BEHAVIOR OF HARMONICS GENERATED BY A DUFFING TYPE EQUATION WITH A NONLINEAR DAMPING: PART I

Abstract

This paper is devoted to the numerical study of a two-dimensional nonautonomous ordinary differential equation with a strong cubic nonlinearity, submitted to an external periodical excitation of period τ = 2π/ω , having an amplitude E. In absence of this excitation the equation of Duffing type does not give rise to self oscillations. It may be a model of a series R-L-C electrical circuit with validity conditions in the parameter space. The purpose is essentially an analysis of the harmonics behavior of the period τ solutions according to points of the parameter plane (ω, E). Let r be the place occupied by a rank-m harmonic from an ordering based on line amplitudes of a frequency spectrum in descending order. Domains of the (ω, E) plane, for which the amplitude of the rank-m harmonic has the place r in the ordering mentioned above, are defined from properties of their boundaries. When r = 2 they are regions of predominance for the rank-m harmonic. When r = 1 they are regions of full predominance for the harmonic m, m = 2,3,4,…, which contain a set of points leading to a resonance for this harmonic. These regions fulfil the following important property: each of them is directly related to a fold bifurcation structure, called rank-mlip (two folds curves joining at two cusps), associated with a well-defined rank-m harmonic. The set of such structures, with m = 2,3,4,…, constitutes an isoordinal lips cascade. As an opening to Part II (to be published) period kτ solutions related to fractional harmonics behavior are also considered for k = 2,3.