A generalization of the concepts of deterministic Morse theory to random dynamical systems is presented. Using the notions of attraction and repulsion in probability, the main building blocks of Morse theory such as attractor–repeller pairs, Morse sets, and the Morse decomposition are obtained for random dynamical systems.
The method of invariant manifolds was originally developed for hyperbolic rest points of autonomous equations. It was then extended from fixed points to arbitrary solutions and from autonomous equations to nonautonomous dynamical systems by either the Lyapunov–Perron approach or Hadamard's graph transformation. We go one step further and study meaningful notions of hyperbolicity and stable and unstable manifolds for equations which are defined or known only for a finite time, together with matching notions of attraction and repulsion. As a consequence, hyperbolicity and invariant manifolds will describe the dynamics on the finite time interval. We prove an analog of the Theorem of Linearized Asymptotic Stability on finite time intervals, generalize the Okubo–Weiss criterion from fluid dynamics and prove a theorem on the location of periodic orbits. Several examples are treated, including a double gyre flow and symmetric vortex merger.
This paper addresses the exponential stability of the trivial solution of some types of evolution equations driven by Hölder continuous functions with Hölder index greater than 1/2. The results can be applied to the case of equations whose noisy inputs are given by a fractional Brownian motion B H with covariance operator Q, provided that H ∈ (1/2, 1) and tr(Q) is sufficiently small.
In this paper we prove that under mild conditions a nonautonomous Young differential equation possesses a unique solution which depends continuously on initial conditions. The proofs use estimates in p-variation norms, the construction of greedy sequence of times, and Gronwall-type lemma with the help of Shauder theorem of fixed points.
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