Narrow Results

Many physical systems are represented by Partial Differential Equations (PDEs), and the study of chaotic dynamics in these systems is interesting and challenging. In this paper, the Li–Yorke chaos of PDEs is studied, and the Li–Yorke chaos is observable in several classes of PDEs, including systems with or without energy injection. For the PDEs without energy injection, three kinds of PDEs are investigated revealing the existence of Li–Yorke chaos, including a type of transport equations, a class of wave equations, and a kind of Navier–Stokes equations. For the PDEs with energy injection, only dissipative type of PDEs are studied, including a model of sound variation of the drum with damping, a kind of reaction–diffusion equations, and a class of two-dimensional Navier–Stokes equations. It is shown that Li–Yorke chaos is well defined for the characterization of the complicated dynamics of these systems. In particular, a physical explanation about chaos in such PDEs is provided, which gives an interesting explanation of acoustic chaos.

In this paper, we introduce fractal interpolation functions (FIFs) and linear FIFs on a post critically finite (p.c.f. for short) self-similar set K. We present a sufficient condition such that linear FIFs have finite energy and prove that the solution of Dirichlet problem -Δ_μ u = f,u|_∂K = 0 is a linear FIF on K if f is a linear FIF.