On Lyapunov exponents for non-smooth dynamical systems with an application to a pendulum with dry friction
This very large article contains an impressive number of very technical concepts and results attempting to construct a general theory of Lyapunov exponents for nonsmooth (measurable) semiflows and to apply it to the example of the pendulum with dry friction that is modelled by discontinuous differential equations of the form: \[ x''+x+sgn(x')=\gamma .\sin (\eta .t), \;\gamma , \eta \geq 0. \] The ``measurable semiflow'' introduced by the author and described by some ten ``axioms'' seems to be, essentially, a finite ``concatenation'' of smooth flows, with smooth ``switching'' times and surfaces that are related in such a way that a reasonable ``canonical cocycle'', \(T(.,.)\), that is replacing the derivative, \(T(t,y)=\partial_y\phi (t,y)\), in the case of a smooth semiflow, and therefore the Lyapunov exponents: \[ \lambda (y_0,y)=\lim_{t\to \infty} {{1}\over {t}}\ln|T(t,y_0)y|, \;y_0,y\in R^d, \] could be defined and the so-called ``Oseledets' multiplicative ergodic theorem'' could be applied. A large part of the paper (some 35pp) is dedicated to the study of the continuous but nonsmooth flow of the pendulum with dry friction, verifying the ten axioms of the general concept of measurable semiflow, proving the existence and stability of periodic solutions and the fact that its Lyapunov exponents are nonpositive and concluding that the system does not show ``chaotic behaviour''.
Oseledets' multiplicative ergodic theorem
scientific article; zbMATH DE number 1453228