Stochastic sensitivity of 3D-cycles

Bashkirtseva, Irina; Ryashko, Lev

doi:10.1016/j.matcom.2004.02.021

Cited by 81 publications

(47 citation statements)

References 16 publications

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“…To analyze the mechanism of stochastic generation of bursting oscillations, we apply the stochastic sensitivity function (SSF) technique [17,19]. An essential theoretical background of SSF technique is given in the Appendix.…”

Section: Stochastic Model Generation Of Bursting Oscillationsmentioning

confidence: 99%

“…Therefore, various asymptotics and approximations are developed [22][23][24]. For the approximation of KFP solutions, a well-known quasipotential method [15,16] and a stochastic sensitivity function (SSF) technique [17][18][19]25] can be applied.…”

Section: Appendixmentioning

confidence: 99%

“…However, a direct usage of this equation is very difficult technically even in simple situations, therefore various asymptotics and approximations were developed. For an approximation of KFP solutions, a well-known quasipotential method [15,16] and a stochastic sensitivity function technique [17,18] can be used.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Stochastic Generation of Bursting Oscillations in the Three-dimensional Hindmarsh–Rose Model

Ryashko¹

2016

J. Sib. Fed. Univ., Math. phys.

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Section: Stochastic Model Generation Of Bursting Oscillationsmentioning

confidence: 99%

Section: Appendixmentioning

confidence: 99%

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Stochastic Generation of Bursting Oscillations in the Three-dimensional Hindmarsh–Rose Model

Ryashko¹

2016

J. Sib. Fed. Univ., Math. phys.

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“…In this paper, we will use asymptotics 35 of this stationary density function based on the quasipotential vðxÞ ¼ Àlim e!0 e 2 log qðx; eÞ. We use Gaussian approximation for the probability density of the bundles of random trajectories, localized near the cycle 36 q % Ke with the covariance matrix e 2 UðcÞ. Here, "þ" means a pseudoinversion.…”

Section: Stochastic Sensitivity Function For 3d Cyclesmentioning

confidence: 99%

Analysis of noise-induced transitions from regular to chaotic oscillations in the Chen system

Bashkirtseva¹,

Chen

Ryashko³

2012

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The stochastically perturbed Chen system is studied within the parameter region which permits both regular and chaotic oscillations. As noise intensity increases and passes some threshold value, noiseinduced hopping between close portions of the stochastic cycle can be observed. Through these transitions, the stochastic cycle is deformed to be a stochastic attractor that looks like chaotic. In this paper for investigation of these transitions, a constructive method based on the stochastic sensitivity function technique with confidence ellipses is suggested and discussed in detail. Analyzing a mutual arrangement of these ellipses, we estimate the threshold noise intensity corresponding to chaotization of the stochastic attractor. Capabilities of this geometric method for detailed analysis of the noiseinduced hopping which generates chaos are demonstrated on the stochastic Chen system. Noise-induced chaos in stochastically perturbed nonlinear deterministic systems with regular attractors has been studied extensively. This phenomenon has received a great deal of interest among researchers from various fields of physics, life sciences, and engineering. Nonlinearity, even in a non-chaotic regime, can imply a non-uniformity of phase portraits, diverse forms of coexicting regular attractors with a complicated geometry of basins of attraction. Under random disturbances, a phase trajectory can cross separatices and induce stochastic hopping between both coexisting regular attractors and their different but close portions. As a consequence of this hopping, the random trajectories with a high probability fall within zones of divergency (local instability), and the dynamics of the perturbed system as a whole becomes chaotic. The positivity of the largest Lyapunov exponent can be used as a measure that this new noise-induced regime is chaotic.In this paper, for the constructive prediction of the noise-induced transitions from regular to chaotic oscillations in a dynamical system, a new stochastic sensitivity function technique is developed and applied. This technique enables to find dispersion ellipses of the random states for the stochastically perturbed dynamical system around deterministic attractors. The sizes of these ellipses are enlarged as noise intensity increases. The constructive method of the dispersion ellipses allows to estimate the threshold of the noise intensity corresponding to the deformation of a regular stochastic attractor to a chaotic one. The effectiveness of this approach is demonstrated by the stochastically perturbed Chen system.

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“…The method of orbital Lyapunov functions with the use of quasipotential is applied in the stability and sensitivity analysis of stochastically perturbed limit cycles [12,13]. This method was used to describe in detail the stochastic attractors in period-doubling bifurcations zone and to analyze the passages caused by the noise [14,15].…”

Section: Introductionmentioning

confidence: 99%