Maximal Lyapunov exponent and rotation numbers for two coupled oscillators driven by real noise

Namachchivaya, N. Sri; Roessel, H. J. Van

doi:10.1007/bf01058437

Cited by 34 publications

(10 citation statements)

References 9 publications

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“…No theorem has had so direct and powerful an influence upon the study of stochastic stability of noisy dynamical systems as the multiplicative ergodic theorem (MET) of Oseledets [1], which established the existence of (typically) finitely many deterministic exponential growth rates called Lyapunov exponents. The stability of linear stochastic systems based on the MET has been well established [2,3] and the top Lyapunov exponent can be evaluated explicitly with relative ease when the noisy perturbations and dissipation are weak [4,5]. The challenge has been to extend the existing techniques in order to explicitly evaluate the top Lyapunov exponent of nonlinear systems with noise, and in particular additive white noise.…”

Section: Introductionmentioning

confidence: 99%

Stability of a Stochastic Two-Dimensional Non-Hamiltonian System

Deville¹,

Namachchivaya²,

Rapti³

2011

SIAM J. Appl. Math.

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Abstract. We study the largest Lyapunov exponent of the response of a two dimensional non-Hamiltonian system driven by additive white noise. The specific system we consider is the third-order truncated normal form of the unfolding of a Hopf bifurcation. We show that in the small-noise limit the top Lyapunov exponent always approaches zero from below (and is thus negative for noise sufficiently small); we also show that there exist large sets of parameters for which the top Lyapunov exponent is positive. Thus the two-point motion can be either stable or unstable, while the one-point motion is always stable.

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Section: Introductionmentioning

confidence: 99%

Stability of a Stochastic Two-Dimensional Non-Hamiltonian System

Deville¹,

Namachchivaya²,

Rapti³

2011

SIAM J. Appl. Math.

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“…Mirroring the method by Baxendale P. and Goukasian L. [17] and the results of an earlier work by Arnold L., Imkeller P. and Namachchivaya N. Sri [19], the top Lyapunov Exponents for the externally excited degree of freedom is found to be positive. We also made use of results on Lyapunov exponents by Arnold, Papanicolaou and Wihstutz [18], Pinsky and Wihstutz [13], Sri Namachchivaya and Van Roessel [14] and Imkeller and Lederer [15]. However, only the first term of the asymptotic expansion was analytically evaluated.…”

Section: Discussionmentioning

confidence: 99%

“…As in [14], a proof that this expansion is, in fact, asymptotic relies on the construction of the PoissonType equations (38) that can be solved and the error is bounded, that is,…”

Section: Asymptotic Form For λ ε Using the Adjoint Methodsmentioning

confidence: 99%

“…In Section 4.2, due to the nilpotent structure of the linear variational equations, the Pinsky and Wihstutz [13] rescaling is used in the linear variational equations to derive the Furstenberg-Khasminskii formula. In Sections 4.3 and 4.4 we appeal to the results of Sri Namachchivaya and Van Roessel [14] and Imkeller and Lederer [15] to evaluate the first term in the asymptotic expansion of the top Lyapunov exponent. In Section 5 the results and discussions of the analysis are presented with some recommendation for possible future work that deals with second variational equation derived in Section 3.…”

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confidence: 99%

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Stability of noisy nonlinear auto-parametric systems

2006

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The purpose of this work is to examine the stationary motion and stability properties of stationary motion of two degree-of-freedom noisy autoparametric systems We shall use analytical techniques to extend the existing results to examine such multidimensional nonlinear systems with noise, and in particular additive white noise. We obtain an approximation for the top Lyapunov exponent, the exponential growth rate, of the response of the so-called singlemode stationary motion. We show analytically that the top Lyapunov exponent is positive, and for small values of noise intensity ε and dissipation ε 2 the exponent grows in proportion with ε 2 3 .

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“…By using the stochastic averaging method, Ariaratnam and Xie [7] obtained the asymptotic expansions of the maximal Lyapunov exponent for two coupled oscillators excited by a real noise. Through a perturbation method, the same system and then a more general four-dimensional linear stochastic system were investigated by Namchchivaya and Rossel [8], Doyle and Namchchivaya [9].…”

Section: Introductionmentioning

confidence: 99%

The maximal Lyapunov exponent of a co-dimension two-bifurcation system excited by a bounded noise

Liu

2012

Acta Mech Sin

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In the present paper, the maximal Lyapunov exponent is investigated for a co-dimension two bifurcation system that is on a three-dimensional central manifold and subjected to parametric excitation by a bounded noise. By using a perturbation method, the expressions of the invariant measure of a one-dimensional phase diffusion process are obtained for three cases, in which different forms of the matrix B, that is included in the noise excitation term, are assumed and then, as a result, all the three kinds of singular boundaries for one-dimensional phase diffusion process are analyzed. Via Monte-Carlo simulation, we find that the analytical expressions of the invariant measures meet well the numerical ones. And furthermore, the P-bifurcation behaviors are investigated for the one-dimensional phase diffusion process. Finally, for the three cases of singular boundaries for one-dimensional phase diffusion process, analytical expressions of the maximal Lyapunov exponent are presented for the stochastic bifurcation system.

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Maximal Lyapunov exponent and rotation numbers for two coupled oscillators driven by real noise

Cited by 34 publications

References 9 publications

Stability of a Stochastic Two-Dimensional Non-Hamiltonian System

Stability of a Stochastic Two-Dimensional Non-Hamiltonian System

Stability of noisy nonlinear auto-parametric systems

The maximal Lyapunov exponent of a co-dimension two-bifurcation system excited by a bounded noise

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