BIFURCATIONS OF HIGHER SUBHARMONICS AND CHAOS IN A FORCED VIBRATORY SYSTEM WITH AN ASYMMETRICAL RESTORING FORCE

In this paper, we describe the behavioral characteristics of a nonline ar oscillator derived from gear-meshing vibration, which exhibits vari ous successive bifurcations ending in chaos. There are two types of su dden change from a chaotic to a periodic attractor, one of which is ca lled ''hysteresis''. In a period-doubling bifurcation, a heteroclinic connection between the outset of an inversely unstable fixed point of period 2n and the inset of an inversely unstable fixed point of period n is necessary to generate an n-band chaotic attractor. If a directly unstable fixed point exists in the vicinity of any band attractor and the attractor collides with the inset of the directly unstable fixed point, the attractor expands due to an ''interior catastrophe'', and t ransforms into a one-band attractor. In the bifurcation diagram, perio dic subharmonic oscillations appear discontinuously as ''windows'' bel onging to the same ''family''. These ''families'' are divided into two types: type- I, in which fold bifurcation occurs repeatedly, and type -II, in which fold bifurcation occurs only at both ends of the solutio n curve. Depending upon whether or not both of the chaotic attractors exist in the same area of the phase plane, either before or after the bifurcation, the discontinuous bifurcation of the chaotic attractor is referred to as an ''interior catastrophe'' or ''hysteresis'', respect ively.