Slow manifolds for dynamical systems with non-Gaussian stable Lévy noise

2019 ◽  
Vol 17 (03) ◽  
pp. 477-511 ◽  
Author(s):  
Shenglan Yuan ◽  
Jianyu Hu ◽  
Xianming Liu ◽  
Jinqiao Duan
Keyword(s):  
Biological Sciences ◽  
Dynamical Systems ◽  
Scale Parameter ◽  
Lévy Noise ◽  
Stochastic Dynamical Systems ◽  
Time Dynamics ◽  
Slow Manifolds ◽  
Long Time ◽  
Non Gaussian ◽  
Levy Noise

This work is concerned with the dynamics of a class of slow–fast stochastic dynamical systems driven by non-Gaussian stable Lévy noise with a scale parameter. Slow manifolds with exponentially tracking property are constructed, and then we eliminate the fast variables to reduce the dimensions of these stochastic dynamical systems. It is shown that as the scale parameter tends to zero, the slow manifolds converge to critical manifolds in distribution, which helps to investigate long time dynamics. The approximations of slow manifolds with error estimate in distribution are also established. Furthermore, we corroborate these results by three examples from biological sciences.

Metastability in a class of hyperbolic dynamical systems perturbed by heavy-tailed Lévy type noise

2015 ◽  
Vol 15 (03) ◽  
pp. 1550019 ◽  
Author(s):  
Michael Högele ◽  
Ilya Pavlyukevich
Keyword(s):  
Dynamical Systems ◽  
Limit Cycles ◽  
Lévy Noise ◽  
Random System ◽  
Finite Dimensional ◽  
Local Attractors ◽  
Heavy Tailed ◽  
Non Gaussian ◽  
Deterministic Dynamical System ◽  
Levy Noise

We consider a finite dimensional deterministic dynamical system with finitely many local attractors Kι, each of which supports a unique ergodic probability measure Pι, perturbed by a multiplicative non-Gaussian heavy-tailed Lévy noise of small intensity ε > 0. We show that the random system exhibits a metastable behavior: there exists a unique ε-dependent time scale on which the system reminds of a continuous time Markov chain on the set of the invariant measures Pι. In particular our approach covers the case of dynamical systems of Morse–Smale type, whose attractors consist of points and limit cycles, perturbed by multiplicative α-stable Lévy noise in the Itô, Stratonovich and Marcus sense. As examples we consider α-stable Lévy perturbations of the Duffing equation and Pareto perturbations of a biochemical birhythmic system with two nested limit cycles.

Asymptotic methods for stochastic dynamical systems with small non-Gaussian Lévy noise

2014 ◽  
Vol 15 (01) ◽  
pp. 1550004 ◽  
Author(s):  
Huijie Qiao ◽  
Jinqiao Duan
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Numerical Simulations ◽  
Asymptotic Methods ◽  
Lévy Noise ◽  
Stochastic Dynamical Systems ◽  
Escape Probability ◽  
Nonlocal Interactions ◽  
Escape Probabilities ◽  
Non Gaussian

The goal of the paper is to analytically examine escape probabilities for dynamical systems driven by symmetric α-stable Lévy motions. Since escape probabilities are solutions of a type of integro-differential equations (i.e. differential equations with nonlocal interactions), asymptotic methods are offered to solve these equations to obtain escape probabilities when noises are sufficiently small. Three examples are presented to illustrate the asymptotic methods, and asymptotic escape probability is compared with numerical simulations.

CHAOTIC AND QUASIPERIODIC MOTIONS OF THREE PLANAR CHARGED PARTICLES

2001 ◽  
Vol 11 (07) ◽  
pp. 1937-1951
Author(s):  
SHU-MING CHANG ◽  
WEN-WEI LIN ◽  
TAI-CHIA LIN
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Charged Particles ◽  
Time Dynamics ◽  
Long Time ◽  
Long Time Dynamics

We study two dynamical systems for the motion of three planar charged particles with charges nj ∈ {±1}, j = 1, 2, 3. Both dynamical systems are parametric with a parameter α ∈ [0, 1] and have the same nonlinear terms. As α = 0, 1, the dynamical systems have no chaos. However, one dynamical system may create chaos as α varies from zero to one. This may provide an example to show that the homotopy deformation of dynamical systems cannot preserve the long-time dynamics even though the dynamical systems have the same nonlinear terms.

Long-time behavior of Lévy-driven Ornstein–Uhlenbeck processes with regime switching

2020 ◽  
Vol 57 (1) ◽  
pp. 266-279
Author(s):  
Zhongwei Liao ◽  
Jinghai Shao
Keyword(s):  
Stationary Distribution ◽  
Regime Switching ◽  
Lévy Noise ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Lévy Measure ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process ◽  
Long Time ◽  
Levy Noise

AbstractWe investigate the long-time behavior of the Ornstein–Uhlenbeck process driven by Lévy noise with regime switching. We provide explicit criteria on the transience and recurrence of this process. Contrasted with the Ornstein–Uhlenbeck process driven simply by Brownian motion, whose stationary distribution must be light-tailed, both the jumps caused by the Lévy noise and the regime switching described by a Markov chain can derive the heavy-tailed property of the stationary distribution. The different role played by the Lévy measure and the regime-switching process is clearly characterized.

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