Secondary resonances and approximate models of routes to chaotic motion in non-linear oscillators

In two types of non-linear oscillators with single equilibrium positions the question of transition to chaos is investigated by making use of a computer simulation and approximate analytical methods to study periodic solutions and their stability. The zone of chaotic motion is found as a transition zone between qT-periodic (sub-or subultraharmonic) solution and the T-periodic solution, thus replacing the classical jump phenomenon. Two routes to chaos are studied: the gentle one via a cascade of period doubling bifurcations which appears to be characteristic in unsymmetrical oscillators, and a sharp route, which proves to accompany a stability limit of a symmetrical periodic solution. An approximate mathematical model of the pre-chaotic motion is proposed.