Characteristic function for the stationary state of a one-dimensional dynamical system with Lévy noise

In this paper we study a one-dimensional dynamical system that provides a model for the evolution of a Fabry–Perot cavity in Laser Physics. We determine the parameter ranges for bistability and chaos in the system. We also examine the bifurcations of the system and the occurrence of various types of cycles.

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Markov covers and finiteness of periodic attractors for diffeomorphisms with a dominated splitting

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700000018 ◽

2000 ◽

Vol 20 (1) ◽

pp. 1-14

Author(s):

MASAYUKI ASAOKA

Keyword(s):

Dynamical System ◽

Dominated Splitting ◽

One Dimensional ◽

Surface Diffeomorphisms ◽

Periodic Attractors ◽

Dimensional Dynamical System

In this paper, we show the existence of Markov covers for $C^2$ surface diffeomorphisms with a dominated splitting under some assumptions. Using a Markov cover, we can reduce the dynamics to a one-dimensional dynamical system having a good metric property. As an application, we show finiteness of periodic attractors for the above diffeomorphisms.

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FIRST EXIT TIMES OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY MULTIPLICATIVE LÉVY NOISE WITH HEAVY TAILS

Stochastics and Dynamics ◽

10.1142/s0219493711003413 ◽

2011 ◽

Vol 11 (02n03) ◽

pp. 495-519 ◽

Cited By ~ 14

Author(s):

ILYA PAVLYUKEVICH

Keyword(s):

Dynamical System ◽

Differential Equations ◽

Heavy Tails ◽

Multiplicative Noise ◽

Exit Time ◽

Lévy Noise ◽

Exit Times ◽

First Exit Time ◽

First Exit Times ◽

Levy Noise

In this paper, we study first exit times from a bounded domain of a gradient dynamical system Ẏt = -∇U(Yt) perturbed by a small multiplicative Lévy noise with heavy tails. A special attention is paid to the way the multiplicative noise is introduced. In particular, we determine the asymptotics of the first exit time of solutions of Itô, Stratonovich and Marcus canonical SDEs.

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Non-existence of wandering intervals and structure of topological attractors of one dimensional dynamical systems: 1. The case of negative Schwarzian derivative

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700005307 ◽

1989 ◽

Vol 9 (4) ◽

pp. 737-749 ◽

Cited By ~ 18

Author(s):

M. Yu. Lyubich

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Limit Cycle ◽

Critical Points ◽

Schwarzian Derivative ◽

Dimensional Manifold ◽

Generic Point ◽

One Dimensional ◽

Manifold With Boundary ◽

Dimensional Dynamical System

AbstractIt is proved that an arbitrary one dimensional dynamical system with negative Schwarzian derivative and non-degenerate critical points has no wandering intervals. This result implies a rather complete view of the dynamics of such a system. In particular, every minimal topological attractor is either a limit cycle, or a one dimensional manifold with boundary, or a solenoid. The orbit of a generic point tends to some minimal attractor.

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