scholarly journals Characterization of the most probable transition paths of stochastic dynamical systems with stable Lévy noise

2019 ◽  
Vol 2019 (6) ◽  
pp. 063204 ◽  
Author(s):  
Yuanfei Huang ◽  
Ying Chao ◽  
Shenglan Yuan ◽  
Jinqiao Duan
Keyword(s):  
Dynamical Systems ◽  
Lévy Noise ◽  
Stochastic Dynamical Systems ◽  
Transition Paths ◽  
Levy Noise

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.

scholarly journals Metastability for discontinuous dynamical systems under Lévy noise: Case study on Amazonian Vegetation

2017 ◽  
Vol 7 (1) ◽  
Author(s):  
Larissa Serdukova ◽  
Yayun Zheng ◽  
Jinqiao Duan ◽  
Jürgen Kurths
Keyword(s):  
Dynamical Systems ◽  
Lévy Noise ◽  
Levy Noise

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.

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