Chaos for one-dimensional linear hyperbolic partial differential equations with variable coefficients

This paper studies the Cauchy problem for the one-dimensional linear hyperbolic partial differential equations (PDEs) with variable coefficients. Chaos of such system is studied in the parameter plane {(a, b)∣b ≥ 0 and (a, b) ≠ (0, 0)}, which is determined by two constant coefficients associated with spatial derivative. Several necessary and sufficient conditions for the existence of chaos are established in (a,b)∣a<0and0≤b≤−a∪(a,b)∣a=0,b>0 by proving the chaotic dynamics of C0-semigroups deduced from this system. Moreover, discretizing the system by the finite difference method induces chaos in the rest of the parameter plane. The mean Lyapunov exponent which quantifies the spatial disorder of discrete PDEs is calculated, and numerical simulations are provided to verify chaos.