In this paper we consider the equation ẍ + x = sin(√2t) + s(x), where s(x) is a piece-wise linear map given by min{5x, 1} if x ≥ 0 and by max{-1, 5x} if x < 0. The existence of chaotic behaviour in the Smale sense inside the instability area is proven. In particular transversal homoclinic fixed point is found. The results follow from the application of topological degree theory the computer-assisted verification of a set of inequalities. Usually such proofs can not be verified by hands due to vast amount of computations, but the simplicity of our system leads to a small set of inequalities that can be verified by hand.

Pokrovskii, A., Rasskazov, O., Visetti, D. (2007). Homoclinic trajectories and chaotic behaviour in a piecewise linear oscillator. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. SERIES B., 8(4), 943-970 [10.3934/dcdsb.2007.8.943].

Homoclinic trajectories and chaotic behaviour in a piecewise linear oscillator

Visetti D
2007

Abstract

In this paper we consider the equation ẍ + x = sin(√2t) + s(x), where s(x) is a piece-wise linear map given by min{5x, 1} if x ≥ 0 and by max{-1, 5x} if x < 0. The existence of chaotic behaviour in the Smale sense inside the instability area is proven. In particular transversal homoclinic fixed point is found. The results follow from the application of topological degree theory the computer-assisted verification of a set of inequalities. Usually such proofs can not be verified by hands due to vast amount of computations, but the simplicity of our system leads to a small set of inequalities that can be verified by hand.
Articolo in rivista - Articolo scientifico
Chaotic behaviour; Topological hyperbolicity; Transversal homoclinic orbit;
English
943
970
28
Pokrovskii, A., Rasskazov, O., Visetti, D. (2007). Homoclinic trajectories and chaotic behaviour in a piecewise linear oscillator. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. SERIES B., 8(4), 943-970 [10.3934/dcdsb.2007.8.943].
Pokrovskii, A; Rasskazov, O; Visetti, D
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/363336
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