Asymptotic Analysis of the Lyapunov Exponent and Rotation Number of the Random Oscillator and Applications

Arnol, Ludwig; Papanicoaou, George; Wihstutz, Volker

doi:10.1137/0146030

Cited by 149 publications

(98 citation statements)

References 20 publications

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“…Hence we reduced the discussion of the stability of the N -dimensional system in (1) to the study of the stability of the eigenmode which is a one dimensional system. Let the eigenvalue λ = a + ib, a, b ∈ R. Clearly, for the eigenmode (2) is stable.…”

Section: Eigenmodesu (T) = −U(t) + λU(t) λ ∈ Cmentioning

confidence: 99%

“…The developments in dynamical systems theory over the last few decades have provided the foundation for understanding the dynamics of networks of coupled dynamical systems [1,2,3,4,5,6,7,8,9,10]. One central aim of the contemporary study of such networks is to understand how emergent activities of a network with nonlinear units arise from their (nonlinear) interactions [11,12,13,14,15,16,17,18,19,20].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Synchronization in networks with random interactions: Theory and applications

Feng

Jirsa²,

Ding³

2006

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Synchronization is an emergent property in networks of interacting dynamical elements. Here we review some recent results on synchronization in randomly coupled networks. Asymptotical behavior of random matrices is summarized and its impact on the synchronization of network dynamics is presented. Robert May's results on the stability of equilibrium points in linear dynamics are first extended to nonlinear systems where the synchronized dynamics can be periodic or chaotic and then to systems with time delayed coupling. Finally, applications of our results to neuroscience, in particular, networks of Hodgkin-Huxley neurons, are included.Behaviour and structures of interacting dynamical systems are complex. Most research on complex networks has been focused so far on networks with deterministic interactions. In this paper we study the behaviour of networks with random interactions. We extend Robert May's results on the stability of equilibrium points in linear dynamics to nonlinear systems where the synchronized dynamics can be periodic or chaotic and then to systems with time delayed coupling. Results are then applied to networks in Neuroscience.

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Section: Eigenmodesu (T) = −U(t) + λU(t) λ ∈ Cmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Synchronization in networks with random interactions: Theory and applications

Feng

Jirsa²,

Ding³

2006

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…Equation (4.3) is the parametrically excited stochastic linear oscillator, which arises when the harmonic parametric forcing in the Mathieu equation (4.1) is replaced by a Gaussian stochastic process x t . For this case, the top Lyapunov exponent was computed by Arnold et al (1986).…”

Section: Small-noise Stability Analysismentioning

confidence: 99%

“…This yields that probabilistic methods have to be used for the evaluation of ship dynamics in realistic seas. Arnold et al (1986) computed the largest Lyapunov exponent for a parametrically excited stochastic linear oscillator, which arises when the harmonic parametric forcing in the Mathieu equation is replaced by a Gaussian stochastic process. A negative maximal Lyapunov exponent yields the almostsure (a.s.) stability of the trivial solution of the oscillators.…”

Section: Introductionmentioning

confidence: 99%

Non-standard stochastic averaging of large-amplitude ship rolling in random seas

2012

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The current intact stability criteria for ships do not account for their dynamic behaviour in irregular waves sufficiently. We provide analytical criteria for ship and sea state parameters that indicate large roll amplitudes or capsizing and determine the mean time for these events. These results can be used to formulate vulnerability criteria for the occurrence of parametric roll. We use reduced order models in order to perform an analytical analysis to this complicated problem. Therefore, we study the dynamics of a ship in random seas as a perturbation of a Hamiltonian system. The random process across the energy levels of the Hamiltonian system is approximated by a Markov process, which is obtained by stochastic averaging. The noise owing to wave excitation is a nonwhite stationary stochastic process. This is an extension to previous results, where white noise excitation was used.

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“…These results are formulated as Theorems 1 and 2 below. To the best of our knowledge the first article to look for the asymptotic behavior of γ(E) and N (E) in the limit E → ∞ is [2]. The best known estimate for the integrated density of states is due to Kirsch and Martinelli [10,Corollary 3.1].…”

Section: Introductionmentioning

confidence: 99%

Global bounds for the Lyapunov exponent and the integrated density of states of random Schrödinger operators in one dimension

Kostrykin¹,

Schrader²

2000

J. Phys. A: Math. Gen.

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ABSTRACT. In this article we prove an upper bound for the Lyapunov exponent γ(E) and a two-sided bound for the integrated density of states N (E) at an arbitrary energy E > 0 of random Schrödinger operators in one dimension. These Schrödinger operators are given by potentials of identical shape centered at every lattice site but with non-overlapping supports and with randomly varying coupling constants. Both types of bounds only involve scattering data for the single-site potential. They show in particular that both γ(E) and N (E) − √ E/π decay at infinity at least like 1/ √ E. As an example we consider the random Kronig-Penney model.

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Asymptotic Analysis of the Lyapunov Exponent and Rotation Number of the Random Oscillator and Applications

Cited by 149 publications

References 20 publications

Synchronization in networks with random interactions: Theory and applications

Synchronization in networks with random interactions: Theory and applications

Non-standard stochastic averaging of large-amplitude ship rolling in random seas

Global bounds for the Lyapunov exponent and the integrated density of states of random Schrödinger operators in one dimension

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