## Differential and Integral Equations

- Differential Integral Equations
- Volume 18, Number 1 (2005), 83-120.

### 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.

#### Article information

**Source**

Differential Integral Equations, Volume 18, Number 1 (2005), 83-120.

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356060238

**Mathematical Reviews number (MathSciNet)**

MR2105341

**Zentralblatt MATH identifier**

1212.34102

**Subjects**

Primary: 34C25: Periodic solutions

Secondary: 34C05: Location of integral curves, singular points, limit cycles 37C27: Periodic orbits of vector fields and flows 37D45: Strange attractors, chaotic dynamics

#### Citation

Zhang, Fu. Existence of a periodic solution in a Chua's circuit with smooth nonlinearity. Differential Integral Equations 18 (2005), no. 1, 83--120. https://projecteuclid.org/euclid.die/1356060238