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Asymptotic stability of periodic solutions for nonsmooth differential equations with application to the nonsmooth van der Pol oscillator. (English) Zbl 1194.34016

The authors study existence, uniqueness and asymptotic stability of periodic solutions of the system

$\stackrel{˙}{x}=\epsilon g\left(t,x,\epsilon \right),$

where $\epsilon >0$ is small and $g\in {C}^{0}\left(ℝ×{ℝ}^{k}×\left[0,1\right],{ℝ}^{k}\right)$ is $T$-periodic in $t$ and locally Lipschitz with respect to $x$.

This generalises the second Bogoliubov’s Theorem [N. N. Bogoliubov, On Some Statistical Methods in Mathematical Physics. Akademiya Nauk Ukrainskoi SSR, Kiev, (1945) (in Russian) Chapter 1, Section 5, Theorem II] where $g$ is assumed to be ${C}^{1}$. The authors give an extensive list of applications of their results, which motivates the study of non-differentiable right-hand sides $g$.

The main results include Theorem 2.1, where no differentiability of $g$ is assumed and sufficient conditions for existence, uniqueness and asymptotic stability of periodic solutions are given. Theorem 2.5 is a consequence of Theorem 2.1; here, $g$ is assumed to be piecewise differentiable with respect to $x$. Finally, Theorem 2.9 gives a sufficient condition on the Brouwer topological degree for the existence of nonasymptotically stable periodic solutions.

In the final section, Theorem 2.5 is used to show the existence of asymptotically stable periodic solutions of the nonsmooth van der Pol oscillator

$\stackrel{¨}{u}+\epsilon \left(|u|-1\right)\stackrel{˙}{u}+\left(1+a\epsilon \right)u=\epsilon \lambda sint·$

The authors construct resonance curves, describing the dependence of the amplitude of these solutions as a function of $a$ and $\lambda$, and compare them to the classical, smooth van der Pol oscillator.

##### MSC:
 34A36 Discontinuous equations 34C29 Averaging method 34C25 Periodic solutions of ODE 47H11 Degree theory (nonlinear operators) 34D20 Stability of ODE