Two-frequency perturbation of a smooth hamiltonian system

Abstract

This work elaborates upon previous studies on the family of smooth continuous and discontinuous two-parameter Hamiltonian systems with a piecewise linear force. For such systems, the Melnikov-Arnold integral is found to be a power and oscillatory function of frequency. In the presence of two primary forcing frequencies, the secondary harmonic with a frequency that is the sum of the primary frequencies may make a major contribution to the formation of a chaotic layer. For the corresponding smooth map, the perturbation parameter ranges where, under strong local chaos, the upper separatrix of fractional resonances is retained while the lower breaks (and vice versa) are determined. It is shown that the zero angle of intersection of the separatrix branches at the central homoclinic point is not a sufficient condition for separatrix retention. Under dynamic conditions, smooth and analytical systems behave in a very different manner.