August  2003, 3(3): 423-438. doi: 10.3934/dcdsb.2003.3.423

Degenerate resonances in forced oscillators

1. 

Department of Mechanical and Systems Engineering, Gifu University, Gifu, Gifu 501-1193, Japan

Received  July 2002 Revised  February 2003 Published  May 2003

Using an extended version of the subharmonic Melnikov method, we discuss resonance behavior in a class of forced nonlinear oscillators when the resonance is degenerate in the following meaning: The frequency of the resonant periodic orbit has a null derivative with respect to the energy level in the unperturbed system without forcing and damping terms. Such an appropriate treatment of degenerate resonances in systems of physical or engineering meaning as performed here was not previously presented. In particular, we show that the degenerate resonances can generally give rise to cusp bifurcations. Moreover, we describe a numerical strategy for the necessary computations for application of the theory. To illustrate our technique, two examples are presented for a nonsymmetric oscillator and a feedback controlled pendulum. Direct numerical bifurcation analysis results are also given and compared with the theoretical results.
Citation: Kazuyuki Yagasaki. Degenerate resonances in forced oscillators. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 423-438. doi: 10.3934/dcdsb.2003.3.423
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