We establish the existence of a bifurcation from an attractive random
equilibrium to shear-induced chaos for a stochastically driven limit cycle,
indicated by a change of sign of the first Lyapunov exponent. This addresses an
open problem posed by Kevin Lin and Lai-Sang Young, extending results by
Qiudong Wang and Lai-Sang Young on periodically kicked limit cycles to the
stochastic context

Engel, Maximilian

Lamb, Jeroen S. W.

Rasmussen, Martin

Communications in Mathematical Physics

English

arXiv

We establish the existence of a bifurcation from an attractive random equilibrium to shear-induced
chaos for a stochastically driven limit cycle, indicated by a change of sign of the first Lyapunov exponent.
This relates to an open problem posed by Kevin Lin and Lai-Sang Young in [11, 16], extending results
by Qiudong Wang and Lai-Sang Young [14] on periodically kicked limit cycles to the stochastic context

Lamb, Jeroen

Spiral - Imperial College Digital Repository

Bifurcation analysis of a stochastically driven limit cycleMaximilian Engel∗ Jeroen S.W. Lamb† Martin Rasmussen†September 20, 2018AbstractWe establish the existence of a bifurcation from an attractive random equilibrium to shear-inducedchaos for a stochastically driven limit cycle, indicated by a change of sign of the first Lyapunov exponent.This relates to an open problem posed by Kevin Lin and Lai-Sang Young in [11, 16], extending resultsby Qiudong Wang and Lai-Sang Young [14] on periodically kicked limit cycles to the stochastic context.Key words. Furstenberg–Khasminskii formula, Lyapunov exponent, Random dynamical system, Shear-induced chaos, Stochastic bifurcation.Mathematics Subject Classification (2010). 37D45, 37G35, 37H10, 37H15.1 IntroductionWe consider the following model of a stochastically driven limit cycledy = −αy dt+ σ∑mi=1 fi(ϑ) ◦ dW it ,dϑ = (1 + by) dt ,(1.1)where (y, ϑ) ∈ R × S1 are cylindrical amplitude-phase coordinates, m ≥ 2 is a natural number, and W itfor i ∈ {1, . . . ,m} denote independent one-dimensional Brownian motions entering the equation as noise ofStratonovich type.In the absence of noise (σ = 0), the ordinary differential equation (1.1) has a globally attracting limitcycle at y = 0 if α > 0. In the presence of noise (σ 6= 0), the amplitude is driven by phase-dependent noise.The real parameter b induces shear: if b 6= 0, the phase velocity dϑdt depends on the amplitude y.The stable limit cycle turns into a random attractor if σ 6= 0. The main question we address in this paperconcerns the nature of this random attractor. The crucial quantity is the sign of the first Lyapunov exponentλ1 = λ1(α, b, σ) with respect to the invariant measure associated to the random attractor. In essence, λ1 isthe dominant infinitesimal asymptotic growth rate of almost all trajectories.The functions fi : S1 ' [0, 1)→ R are assumed to be C2,δ for some 0 < δ ≤ 1 to guarantee differentiabilityof the random dynamical system (see [1, Theorem 2.3.32]), and, to facilitate the analysis, we requirem∑i=1f ′i(ϑ)2 = 1 for all ϑ ∈ S1 . (1.2)A simple example is given bym = 2, f1(ϑ) = cos(ϑ), f2(ϑ) = sin(ϑ) . (1.3)Using an explicit formula of Furstenberg–Khasminskii type for the top Lyapunov exponent λ1 derived byPeter Imkeller and Christian Lederer [7], we obtain the following bifurcation result with such choices of theamplitude-phase coupling.∗Zentrum Mathematik der TU München, Boltzmannstr. 3, D-85748 Garching bei München†Department of Mathematics, Imperial College London, 180 Queens Gate, London SW7 2AZ, United Kingdom1Theorem 1.1. Consider the SDE (1.1) with m ≥ 2 and fi, i ∈ {1, . . . ,m}, satisfying assumption (1.2).Then there is c0 ≈ 0.2823 such that for all α > 0 and b 6= 0, the number σ0(α, b) = α3/2c1/20 |b|> 0 is the uniquevalue of σ where the top Lyapunov exponent λ1(α, b, σ) of (1.1) changes sign:λ1(α, b, σ)< 0 if 0 < σ < σ0(α, b) ,= 0 if σ = σ0(α, b) ,> 0 if σ > σ0(α, b) .As long as b, σ 6= 0, the amplitude variable y can be rescaled so that the shear parameter b becomesequal to 1 and the effective noise amplitude becomes σb. Hence, the above result also holds with the rolesof σ and b interchanged. Note that σ0(α, b) is an increasing function of α. If σ = 0, we clearly have λ1 = 0for all α > 0. The case α = 0 is obviously not of any interest in our model.If the top Lyapunov exponent is negative, it turns out that the (weak) random point attractor is anattracting random equilibrium, i.e. its fibers are singletons almost surely. Properties of random attractorswith positive top Lyapunov exponents are not yet well understood, apart from the fact that such attractorsare not random equilibria. They are sometimes referred to as random strange attractors [10, 15]. Theorem 1.1confirms numerical results by Lin & Young [11] for a similar model. The mechanism, whereby a combinationof shear and noise causes stretching and folding leading to a positive Lyapunov exponent, was coined withthe term shear-induced chaos by Lin & Young. Wang & Young [13, 14] and Ott & Stenlund [12] havedemonstrated analytically the validity of this mechanism in the case of periodically kicked limit cycles,including probabilistic characterizations of the dynamics. An analytical proof of shear-induced chaos in thestochastic setting, as presented in this paper, had remained an open problem.The remainder of the paper is organized as follows. Section 2 provides the analysis of Lyapunov exponentsfor our model. The main result concerning the change of sign of λ1 is proven in Section 3 and its consequencesare discussed. We give illustrations of λ1 in dependence on the parameters and confirm a scaling conjectureby Lin and Young. We conclude with a short summary and outlook in Section 4.2 Explicit formula for the Lyapunov exponentsConsider the stochastic differential equation of Stratonovich type (1.1) fulfilling condition (1.2), and assumethat the three parameters fulfill α > 0, σ > 0 and b ∈ R. Note that the equation reads the same in Itô formaccording to the Itô–Stratonovich conversion formula.Since the drift and diffusion coefficients are Lipschitz continuous and satisfy linear growth conditions, theSDE (1.1) generates a continuous random dynamical system (θ : R×Ω→ Ω, ϕ : R+0 ×Ω×R× S1 → R× S1)[1, Definition 1.1.2]. The map θ is B(R) ⊗ F-measurable and preserves the ergodic probability measureP : F → [0, 1]. The map ϕ represents the stochastic flow, i.e. ϕ is a B(R+0 ) ⊗ F ⊗ B(R × S1)-measurablemapping such that (t, x) 7→ ϕ(t, ω)x is continuous for every ω ∈ Ω. We haveϕ(0, ω) = id , and the cocycle property ϕ(t+ s, ω) = ϕ(t, θsω) ◦ ϕ(s, ω)for almost all ω ∈ Ω and t, s ∈ R+0 .In the case of (1.1), the variational equation [1, Theorem 2.3.32] on the tangent space Tx(R× S1) ∼= R2takes the formdv =(−α 0b 0)︸ ︷︷ ︸=:Av dt+ σm∑i=1(0 f ′i(ϑ)0 0)︸ ︷︷ ︸=:Biv ◦ dW it . (2.1)Note that we omit the (t, ω)-dependence of ϑ and B. According to [1, Theorem 2.3.32], the cocycle ϕ isdifferentiable in x and its partial derivative Dϕ(t, ω, x) with respect to x applied to an initial conditionv0 ∈ R2 solves uniquely the variational equation (2.1). The random dynamical system (θ, ϕ) has an ergodicinvariant measure ν associated with the stationary measure ρ for (1.1) and clearly satisfies the integrabilityconditionsup0≤t≤1log+ ‖Dϕ(t, ω, x)‖ ∈ L1(ν).2Denoting the skew product flow Θ = (θ, ϕ) on Ω ×X, it is a direct consequence of the chain rule and thecocycle property of ϕ that we haveDϕ(t+ s, ω, x) = Dϕ(t,Θs(ω, x))Dϕ(s, ω, x) (2.2)for all t, s ∈ R+0 , x ∈ X and almost all ω ∈ Ω. Hence, the Furstenberg–Kesten Theorem [1, Theorem 3.3.3]gives the existence of at least one λ1 and at most two Lyapunov exponents λ1 ≥ λ2 for ϕ via its derivativeDϕ such thatlimt→∞1tlog ‖Dϕ(t, ω, x)v‖ ∈ {λ1, λ2} ,for all v ∈ R2 and ν-almost all (ω, x) ∈ Ω× R× S1.We observe thatUt :=m∑i=1∫ t0f ′i(ϑs)dWis (2.3)defines a standard Brownian motion under condition (1.2). Hence, the variational equation (2.1) can bereplaced bydv =(−α 0b 0)v dt+(0 σ0 0)v ◦ dUt , (2.4)and the Lyapunov exponents for ϕ can be computed from this equation.The following theorem is now a direct corollary of a result by Imkeller and Lederer [7, Theorem 3].Theorem 2.1. Consider the stochastic differential equation (1.1) with m ≥ 2 and fi, i ∈ {1, . . . ,m},satisfying assumption (1.2). Then the two Lyapunov exponents are given byλ1(α, b, σ) = −α2+|bσ|2∫ ∞0v mσ,b,α(v) dv , (2.5)λ2(α, b, σ) = −α2− |bσ|2∫ ∞0v mσ,b,α(v) dv , (2.6)wheremσ,b,α(v) =1√vexp(− |bσ|6 v3 + α22|bσ|v)∫∞01√uexp(− |bσ|6 u3 +α22|bσ|u)du. (2.7)Proof. Replacing v =(v1v2)by v̂ =(v2v1σ)leaves the Lyapunov exponents invariant and transforms (2.4) intothe equationdv =(0 σb0 −α)v dt+(0 01 0)v ◦ dW 1t . (2.8)The matrices in this equation satisfy the assumptions of [7, Theorem 3] which gives the formulas (2.5)and (2.6).3 Bifurcation from negative to positive top Lyapunov exponentWe now use Theorem 2.1 to prove Theorem 1.1, which asserts that there is a bifurcation from negative topositive Lyapunov exponent for the stochastic differential equation (1.1).Proof of Theorem 1.1. We fix α > 0 and b 6= 0. Introducing the change of variables v = α|bσ|u in (2.7), weobtainλ1(α, b, σ) =α2(∫ ∞0u m̃σ,b,α(u) du− 1), (3.1)wherem̃σ,b,α(u) =1√uexp(− α3σ2b2[16u3 − 12u])∫∞01√wexp(− α3σ2b2[16w3 − 12w])dw.3Defining c := α3σ2b2 , we observe that λ1(α, b, σ) has the same sign as the function G : (0,∞)→ R given byG(c) :=∫ ∞0(√u− 1√u)exp(−c[16u3 − 12u])du . (3.2)Using dominated convergence, we may interchange the order of differentiation and integration and considerG′(c) =∫ ∞0h1(u)h2(u) exp (c h2(u)) du , h1(u) =√u− 1√u, h2(u) = −16u3 +12u .Note that h1h2, and thereby the integrand, has positive sign on the interval (1,√3) and negative sign on(0, 1) and (√3,∞). Moreover, |h1(1− δ)| > h1(1 + δ) and h2(1 − δ) > h2(1 + δ) for all δ ∈ (0,√3 − 1) sothatG′(c) <∫ √32−√3h1(u)h2(u) exp (c h2(u)) du < 0 for all c ∈ (0,∞) .Hence, G is strictly decreasing. Furthermore, we observe that G(c) → ∞ as c ↘ 0 (using monotoneconvergence on [√3,∞)) and that G(c)→ −∞ as c→∞ (using similar arguments as for G′ and monotoneconvergence on (0, 2−√3)).Combining these observations, we conclude that there is a unique c0 such that G(c0) = 0, G(c) > 0 forall c ∈ (0, c0) and G(c) < 0 for all c ∈ (c0,∞). This proves the claim with σ0(α, b) = α3/2c1/20 |b|. Numericalintegration gives c0 ≈ 0.2823.Note that, as explained in the Introduction in more detail, the same result holds if we interchange theroles of σ and b. This can be seen also directly from the proof above.Remark 3.1. The random dynamical system induced by (1.1) has a random set attractor {Ã(ω)}ω∈Ω (see [9,Definition 14.3] for a formal definition) for all parameter values, as can be seen similarly to [8]. Furthermore,the random dynamical system possesses an ergodic invariant Markov measure µ which is associated to theunique invariant measure (also called stationary measure) for the corresponding Markov semigroup, see [3].The disintegrations µω of the ergodic invariant measure µ are supported on subsets of the fibers Ã(ω),i.e. A(ω) := supp(µω) ⊂ Ã(ω). In fact, the measurable random compact set {A(ω)}ω∈Ω is a minimal (weak)random point attractor of (1.1) by [6, Proposition 2.20 (1)].The fact that {A(ω)}ω∈Ω is a singleton almost surely if λ1 < 0, follows from a slightly modified reasoningalongside [6, Theorem 2.23] and its proof. In the case of λ1 > 0, we deduce that µω is atomless almost surelyas in the proof of [2, Remark 4.12]. Hence, Theorem 1.1 implies the bifurcation from an attractive randomequilibrium to an atomless random point attractor (also called random strange attractor).The positive top Lyapunov exponent is the only characterization of chaos we can give in this case asan analysis in the sense of [14] seems not feasible for white noise. However, the geometric mechanism ofshear-induced chaos can still be understood along the same lines: the Brownian motions drive some pointsof the deterministic limit cycle up and some down on the cylinder. Due to the phase amplitude coupling b,the points with larger y-coordinates move faster in the ϑ-direction. At the same time, the dissipation forcewith strength α attracts the curve back to the limit cycles. This provides a mechanism for stretching andfolding characteristic of chaos. The transition to chaos in the continuous time stochastic forcing is muchfaster than in the case of periodic kicks due to the effect of large deviations [11]. This is due to the fact thatpoints end up in areas with arbitrarily large values of y with positive probability. Hence, not so much shearis needed to generate the described stretching and folding due to phase amplitude coupling. However, forvery small shear and noise, the dissipation leads to sinks being formed between these large deviation events,and the attractor ends up to be a singleton.In Figure 1(a), we show the top Lyapunov exponent as a function of σ for fixed b and α according toformula (2.5), illustrating the bifurcation of the sign of λ1. Figure 1(b) displays the (σ, α)-parameter spacebeing separated by the curve {(σ0(α, b), α)} for fixed b. The choice of parameters coincides with an examplein [11], where Lin and Young investigate numerically the Lyapunov exponents of the system given bydy = −αydt+ σ sin(2πϑ) ◦ dWt ,dϑ = (1 + by) dt ,(3.3)4taking ϑ ∈ [0, 2π].Note that, in particular, the situation of (1.3) is similar to model (3.3) but differs in form of an additionalBrownian motion with the cosine function as diffusion coefficient. This variation does not seem to change thequalitative behaviour and shows the same stability results as Lin and Young’s numerical investigations suggestfor model (3.3), modulo the scaling factor 2π. For example, this can be seen by comparing Figure 1(a) with[11, Figure 6]. However, note that the insertion of m ≥ 2 Brownian motions, as in model (1.1), establishesa major change to the noise correlation compared to model (3.3). Hence, we advise caution not to identifyboth models.Nevertheless, we observe that Lin and Young’s conjecture concerning a scaling property of λ1 withrespect to the parameters is satisfied in our setting. For model (3.3) they observed numerically that, underthe transformations α 7→ kα, b 7→ kb and σ 7→√kσ, the top Lyapunov exponent λ1 transforms approximatelyas λ1 7→ kλ1. This scaling property holds exactly for our model (1.1).Proposition 3.2. Consider the stochastic differential equation (1.1) with m ≥ 2 and fi, i ∈ {1, . . . ,m},satisfying assumption (1.2). Then the top Lyapunov exponent λ1 as given by (2.5) satisfiesλ1(kα, kb,√kσ) = kλ1(α, b, σ) for all k ∈ R+ \ {0} . (3.4)Proof. Recall that λ1 can be calculated as the top Lyapunov exponent of the linear system (2.4). Then theclaim follows immediately if we conduct the time change t 7→ kt in this equation.(a) (b)Figure 1: In Figure 1(a) the top Lyapunov exponent λ1, calculated according to (2.5) using numerical integration, isshown as a function of σ for fixed b and α. Figure 1(b) shows the areas of positive and negative λ1 in the (σ, α)-parameter space being separated by the curve {(σ0(α, 3), α)} as a function of α for fixed b = 3, using the formula forσ0 in Theorem 1.1.4 Summary and outlookWe have investigated systems with a single limit cycle on a cylinder in the presence of white noise andestablished a transition from negative to positive top Lyapunov exponents for fixed dissipation parameter αand big enough noise σ and/or shear b. This implies a bifurcation from a random equilibrium to randomstrange attractor.Our results depend on a particular choice of the diffusion which makes an analytical treatment possible.In accordance with Lin and Young [11], we anticipate that the appearance of chaotic behaviour expressedby a positive Lyapunov exponent applies to a more general class of stochastically driven limit cycles withlarge enough shear force and hope that more general proofs can be found in the future.Optimally, this would also extend to the problem of stochastic Hopf bifurcation, where numerical studiesindicate a transition from negative to positive Lyapunov exponent [4, 5, 15]. We hope that our results in asimpler setting are a step towards rigorous results for this challenging open problem.5AcknowledgmentsThe authors would like to thank the referee for constructive comments which substantially helped improvingthe quality of the paper, in particular strengthening the main result. We also express our gratitude to MartinHairer and Nils Berglund for suggesting the suitable generalization of the original model. In addition, theauthors thank Alexis Arnaudon, Darryl Holm, Aleksandar Mijatovic, Nikolas Nüsken, Grigorios Pavliotisand Sebastian Wieczorek for useful discussions. Maximilian Engel was supported by a Roth Scholarship fromthe Department of Mathematics at Imperial College London and the SFB Transregio 109 ”Discretizationin Geometry and Dynamcis” sponsored by the German Research Foundation (DFG). Jeroen S.W. Lambacknowledges the support by Nizhny Novgorod University through the grant RNF 14-41-00044, and MartinRasmussen was supported by an EPSRC Career Acceleration Fellowship EP/I004165/1. This researchhas also been supported by EU Marie-Curie IRSES Brazilian-European Partnership in Dynamical Systems(FP7-PEOPLE-2012-IRSES 318999 BREUDS) and EU Marie-Sk lodowska-Curie ITN Critical Transitions inComplex Systems (H2020-MSCA-2014-ITN 643073 CRITICS).References[1] L. Arnold. Random Dynamical Systems. Springer, Berlin, 1998.[2] P.H. Baxendale. Statistical equilibrium and two-point motion for a stochastic flow of diffeomorphisms.In Spatial stochastic processes, volume 19 of Progress in Probability, pages 189–218. Birkhäuser Boston,Boston, MA, 1991.[3] H. Crauel. Markov measures for random dynamical systems. Stochastics and Stochastics Reports,37(3):153–173, 1991.[4] L. Deville, N. Sri Namachchivaya, and Z. Rapti. Stability of a stochastic two-dimensional non-Hamiltonian system. Journal of Applied Mathematics, 71(4):1458–1475, 2011.[5] T.S. Doan, M. Engel, J.S.W. Lamb, and M. Rasmussen. Hopf bifurcation with additive noise. Nonlin-earity, 31(10):4567–4601, 2018.[6] F. Flandoli, B. Gess, and M. Scheutzow. Synchronization by noise. Probability Theory and RelatedFields, 168(3–4):511–556, 2017.[7] P. Imkeller and C. Lederer. Some formulas for Lyapunov exponents and rotation numbers in twodimensions and the stability of the harmonic oscillator and the inverted pendulum. Dynamical Systems,16(1):29–61, 2001.[8] H. Keller and B. Schmalfuß. Attractors for stochastic differential equations with nontrivial noise. Bulet-inul A.S. a R.M. Matematica, 26(1):43–54, 1998.[9] P.E. Kloeden and M. Rasmussen. Nonautonomous dynamical systems, volume 176 of MathematicalSurveys and Monographs. American Mathematical Society, Providence, RI, 2011.[10] F. Ledrappier and L.-S. Young. Entropy formula for random transformations. Probability Theory andRelated Fields, 80:217–240, 1988.[11] K. K. Lin and L.-S. Young. Shear-induced chaos. Nonlinearity, 21:899–922, 2008.[12] W. Ott and M. Stenlund. From limit cycles to strange attractors. 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Bifurcation Analysis of a Stochastically Driven Limit Cycle

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