In this talk, we consider the one-and two-dimensional wave equation on the unit interval [0, 1] with a van der Pol type condition. This nonlinear boundary condition behaves like a van der Pol oscillator, causing the total energy to rise and fall within certain bounds regularly or irregularly. We formulate the problem in terms of an equivalent first order hyperbolic system and use the method of characteristics to derive a nonlinear reflection relation caused by the nonlinear boundary conditions. Qualitative and numerical techniques are developed to tackle the cubic nonlinearities and the chaotic regime is determined. Numerical simulations and visualizations of chaotic vibrations are illustrated by computer graphics.
Zhaosheng Feng is a full-professor at the School of Mathematical and Statistical Sciences of University of Texas-RGV, Edinburg, Texas 78539, USA. His research interests include nonlinear analysis, dynamical systems, computational methods, mathematical physics and mathematical biology etc.