Chaotic behavior in differential equations driven by a Brownian motion

Kening Lu; Qiudong Wang

doi:10.1016/j.jde.2011.05.032

Chaotic behavior in differential equations driven by a Brownian motion

Kening Lu, Qiudong Wang

Research output: Contribution to journal › Article › peer-review

47 Scopus citations

Abstract

In this paper, we investigate the chaotic behavior of ordinary differential equations with a homoclinic orbit to a saddle fixed point under an unbounded random forcing driven by a Brownian motion. We prove that, for almost all sample paths of the Brownian motion in the classical Wiener space, the forced equation admits a topological horseshoe of infinitely many branches. This result is then applied to the randomly forced Duffing equation and the pendulum equation.

Original language	English (US)
Pages (from-to)	2853-2895
Number of pages	43
Journal	Journal of Differential Equations
Volume	251
Issue number	10
DOIs	https://doi.org/10.1016/j.jde.2011.05.032
State	Published - Nov 15 2011

Keywords

Brownian motion
Cantor sets
Chaotic behavior
Duffing equation
Pendulum equation
Random melnikov function
Topological horseshoe
Unbounded stochastic forcing
Wiener shift

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1016/j.jde.2011.05.032

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title = "Chaotic behavior in differential equations driven by a Brownian motion",

abstract = "In this paper, we investigate the chaotic behavior of ordinary differential equations with a homoclinic orbit to a saddle fixed point under an unbounded random forcing driven by a Brownian motion. We prove that, for almost all sample paths of the Brownian motion in the classical Wiener space, the forced equation admits a topological horseshoe of infinitely many branches. This result is then applied to the randomly forced Duffing equation and the pendulum equation.",

keywords = "Brownian motion, Cantor sets, Chaotic behavior, Duffing equation, Pendulum equation, Random melnikov function, Topological horseshoe, Unbounded stochastic forcing, Wiener shift",

author = "Kening Lu and Qiudong Wang",

note = "Funding Information: ✩ This work was partially supported by NSF0401708 and NSF0909400 (K. Lu), NSF0505594 (Q. Wang).",

year = "2011",

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doi = "10.1016/j.jde.2011.05.032",

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T1 - Chaotic behavior in differential equations driven by a Brownian motion

AU - Lu, Kening

AU - Wang, Qiudong

N1 - Funding Information: ✩ This work was partially supported by NSF0401708 and NSF0909400 (K. Lu), NSF0505594 (Q. Wang).

PY - 2011/11/15

Y1 - 2011/11/15

N2 - In this paper, we investigate the chaotic behavior of ordinary differential equations with a homoclinic orbit to a saddle fixed point under an unbounded random forcing driven by a Brownian motion. We prove that, for almost all sample paths of the Brownian motion in the classical Wiener space, the forced equation admits a topological horseshoe of infinitely many branches. This result is then applied to the randomly forced Duffing equation and the pendulum equation.

AB - In this paper, we investigate the chaotic behavior of ordinary differential equations with a homoclinic orbit to a saddle fixed point under an unbounded random forcing driven by a Brownian motion. We prove that, for almost all sample paths of the Brownian motion in the classical Wiener space, the forced equation admits a topological horseshoe of infinitely many branches. This result is then applied to the randomly forced Duffing equation and the pendulum equation.

KW - Brownian motion

KW - Cantor sets

KW - Chaotic behavior

KW - Duffing equation

KW - Pendulum equation

KW - Random melnikov function

KW - Topological horseshoe

KW - Unbounded stochastic forcing

KW - Wiener shift

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DO - 10.1016/j.jde.2011.05.032

M3 - Article

AN - SCOPUS:80051935034

SN - 0022-0396

VL - 251

SP - 2853

EP - 2895

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 10

ER -

Chaotic behavior in differential equations driven by a Brownian motion

Abstract

Keywords

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