Dynamical systems in the theory of solitons in the presence of nonlocal interactions

1992 ◽  
Vol 2 (4) ◽  
pp. 565-570 ◽  
Author(s):  
G. L. Alfimov ◽  
V. M. Eleonsky ◽  
N. E. Kulagin
Keyword(s):  
Dynamical Systems ◽  
Nonlocal Interactions

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.

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