ISSN:
1572-9222
Keywords:
Lyapunov exponents
;
cocycle
;
Oseledets' Multiplicative Ergodic Theorem
;
non-smooth mechanical pendulum
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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.”
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1009046702601
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