On the global attractivity of monotone random dynamical systems

Abstract

Abstract. Suppose that (θ, ϕ) is a monotone (order-preserving) random dynamical system (RDS for short) with state space V , where V is a real separable Banach space with a normal solid minihedral cone V + . It is proved that the unique equilibrium of (θ, ϕ) is globally attractive if every pull-back trajectory has compact closure in V .

Random attractors for locally monotone stochastic partial differential equations

“…The dynamical behavior of RDS can be significantly simpler if the RDS preserves a partial order structure on the state space H. For example this idea has been used in [3,12,20,21,42]. A closed, convex cone H + ⊆ H satisfying H + ∩ (−H + ) = {0} defines a partial order relation on H which is compatible with the vector structure on H by defining x ≤ y iff y − x ∈ H + .…”

A Small-Gain Theorem for Random Dynamical Systems with Inputs and Outputs

“…Recall that a subset V + of a real topological vector space V is said to be a cone if (C1) V + is closed (not typically part of the definition [20], but a standard assumption in the theory of monotone RDS [6,5], and also needed in our preliminary results); (C2)…”

“…The nonlinearity implied in the h i 's makes this system difficult to study directly. Since (1.2) is not cooperative with respect to any orthant cone-induced partial order, one cannot directly analyze (1.2) using global convergence results from the theory of monotone RDS [6,5]. To overcome these difficulties, we present a decompositionbased alternative inspired on the works of Angeli, Enciso, and Sontag on deterministic systems in [1,12].…”

Translation-invariant monotone systems II: Almost periodic/automorphic case

“…Together with the stability property with induction, one can choose a sequence of positive limit sets in a given positive limit set such that the corresponding sequence of Lyapunov integer-valued functions is strictly decreasing if this given positive limit set is not a singleton. In this way, it was proved that an equilibrium is globally attractive if and only if every forward orbit has compact closure (see [21,23,30,9]). This technique was further exploited in the multiple equilibria/fixed points case to prove global convergent results in monotone (without stronger notion) dynamical systems with every equilibrium/fixed point stable [25,26,27], sublinear nonlinearity [24] and possessing a positive gradient first integral [28].…”