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) |
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Pages (from-to) | 2853-2895 |
Number of pages | 43 |
Journal | Journal of Differential Equations |
Volume | 251 |
Issue number | 10 |
DOIs | |
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