GROWTH RATES OF TOTAL VARIATIONS OF SNAPSHOTS OF THE 1D LINEAR WAVE EQUATION WITH COMPOSITE NONLINEAR BOUNDARY REFLECTION RELATIONS

Abstract

The linear wave equation on an interval with a van der Pol nonlinear boundary condition at one end and an energy-pumping condition at the other end is a useful model for studying chaotic behavior in distributed parameter system. In this paper, we study the dynamics of the Riemann invariants (u, v) of the wave equation by means of the total variations of the snapshots on the spatial interval. Our main contributions here are the classification of the growth of total variations of the snapshots of u and v in long-time horizon, namely, there are three cases when a certain parameter enters a different regime: the growth(i) remains bounded;(ii) is unbounded (but nonexponential);(iii) is exponential,for a large class of initial conditions with finite total variations. In particular, case (iii) corresponds to the onset of chaos. The results here sharpen those in an earlier work [Chen et al., 2001].