Bifurcations of Higher Subharmonics and Chaos in a Forced Vibratory System with an Asymmetrical Restoring Force

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In this paper, we describe the behavioral characteristics of a nonlinear oscillator derived from gear-meshing vibration, which exhibits various successive bifurcations ending in chaos. There are two types of sudden change from a chaotic to a periodic attractor, one of which is called "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 transforms into a one-band attractor. In the bifurcation diagram, periodic subharmonic oscillations appear discontinuously as "windows" belonging 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 solution 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", respectively.
一般社団法人日本機械学会の論文
1996-12-15