Mathematical Problems in Engineering
Volume 2011 (2011), Article ID 248328, 16 pages
Research Article

Nonlinear Dynamics of a Periodically Driven Duffing Resonator Coupled to a Van der Pol Oscillator

1School of Mechanical Engineering, University of Birmingham, Birmingham B4 7ET, UK
2Nonlinearity and Complexity Research Group, Aston University, Birmingham B15 2TT, UK

Received 25 July 2010; Accepted 12 September 2010

Academic Editor: Marco Antonio Teixeira

Copyright © 2011 X. Wei 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 explore the dynamics of a periodically driven Duffing resonator coupled elastically to a van der Pol oscillator in the case of 1 : 1 internal resonance in the cases of weak and strong coupling. Whilst strong coupling leads to dominating synchronization, the weak coupling case leads to a multitude of complex behaviours. A two-time scales method is used to obtain the frequency-amplitude modulation. The internal resonance leads to an antiresonance response of the Duffing resonator and a stagnant response (a small shoulder in the curve) of the van der Pol oscillator. The stability of the dynamic motions is also analyzed. The coupled system shows a hysteretic response pattern and symmetry-breaking facets. Chaotic behaviour of the coupled system is also observed and the dependence of the system dynamics on the parameters are also studied using bifurcation analysis.