*(English)*Zbl 1194.34016

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

where $\epsilon >0$ is small and $g\in {C}^{0}(\mathbb{R}\times {\mathbb{R}}^{k}\times [0,1],{\mathbb{R}}^{k})$ 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

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 |