Chaotic Vibrations of the One-Dimensional Mixed Wave System

In this paper, we consider the initial-boundary value problem of the one-dimensional linear mixed wave equation ωtt - dωtx - c2ωxx = 0 (d ∈ ℝ, c > 0) on an interval, where the boundary condition at the left endpoint is linear, pumping energy into the system, while the boundary condition at the right endpoint has odd-degree nonlinearity. This problem is said to be the one-dimensional mixed wave system. The solution of the one-dimensional mixed wave system corresponds to the iteration of an interval map h. Thus, the mixed wave system is said to be chaotic if the interval map h is chaotic in the sense of Li–Yorke. In this paper, we show that the mixed wave system is chaotic under some conditions.