Probabilistic approach to homoclinic chaos |
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Authors: | D Daems G Nicolis |
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Institution: | (1) Faculté des Sciences and Center for Nonlinear Phenomena and Complex Systems, Université libre de Bruxelles, 1050 Bruxelles, Belgium |
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Abstract: | Three-dimensional systems possessing a homoclinic orbit associated to a saddle focus with eigenvalues ±i , – and giving rise to homoclinic chaos when the Shil'nikov condition < is satisfied are studied. The 2D Poincaré map and its 1D contractions capturing the essential features of the flow are given. At homoclinicity, these 1D maps are found to be piecewise linear. This property allows one to reduce the Frobenius—Perron equation to a master equation whose solution is analytically known. The probabilistic properties such as the time autocorrelation function of the state variablex are explicitly derived. |
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Keywords: | Homoclinic chaos discrete maps Frobenius-Perron equation generalized coarse-graining master equation time autocorrelation function |
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