Global Lipschitz invariant center manifolds for ODEs with generalized trichotomies

Abstract

In a Banach space, assuming that a linear nonautonomous differential equation v ′ = A(t)v admits a very general type of trichotomy, we establish conditions for the existence of global Lipschitz invariant center manifold of the perturbed equation v ′ = A(t)v + f (t, v). Our results not only improve results already existing in the literature, as well include new cases.

Backwards theory supports modelling via invariant manifolds for non-autonomous dynamical systems

“…Crucial modelling properties often hold over domains usefully larger I = R" that unstable manifolds exist, [p.441] the linear operator "is bounded", and [p.437] the nonlinearity function "F is uniformly C m -bounded". Bento & da Costa (2017) in their non-uniform analysis similarly require "for all t ∈ R" in their Theorem 3.1. Barreira & Valls (2007) [p.172] require the nonlinearity function in the system to decay exponentially quickly in time, "for every t ∈ R".…”

Invariant manifolds for difference equations with generalized trichotomies

“…In [11] it was introduced, for linear differential equations, a very general type of trichotomies that include as particular cases all the notions of trichotomies mentioned above, as well as new cases. Despite of this generality, it was possible to prove the existence of central invariant Lipschitz manifolds for sufficiently small Lipschitz perturbations of the linear differential equations that admit this type of generalized trichotomy.…”

“…Despite of this generality, it was possible to prove the existence of central invariant Lipschitz manifolds for sufficiently small Lipschitz perturbations of the linear differential equations that admit this type of generalized trichotomy. This paper is a discrete time counterpart of [11]. We are going to consider for linear difference equations the same general type of trichotomies.…”

“…In the mentioned papers the authors use two applications of the Banach's fixed point theorem, the first one to obtain the solutions of the perturbed equation along the stable/center direction and the other to obtain the solutions of the perturbed equation in the other directions. In this paper, as in [11], we use only one application of the Banach's fixed point theorem to obtain the solutions of the perturbed equation in all directions.…”

“…[11, Lemma 5.1]). If σ and ω are positive numbers such that σ + ω < 1/2, then there are C ∈ ]1, 2[ and D ∈ ]0, 1[ such that…”