Colored-noise-induced parametric resonance

Bobryk, Roman V.; Chrzeszczyk, A.

doi:10.1016/s0378-4371(02)01312-2

“…There are more recent works that studied equation (1.2) for the case of coloured Gaussian noise to include the effects of memory (Bobryk & Chrzeszczyk 2002, 2004Lee et al 2004). There have also been studies of a stochastic Mathieu's equation using non-Gaussian noise.…”

Section: Introductionmentioning

confidence: 99%

The stochastic Mathieu's equation

Poulin

¹

,

Flierl

²

2008

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In this manuscript, we consider generalizations of the classical Mathieu's equation to stochastic systems. Unlike previous works, we focus on internal frequencies that vary continuously between periodic and stochastic variables. By numerically integrating the system of equations using a symplectic method, we determine the Lyapunov exponents for a wide range of parameters to quantify how the growth rates vary in parameter space.In the nearly periodic limit, we recover the same growth rates as the classical Mathieu's equation. As the stochasticity increases, the maximum growth decreases and the unstable region broadens, which signifies that the stochasticity can both stabilize and destabilize a system in relation to the deterministic limit. In addition to the classical parametric modes, there is also an unstable stochastic mode that arises only for moderate stochasticity. The power spectrum of the solutions shows that an increase in stochasticity tends to narrow the width of the subharmonic peak and increase the decay away from this peak.

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“…2 ω 2 (also called multiplicative noise) for which a vast literature exists [30,[39][40][41][42][43]. The focus of the present work is to study a "generalised" random oscillator including the term 2γP ) which accounts for the nonlinear effects, where the random frequency is modeled by the non-stationary stochastic processes described by Eq.…”

Section: Modulational Instability In Randomly Kicked Fibersmentioning

confidence: 99%

Modulational instability in optical fibers with randomly kicked normal dispersion

Dujardin

¹

,

Armaroli

²

,

Nodari

³

et al. 2021

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We study modulational instability (MI) in optical fibers with random group velocity dispersion (GVD) generated by sharply localized perturbations of a normal GVD fiber that are either randomly or periodically placed along the fiber and that have random strength. This perturbation leads to the appearance of low frequency MI side lobes that grow with the strength of the perturbations, whereas they are faded by randomness in their position. If the random perturbations exhibit a finite average value, they can be compared with periodically perturbed fibers, where Arnold tongues appear. In that case, increased randomness in the strengths of the variations tends to affect the Arnold tongues less than increased randomness in their positions.

show abstract

“…2 ω 2 (also called multiplicative noise) for which a vast literature exists [30,[39][40][41][42][43]. The focus of the present work is to study a "generalised" random oscillator including the term 2γP ) which accounts for the nonlinear effects, where the random frequency is modeled by the non-stationary stochastic processes described by Eq.…”

Section: Modulational Instability In Randomly Kicked Fibersmentioning

confidence: 99%

Modulational instability in optical fibers with randomly-kicked normal dispersion

Dujardin,

Armaroli,

Nodari

et al. 2021

Preprint

0

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We study modulational instability (MI) in optical fibers with random group velocity dispersion (GVD) generated by sharply localized perturbations of a normal GVD fiber that are either randomly or periodically placed along the fiber and that have random strength. This perturbation leads to the appearance of low frequency MI side lobes that grow with the strength of the perturbations, whereas they are faded by randomness in their position. If the random perturbations exhibit a finite average value, they can be compared with periodically perturbed fibers, where Arnold tongues appear. In that case, increased randomness in the strengths of the variations tends to affect the Arnold tongues less than increased randomness in their positions.

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Colored-noise-induced parametric resonance

Cited by 14 publications

References 21 publications

The stochastic Mathieu's equation

The stochastic Mathieu's equation

Modulational instability in optical fibers with randomly kicked normal dispersion

Modulational instability in optical fibers with randomly-kicked normal dispersion

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