非対称復元力を有する強制振動系の高分数調波振動の分岐現象とカオス : 第1報,1次共振点以下における周期解から1バンドアトラクタへの分岐構造

抄録

This paper describes a nonlinear oscillator derived from the gear meshing vibration, which exhibits successive different bifurcations ending in chaos below the first resonance. These bifurcations are examined in detail using bifurcation diagrams, Lyapunov exponents, winding numbers, invariant curves, and Poincare maps. There are two types of sudden changes from a chaotic to a periodic attractor. One is the case, named "hysteresis", in which a chaotic attractor is transformed into a periodic one far from the chaotic one, and the other is the case in which a chaotic attractor transforms into a periodic one which lies inside the chaotic one. When an n-periodic attractor transforms into a chaotic one through period-doubling, a heteroclinic connection between the outset of a 2n-periodic inversely unstable fixed point and the inset of an n-periodic inversely unstable fixed point is necessary to generate an n-band chaotic attractor. If a directly unstable fixed point exists in the vicinity of any band attractors and the attractor touches the inset of the directly unstable fixed point on the above-mentioned sequence, the attractor expands due to "interior catastrophe", and immediately transforms into a one-band attractor.