微小外力を含むBVP型拘束方程式に見られる微小振動の崩壊と安定な弛張振動解

摘要

本研究では，超微小な周期外力の印加された4区分線形拘束BVP方程式に見られる分岐現象を解析した. この方程式には，周期外力の印加のない場合，安定なフォーカスと安定な弛張振動解が共存する.このとき，微小周期外力を印加するとカオスが発生するが，その振幅を僅かに大きくするとカオス状態が消滅し，弛張振動解が安定となることが分かった.この現象を，写像法によって導かれる一次元写像を解析することにより明らかにした.%This paper rigorously analyzes a four-segment piecewise linear constraint BVP oscillator under weak periodic perturbation. The parameter values are chosen such that stable focus and stable relaxation oscillation coexist in the absence of perturbation. When perturbation is applied, the stable relaxation oscillation disappears in a wide parameter range. However, the solution that stays near the stable focus disappears when the amplitude of periodic perturbation becomes large. Prom a degenerate dynamical system, Poincare map is derived rigorously as a one-dimensional map. By the analysis of the one-dimensional map, it is shown that "interval to solution near the stable focus" vanishes. It explains the disappearance of the solution near the stable focus and the generation of the stable relaxation oscillation. The manner of this disappearance is very complex, even if the dynamics is very simple.