Introduction to Some Methods of Chaos Analysis and Control for PDEs

Following the development of the research on chaos and controlling chaos for ODEs, some methods and results of that for PDEs were developed in last decade. In this chapter, in addition to give a summary account in part, we present some results on controlling chaos for a class of parabolic type PDEs by applying the invariant manifold and structure stability theory. 1 Some Typical Models of PDEs In recent years the chaotic behavior of the systems described by some PDE models are often studied. The subject of this section is devoted to present mathematical setting of some of these models and brief in their physical background. 1.1 The generalized complex Ginzburg-Landau equation (CGLE) ut = ρu+ (1 + ic1)∆u+ (1 + ic2)|u|u (1) that describes the evolution of a complex-valued u = u(x, t). It has a long history in physics as a generic amplitude equation near the onset of instabilities that lead to chaotic dynamics in fluid mechanical systems, as well as in the theory of phase transitions and superconductivity. It is a particularly interesting model because it is important for the study of turbulent problems and spatiotemporal structure. So, the long time and finite dimensional behavior, such as the global attractor and approximate inertial manifolds (AIM), for CGLE are discussed in some papers. Especially, the turbulent behavior and control of spatiotemporal chaos as well as spatiotemporal patterns for CGLE are studied (see [1][3]). Moreover, the convergence of chaotic attractor with increased partial resolution is done for CGLE (see [4]). 1.2 The perturbed nonlinear Schrödinger equation (NSL) iqt = qxx + 2[|q| − w]q + iε(aq − bqxx + r), (2)