PERTURBATION OF THE CHAOTIC 1D MIXED WAVE SYSTEM CAUSES THE UNBOUNDED GROWTH OF THE SNAPSHOTS OF THE GRADIENT

Abstract

Consider the one-dimensional mixed 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. First, we show a certain parameter range of the chaotic vibration of the system. Furthermore, if the right-end van der Pol boundary contains an extra linear displacement feedback term, then we show that under some suitable conditions, the system is still chaotic in the sense of unbounded growth of the snapshots of the gradient.