Speaker：Zhaosheng Feng，University of Texas-Rio Grande Valley
Title：Chaotic Vibration of the Wave Equation with a van der Pol Boundary Condition |
Abstract：In this talk, we consider the one-dimensional wave equation on the unit interval [0, 1]. At the left end x = 0, an energy injecting boundary condition is posed, and at the right end, x = 1, the boundary condition is a cubic nonlinearity, which is 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. Since the solution of the first order hyperbolic system completely depends on this nonlinear relation and its iterations, the problem is reduced to a discrete iteration problem. Following Devaney’s definition of chaos, we say that the PDE system is chaotic if the corresponding mapping is chaotic as an interval map. 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. |
