CHAOTIC OSCILLATIONS OF LINEAR HYPERBOLIC PDE WITH VARIABLE COEFFICIENTS AND IMPLICIT BOUNDARY CONDITIONS

In this paper, the chaotic oscillations of the initial-boundary value problem of linear hyperbolic partial differential equation (PDE) with variable coefficients are investigated, where both ends of boundary conditions are nonlinear implicit boundary conditions (IBCs). It separately considers that IBCs can be expressed by general nonlinear boundary conditions (NBCs) and cannot be expressed by explicit boundary conditions (EBCs). Finally, numerical examples verify the effectiveness of theoretical prediction. 1. Introduction. Over the past decades, there has been a great deal of interest in the research on chaos of dynamical systems. The theory of chaos in finitedimensional dynamical systems, including both discrete maps and systems governed by ordinary differential equations, has been well-developed [8, 16, 19]. However, there is few theory of chaos in systems governed by partial differential equations (PDEs). This is partly because the rigorous proof of the existence of their chaotic behaviors is challenging. For more results on the chaos in infinite-dimensional dynamical systems, we refer to [7, 15, 21, 22, 23, 24] and the references therein. In recent twenty years, there have been lots of papers studying the chaotic oscillations in the systems governed by one-dimensional wave equations, see [1, 2, 3, 4, 5, 9, 10, 11, 12, 14] and the references therein. Interestingly, Chen et al. [6] studied chaotic dynamics of hyperbolic PDE with constant coefficients and van der Pol boundary condition. Li et al. [13] characterized the chaotic oscillations of hyperbolic PDE, which is factorizable but noncommutative, with van der Pol boundary condition. Recently, chaotic oscillations of second-order linear hyperbolic