Existence of a periodic solution in a Chua's circuit with smooth nonlinearity

Abstract

In this paper, we consider Chua's circuit: $$ \varepsilon u'=z+f(u) , \ \ z'=u+w-z, \ \ w'=-\beta z-\gamma w, $$ where $f(u)$ is chosen as a cubic function, $\beta>0$ and $\gamma\geqslant 0$ are constants, and $\varepsilon>0$ is a small parameter. We prove that the flow defines a Poincar$\acute{e}$ map from a compact set which is homeomorphic to the unit disk to itself and then apply Brouwer's fixed-point theorem to conclude that the system has a ``big'' periodic solution. This global analysis is viewed as a step towards understanding chaos in this model analytically.