The derivation method of periodic solutions for piecewise linear systems: proposition of the correction extension method of single direction solution

摘要

The piecewise linear system is the one of the most fundamental nonlinear system, and its chaotic behaviour and bifurcation analysis have been studied in detail. In this paper, we propose the exact derivation method of all periodic solutions for forced piecewise linear systems, and clarified its mathematical structure. The key idea is to determine initial condition for periodic solution that enable the solution arising any current time to the future or past make periodic. By means of substitute derived initial condition to initial condition of general solution and evaluate it, periodic solution can be globaly represented by superposition of periodic stationary solution of linear system and finite number of periodic functions which is induced from evaluate of infinite summation of periodic pseudo feedback responses in terms of impact nonlinearity. According to comparing this result and our former result that was induced by feedback superposition method, new insight about mathmatical structure of periodic solutions is derived.