The Conley index and bifurcation points

Discrete Conley index and bifurcation points

“…Similar to [5], if there does not exist a nonempty prime isolated invariant set or if there are infinitely many prime isolated invariant sets, then we define the extreme maximal isolated invariant set for f by T f = ∅. Thus, for every continuous map f , there is a unique extreme maximal isolated invariant set.…”

“…The definition is given in abstract forms, because that is sufficient for theoretical discussion. In Section 2, we extend all the concepts introduced in [5] to discrete cases. The corresponding sufficient condition for the existence of bifurcation points will then be presented in Section 3.…”

A sufficient condition for bifurcation in random dynamical systems

“…Suppose ϕ has only finitely many prime random isolated invariant sets S 1 (ω), S 2 (ω), · · · , and S r (ω), then T f (ω) = r i=1 S i (ω) is called an extreme maximal random isolated invariant set for ϕ. For the deterministic case see [6]. Lemma 3.1.…”

“…By Lemmas 3.1 and 2.3, we see that an extreme maximal random isolated invariant set is indeed a random isolated invariant set. As in [6], if there exists no nonempty prime random isolated invariant set or if there are infinitely many prime random isolated invariant sets, then we define the extreme maximal random isolated invariant set for ϕ by T ϕ (ω) = ∅. Thus, for every random homeomorphism ϕ, there exists a unique extreme maximal random random isolated invariant set.…”

“…The Conley index is a topological tool for investigating dynamical systems [4,10,13,14]. In particular, the Conley index has been used to detect bifurcation in deterministic dynamical systems [2,6,7,8,9,11]. Recently, the Conley index is defined for discrete-time random dynamical systems [12].…”