rection. On the other hand, along the expanding direction r , thecondition writes λ2 < 0, that is only marginally verified. The

98 A. Politi et al. / Physica D 224 (2006) 90–101

Fig. 5. Second Lyapunov exponent for the Generalized Special Flow (25) as afunction of detuning K . The dashed line represent the analytical result, whiledots (with the bars indicating one standard deviation) results of direct numericalsimulations of the GSF. Parameter values have been fixed to: ∆ = −0.18,α = −π/4, b = −0.6, c = 2.8. Ordinates have been rescaled by a factor ε2 toevidentiate the second order coefficient. The inset magnifies a part of the largergraph.

apparent roughness exhibited by L(r) and R(r) can thereforebe a manifestation of the expected non-complete smoothness.It is, however, also important to stress the role played by thefunctions we have specifically considered in the GSF. In fact,the tent map induces a discontinuity in the tangent space thatpropagates everywhere, though significantly squeezed. Luckilyenough, as can be appreciated in Appendix B, such singularitiesare integrated out when determining the leading contribution tothe Lyapunov exponent which is therefore substantially insen-sitive and can be computed without much harm.

6. Numerical results and conclusions

In this paper we have introduced a novel approach todetermine analytically the Lyapunov exponent and applied itto the specific case of a periodically forced chaotic oscillatordescribed by a model (the generalized special flow) whichis also introduced here starting from the specific case of theRossler oscillator.

Given the many technical difficulties that are necessary toovercome in order to finally obtain the numerical value ofthe quadratic coefficient, it is wise to compare the analyticexpression with the direct numerical computation of the secondLE performed for small enough values of ε. In Fig. 5, thesecond order coefficient λ/ε2 is determined from the analyticexpression (41) and by directly simulating the GSF for ε-values in the range [10−4, 10−2

]. The good agreement obtainedfor all K values confirms the correctness of the analyticalcalculations. The relatively strong negative peak around K = 0is a manifestation of a resonance phenomenon. The LE tendsto be more negative when the forcing frequency is close to theaverage frequency of the chaotic attractor.

It is also important to stress that our results are validfor arbitrarily small ε, i.e. below the transition to phasesynchronization (if there is any) and therefore tells us that

Fig. 6. Second Lyapunov exponent for the Generalized Special Flow (25) asa function of detuning K . Ordinates have been rescaled by a factor ε2 toevidentiate the second order coefficient. Symbols and parameters values arethe same as in Fig. 5, except for b = 0.

the LE corresponding to the phase dynamics is immediatelydifferent from zero, as soon as the coupling is switched on.

Another important point concerns the sign of the LE: naiveconsiderations might suggest that the coupling tends to stabilizethe phase dynamics and thereby to give a negative LE. However,the left tail in Fig. 5 (see also the inset) definitely shows apositive exponent. It is desirable to find some simple heuristicarguments to understand when and why the phase dynamics isstable, but this does not seem to be an easy task and is left as anopen problem for future investigations.

The major difference between the GSF, we analyse in thismanuscript and the special flow introduced in [21] is the termproportional to c in the equation for r in Eq. (25). Such a termprevents the possibility of further dimension reductions andrequires setting up the machinery we have developed in thispaper. It is therefore interesting to quantify its direct effect onthe actual value of the LE. This can be simply done, by settingthe other coupling term b = 0, an assignment that is basicallycomplementary to what done in the standard special flow. Theresults, reported in Fig. 6, show a sort of “dispersive” behaviourfor the LE which also tends to be positive. This suggests thatthe back coupling of the phase dynamics onto the amplitudeevolution maybe responsible for an eventually positive LE.

While Eq. (41) cannot by any means capture the quantitativebehavior of the original Rossler system, still the quadraticbehaviour of the second LE seems to be a very generalfeature even though we can imagine that the lack of structuralstability of generic oscillators may mask the overall behaviourwith the presence of additional stability windows. We havetherefore computed directly the second Lyapunov exponent forthe periodically forced Rossler system choosing the same setof parameters (a0 = 0.2, a1 = 1 and a2 = 9) considered inRef. [21].

When the frequency of the periodic force is close to thenatural frequency of the oscillator, ν0 = 1.0158(1) for ourchoice of parameters, we are able to detect the quadraticbehavior with a good accuracy, as demonstrated in Fig. 7. Since

A. Politi et al. / Physica D 224 (2006) 90–101 99

Fig. 7. Second Lyapunov exponent for the periodically forced Rossler systemclose to the resonance. Full circles refer to a forcing frequency Ω = 1.007(the Rossler natural average frequency is ν0 = 1.0158(1)), while open squarescorrespond to Ω = 1. Rossler parameters have been fixed as a0 = 0.2, a1 = 1,a2 = 9, while the integration interval is about t = 108. The inset shows thequadratic relation between λ(2) and ε.

the coupling strengths we have reached are much below theonset of phase synchronization, as can bee seen in Ref. [21],these numerical results confirm our theoretical conclusions thatthe Lyapunov exponent corresponding to the phase dynamicsdeviates from zero as soon as the coupling is switched on. Itwould now be interesting to extend the analysis carried outin this paper to two coupled chaotic oscillators, perhaps byinvestigating suitable discrete-time models such as that oneintroduced in [24]. However, while one can presumably learnsomething on the phenomenon of phase synchronization, wedo not expect qualitative changes for the behaviour of theLyapunov spectrum. This is supported by the recently reportedquadratic growth of the eighth LE (the first one correspondingto phase dynamics) in a system of seven coupled Rossleroscillators [25].

Altogether, the numerical and analytical results presented inthis paper clearly show that the onset of phase synchronizationis not signalled by the second LE (or, more generally, theLE associated to the phase dynamics) turning negative, butit is rather associated to a change in the structure of thedynamical attractor [22,26] that is not directly related tothe sign of the “phase exponent”. On the other hand, thequadratic dependence on the coupling strength makes it difficultto numerically appreciate deviations from zero (especiallybecause of the statistical fluctuations that necessarily affectnumerical simulations) and explains why in earlier studies, theLE has been mistakenly regarded as a proper order parameterto characterize the transition to a phase-synchronized regime.

Another important point concerns the sign of the secondLyapunov exponent. In fact, it was formerly believed that phasechaos (i.e. a positive LE) can only occur in the presence ofa specific structure of the underlying chaotic attractor (seee.g. [27]). On the other hand, our analytical results show that thesecond LE can be positive even in a context where no peculiaramplitude evolution has to be invoked. However, our approach

does not give any physical insight about the expected sign of theLE. It will be certainly useful to find under which conditions achaotic phase dynamics may arise.

Finally another major achievement of this paper is thatLyapunov exponents can be effectively determined fromensemble averages, passing through the determination of thelocal direction of the corresponding Lyapunov vectors. From apurely numerical point of view, there is no conceptual difficultyin applying this method for a more detailed characterization ofhigh-dimensional chaos [20]. However, in the perspective ofobtaining more general analytical results, it is desirable to gobeyond systems with finite Markov partitions.

Acknowledgment

Part of this project has been carried out with the helpof financial support provided by the Collaborative ResearchNATO grant PST.CLG.979410.

Appendix A. From continuous to discrete time

In this appendix we present the detailed calculations relativeto the determination of the Poincare mapping (21) for theperiodically forced Rossler oscillator (18). Notice, however,that the methodology is quite general and is indeed applicableto a generic periodically perturbed system, as long as it can bewritten in the form (19).

Let us start by introducing some useful notations

U(r, z, ψ; t) ≡ (Φ(r, z, ψ; t), R(r, z, ψ; t), Z(r, z, ψ; t))

(A.1)

denotes the phase point in Sc at time t of a trajectory started in(0, r, z) at time 0 and with an initial phase of the forcing termequal to ψ (pay attention to the fact that the triple (r, z, ψ) ∈

Sd). The crossing time with the Poincare surface is determinedby imposing that the phase Φ has increased by 2π , i.e.,

Φ(r, z, ψ; T ) = 2π. (A.2)

As we are interested in the small coupling regime, we canexpand U in powers of ε and retain just the first order term,

U(t) = U(0)(t)+ εU(1)(t). (A.3)

In particular, from Eq. (A.2), we obtain

2π = Φ(0)(T )+ εΦ(1)(T ) = Φ(0)(T (0))

+ εΦ(1)(T (0))+ ε∂Φ(0)

∂t(T (0))T (1) (A.4)

where we have expanded T as well, assuming that T = T (0) +

εT (1). Since Φ(0)= 2π , we conclude that

T (1) = −Φ(1)(T (0))

f1(A.5)

where f1 =∂Φ(0)(T (0))

∂t is determined by the right-hand side of(19) with ε = 0, namely, it is the first component of F.

100 A. Politi et al. / Physica D 224 (2006) 90–101

It can be easily seen that the zeroth and first ordercomponents satisfy the differential equations

U(0)

= F(U(0)) (A.6)

U(1)

= DF(U(0))U(1) + G(U(0),Ω t + ψ) (A.7)

where DF denotes the Jacobian of the velocity field F and wehave introduced an explicit dependence on the phase ψ , as itchanges in going from one section to the next. The equation forthe first order correction can be formally solved,

U(1) =

∫ T (0)

0dτW(T (0), τ )G(U(0)(τ ),Ωτ + ψ) (A.8)

where W(t, τ ) is the matrix of fundamental solutions ofthe equation U = DF(U(0))U. Since G contains only firstharmonics in ψ , it can be decomposed into sine and cosinecomponents,

G(U(0),Ωτ + ψ) = Gc(U(0),Ωτ) cosψ

+ Gs(U(0),Ωτ) sinψ (A.9)

where

Gc(U(0),Ωτ)

=

− sin2 Φ(0) cos Ωτ − cos2 Φ(0) sin ΩτR(0) sin 2Φ(0) cos(Ωτ + π/4)/

√2

0

(A.10)

and

Gs(U(0),Ωτ)

=

sin2 Φ(0) sin Ωτ − cos2 Φ(0) cos Ωτ−R(0) sin 2Φ(0) sin(Ωτ + π/4)/

√2

0

. (A.11)

Accordingly, as follows from (A.8), U(1) can be decomposedin the same way

U(1) = Mc(r, z) cosψ + Ms(r, z) sinψ (A.12)

where

Mc(r, z) =

∫ T (0)

0dτW(T (0), τ )Gc(U(0)(τ ),Ωτ) (A.13)

and a similar equation holds for Ms(r, z).The first component of Eq. (A.12) gives Φ(1). After

substituting it into Eq. (A.5), we obtain,

T = T (0)(r, z)−ε

f1(Mc

1(r, z) cosψ + Ms1(r, z) sinψ) (A.14)

where the subscripts indicate once more the component of thevector. Accordingly, the new phase ψ ′ is

ψ ′= ψ + ΩT = ψ + ΩT (0)(r, z)

−εΩf1

[Mc1(r, z) cosψ + Ms

1(r, z) sinψ]. (A.15)

On the other hand, from the second and third components ofEq. (A.3) we obtain the new values r ′ and z′ by also expanding

the expression of T around T (0),

U(T ) = U(0)(T (0))+ εU(1)(T (0))+ εF(U(0))T (1). (A.16)

Straightforward but tedious calculations lead to

ψ ′= ψ + 〈T (0)〉Ω + A1 + ε(Bc

1 cosψ + Bs1 sinψ) (A.17)

r ′= A2 + ε(Bc

2 cosψ + Bs2 sinψ) (A.18)

z′= A3 + ε(Bc

3 cosψ + Bs3 sinψ) (A.19)

where 〈T (0)〉 is the average period of the unperturbed Rossleroscillator. The functions Ai can be determined by integratingthe unperturbed equations

A1(r, z) = [T (0)(r, z)− 〈T (0)〉]Ω

A2(r, z) = R(0)(r, z; T (0))

A3(r, z) = Z (0)(r, z; T (0))

(A.20)

while the functions Bci read as

Bc1(r, z) = −Ω Mc

1(r, z)/ f1

Bc2(r, z) = Mc

2(r, z)− f2 Mc1/ f1

Bc3(r, z) = Mc

3(r, z)− f3 Mc1/ f1

(A.21)

and similar equations hold for the sine components.

Appendix B. Second order contribution to the Lyapunovvector

In this appendix we derive a closed expression for the termI2, which accounts for the contribution to the LE arising fromsecond order corrections to the LV direction (see Eq. (40)). Tothis purpose, we have to consider all terms of order ε in Eq.(57), starting from v′

1(r′, ψ ′) ≡ v′

1(r′

0, ψ′

0)+ εδv′

1, with

δv′

1 = +bc[L(r ′

0) cos(ψ ′

0 + α)− R(r ′

0) sin(ψ ′

0 + α)]

cosψ

+ 2c2g(r)[Lr (r

′

0) sin(ψ ′

0 + α)

+ Rr (r′

0) cos(ψ ′

0 + α)]

cos(ψ + α) (B.1)

where r ′

0 = f (r) and ψ ′

0 = ψ + K + ∆r are the iterates of theunperturbed GSF as defined in Section 5. In the following weshall not care about the possible lack of differentiability alongthe direction r for two reasons: (i) we have verified that settingup a more accurate procedure leads to the same results, but itspresentation would be more cumbersome; (ii) the procedure isin itself correct, because in the end we are interested in theintegral that is insensitive to the presence of singularities.

The recursive equation for the second order term writes as

v2(r, ψ) =1

fr (r)v′

2(r′

0, ψ′

0)+ s(r, ψ) (B.2)

where we have introduced the source term

s(r, ψ) =1

fr (r)

[− 2cgr (r) cos(ψ + α)v1(r, ψ)

+ δv′

1 + v′

1(r′

0, ψ′

0)(∆v1 − b sinψ)]

(B.3)

and v′

1 is given by Eq. (60).

A. Politi et al. / Physica D 224 (2006) 90–101 101

Being interested in the integral I2 (see Eq. (40)), we see thatthe integration over ψ can be easily performed since ψ ′

0 rangesover [0, 2π ]. More delicate is the integral over r because ofthe folding of r ′

0. However, one can still solve the problem byseparately integrating over the negative and positive values of r ,i.e. the two atoms of the finite Markov partition. By introducingthe integral over negative r -values

I −

2 =

∫ 0

−1dr

∫ 2π

0dψ v2(r, ψ) (B.4)

and analogously defining I +

2 , we obtain from Eq. (B.2)

I −

2 =I −

2 + I +

2

4+ S

I +

2 = −I −

2 + I +

2

4+ S (B.5)

where S is the integral of s(r, ψ)

S =

∫ 1

−1dr

∫ 2π

0dψ s(r, ψ). (B.6)

We thus eventually obtain

I2 = I −

2 + I +

2 = S. (B.7)

It is now convenient to express s(r, ψ) as a function of v′

1 only,

s(r, ψ) =1

fr (r)

− bv′

1 sinψ + δv′

1

+∆v′

1 − 2cgr cos(ψ + α)

fr (r)

× [v′

1 + 2cg(r) sin(ψ + α)]

. (B.8)

Upon integrating over ψ , we obtain,

S = πc2 ∆4

∫ 1

−1dy[L2(y)+ R2(y)]

− 2πc2∫ 1

0dr (r2

− r) sin(∆r)

×[Lr (1 − 2r) cos K − Rr (1 − 2r) sin K ]

+πc2∆∫ 1

0dr (r2

− r) cos(∆r)

×[L(1 − 2r) cos K − R(1 − 2r) sin K ]

+πc2∫ 1

0dr (1 − 2r) sin(∆r)

× [L(1 − 2r) cos K − R(1 − 2r) sin K ]. (B.9)

After integrating by parts the integral involving Lr and Rrand rescaling the dummy variable, we finally arrive at thedesired result:

I2 = πc2 ∆4

∫ 1

−1dy[L2(y)+ R2(y)] (B.10)

+ πc2∫ 1

−1dy sin

∆(1 − y)

2y[L(y) cos K − R(y) sin K ].

(B.11)

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