The random case of Conley's theorem: II. The complete Lyapunov function

Abstract

Conley in [6] constructed a complete Lyapunov function for a flow on compact metric space which is constant on orbits in the chain recurrent set and is strictly decreasing on orbits outside the chain recurrent set. This indicates that the dynamical complexity focuses on the chain recurrent set and the dynamical behavior outside the chain recurrent set is quite simple. In this paper, a similar result is obtained for random dynamical systems under the assumption that the base space (Ω, F , P) is a separable metric space endowed with a probability measure. By constructing a complete Lyapunov function, which is constant on orbits in the random chain recurrent set and is strictly decreasing on orbits outside the random chain recurrent set, the random case of Conley's fundamental theorem of dynamical systems is obtained. Furthermore, this result for random dynamical systems is generalized to noncompact state spaces.

Random chain recurrent sets for random dynamical systems

“…See Definition 2 below for the meaning of pre-attractors and local attractors. Liu [16] further extended his result to noncompact Polish spaces (i.e., separable complete metric spaces) under an additional absorbing condition: the pre-attractor U(ω) is assumed to be an absorbing set. Obviously this absorbing condition depends on the underlying metric in the state space X.…”

The random case of Conley's theorem: III. Random semiflow case and Morse decomposition

“…3) Since backward orbits are not necessarily unique, τ (ω, x) is not well defined. To bypass the obstacles mentioned above, we will construct a new complete Lyapunov function following Conley [9] as well as Arnold and Schmalfuss [3], which has weaker properties than the complete Lyapunov function in [26]. Morse decomposition theorem, originated from Conley [9], is very useful in studying the inner structure of invariant sets, e.g.…”

“…In [25,26], ϕ is defined for every t ∈ R (i.e. random flow), which is usually true for the RDS generated by finite dimensional random and stochastic differential equations, i.e.…”

“…Afterwards, when we say backward orbits, we refer those lying on S unless otherwise stated (since there may be backward orbit not lying on S but lying on the entire state space -X). [26], without loss of generality, we can assume that U is forward invariant.) The basin of attraction of A is defined by B(A)(ω) := {x ∈ S(ω)|ϕ(t, ω)x ∈ U (θ t ω) for some t ≥ 0} and the repeller R corresponding to A is defined by…”

Gradient Infinite-Dimensional Random Dynamical Systems

“…Recently, Liu has introduced a random version of Morse decomposition theory in Conley [13] adapted to random invariant compact sets for flows or even semiflows (see Liu [27,28,29] and Liu, Ji, and Su [30]). In particular, given a random attractor, it is first possible to define a random attractor-repeller pair associated to a random dynamical system, from which to describe a finite family {M i (ω), i = 1, .…”