Analyzing displacement term's memory effect in a nonlinear boundary value problem to prove chaotic vibration of the wave equation

Consider a nonlinear boundary value problem for the one-dimensional wave equation with the left-end boundary condition containing an extra linear displacement feedback term, which induces a memory effect. Technical difficulty arises as to how to determine chaotic vibrations of the system. In this paper, we build new Riemman invariants, one of which contains time-delay term, and utilize method of characteristics to rigorously prove the onset of chaos in the sense of exponential growth of total variations of the bounded gradient.