Analyzing Displacement Term's Memory Effect in a van der Pol Type Boundary Condition to Prove Chaotic Vibration of the Wave equation

Consider the one-dimensional wave equation on a unit interval, where the left-end boundary condition is linear, pumping energy into the system, while the right-end boundary condition is self-regulating of the van der Pol type with a cubic nonlinearity. Then for a certain parameter range it is now known that chaotic vibration occurs. However, if the right-end van der Pol boundary condition contains an extra linear displacement feedback term, then it induces a memory effect and considerable technical difficulty arises as to how to define and determine chaotic vibration of the system. In this paper, we take advantage of the extra margin property of the reflection map and utilize properties of homoclinic orbits coupled with a perturbation approach to show that for a small parameter range, chaotic vibrations occur in the sense of unbounded growth of snapshots of the gradient. The work also has significant implications to the occurrence of chaotic vibration for the wave equation on a 3D annular domain.