Mathematical Problems in Engineering
Volume 2009 (2009), Article ID 149563, 26 pages
Research Article

Bifurcation Analysis of a Van der Pol-Duffing Circuit with Parallel Resistor

1Instituto de Sistemas Elétricos e Energia, Universidade Federal de Itajubá, Avenida BPS 1303, Pinheirinho, 37500-903 Itajubá, MG, Brazil
2Instituto de Ciências Exatas, Universidade Federal de Itajubá, Avenida BPS 1303, Pinheirinho, 37500-903 Itajubá, MG, Brazil
3Departamento de Matemática, Estatística e Computação, Faculdade de Ciências e Tecnologia, UNESP—Univ Estadual Paulista, Cx. Postal 266, 19060-900 Presidente Prudente, SP, Brazil

Received 29 April 2009; Accepted 16 August 2009

Academic Editor: Dane Quinn

Copyright © 2009 Denis de Carvalho Braga et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We study the local codimension one, two, and three bifurcations which occur in a recently proposed Van der Pol-Duffing circuit (ADVP) with parallel resistor, which is an extension of the classical Chua's circuit with cubic nonlinearity. The ADVP system presents a very rich dynamical behavior, ranging from stable equilibrium points to periodic and even chaotic oscillations. Aiming to contribute to the understand of the complex dynamics of this new system we present an analytical study of its local bifurcations and give the corresponding bifurcation diagrams. A complete description of the regions in the parameter space for which multiple small periodic solutions arise through the Hopf bifurcations at the equilibria is given. Then, by studying the continuation of such periodic orbits, we numerically find a sequence of period doubling and symmetric homoclinic bifurcations which leads to the creation of strange attractors, for a given set of the parameter values.