Amplitude modulated chaos in two degree-of-freedom systems with quadratic nonlinearities

TLDR: In this paper, two degree-of-freedom systems with weak quadratic nonlinearities were studied under weak external and parametric excitations respectively, and the method of averaging was used to obtain a set of four first-order amplitude equations that govern the dynamics of the firstorder asymptotic approximation to the response.

Abstract: Two degree-of-freedom systems with weak quadratic nonlinearities are studied under weak external and parametric excitations respectively. All six possible cases, that arise in the presence of 1∶2 internal resonance, are investigated. The method of averaging is used to obtain a set of four first-order amplitude equations that govern the dynamics of the first-order asymptotic approximation to the response. An analytical technique, based on Melnikov's method is used to predict the parameter range for which chaotic dynamics exists in the undamped averaged system. Numerical studies show that such chaotic responses are quite common in these quadratic systems, and they seem to persist even in the presence of damping.