Applied Nonautonomous and Random Dynamical Systems

Abstract

PrefaceDuring the past two decades, the theory of nonautonomous dynamical systems and random dynamical systems has made substantial progress in studying the long term dynamics of open systems subject to time-dependent or random forcing. However, most of the existing pertinent literatures are fairly technical and almost impenetrable for a general audience, except the most dedicated specialists. Moreover, the concepts and methods of nonautonomous and random dynamical systems, though well-established, are extremely nontrivial to apply to real-world problems. The aim of this work is to provide an accessible and broad introduction to the theory of nonautonomous and random dynamical systems, with an emphasis on applications of the theory to problems arising in the applied sciences and engineering.The book starts with basic concepts in the theory of autonomous dynamical systems which are easier to understand, and used as the motivation for the study of non-autonomous and random dynamical systems. Then the framework of nonautonomous dynamical systems is set up, including various approaches to analyze the long time behavior of non-autonomous problems. The major emphasis is given to the novel theory of pullback attractors, as it can be regarded as a natural extension of the autonomous theory and allows a larger variety of time-dependence forcing than other alternatives such as skew-product flows or cocycles. In the end the theory of random dynamical systems and random attractors is introduced and shown to fairly informative to the study of long term behavior of stochastic systems with random forcing.Each set of theory is illustrated by being applied to three different models, the chemostat model, the SIR epidemic model, and the Lorenz-84 model, in their autonomous, nonautonomous, and stochastic formulations, respectively. The techniques and methods adopted can be applied to the study of the long term behavior of a wide range of applications arising in applied sciences and engineering.

Global and cocycle attractors for non-autonomous reaction-diffusion equations. The case of null upper Lyapunov exponent

“…We now state the definitions of global attractor and cocycle attractor for skewproduct semiflows. The books by Caraballo and Han [5], Carvalho et al [9] and Kloeden and Rasmussen [22] are good references for this topic.…”

Dynamics of a continuous Hénon model

“…Proof. Thanks to theorem 2.7 in Caraballo and Han, 12 we obtain the existence of a compact attractor by the existence of a compact absorbing set.…”

“…On the basis of the above analysis, we have the following results on the stability of the equilibria ðx þ * ; y þ * Þ and ðx − * ; y − * Þ. Theorem 1. Let τ 0 2;0 be defined as in (12).…”

Topologies of continuity for Carathéodory delay differential equations with applications in non-autonomous dynamics

“…It is worth noticing, however, that, as the problem becomes essentially infinitedimensional, this is not a trivial extension of the previous results. Such framework opens a wide range of dynamical scenarios in which it is possible to combine techniques of continuous skew-product flows, processes and random dynamical systems (see Arnold [1], Aulbach and Wanner [5], Caraballo and Han [6], Carvalho et al [7], Johnson et al [13], Pötzsche and Rasmussen [21], Sacker and Sell [22], Sell [24], Shen and Yi [25] and the references therein). As a consequence, it is possible to perform a richer qualitative analysis of the local and global behavior of the solutions for such Carathéodory delay differential equations.…”