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On Lyapunov Exponents for Non-Smooth Dynamical Systems with an Application to a Pendulum with Dry Friction

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Abstract

We consider dynamical systems from mechanics for which, due to some non-smooth friction effects, Oseledets' Multiplicative Ergodic Theorem cannot be applied canonically to define Lyapunov exponents. For general non-smooth systems which fit into a natural formal framework, we construct a suitable cocycle which lives on a “good” invariant set of full Lebesgue measure. Afterwards, this construction is applied to investigate a pendulum with dry friction, described through the equation \(\ddot x + x + \operatorname{sgn} \dot x = \gamma \sin (\eta t)\). The Lyapunov exponents obtained by our construction show a good agreement with the dynamical behaviour of the system, and since we will prove that these Lyapunov exponents are always non-positive, we conclude that the system does not show “chaotic behaviour.”

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  1. Mathematisches Institut, Universität Köln, Weyertal 86, D-50931, Köln, Germany

    Markus Kunze

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  1. Markus Kunze
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Kunze, M. On Lyapunov Exponents for Non-Smooth Dynamical Systems with an Application to a Pendulum with Dry Friction. Journal of Dynamics and Differential Equations 12, 31–116 (2000). https://doi.org/10.1023/A:1009046702601

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