arXiv subjects

José A. Langa

Publications and source records attributed to José A. Langa.

6 recordsLinked to original sources

arXiv2024

Monotone Multivalued Nonautonomous Dynamical Systems

This paper is devoted to the study of nonautonomous multivalued semiflows and their associated pullback attractors. For this kind of dynamical systems we are able to characterize the upper and lower bounds of the attractor as complete trajectories belonging to the attractor, so that all the internal dynamics is confined in this region, which can be described as an interval due to the orderly nature of the processes. Thus, we are able to generalize to this framework previous general results in literature for autonomous multivalued flows or nonautonomous differential equations. We apply our results to a partial differential inclusion with a nonautonomous term, also proving the upper semicontinuity dependence of pullback and global attractors when the time dependent term asymptotically converges to an autonomous multivalued term.

math.DS↗

arXiv2024

Finite Fractal Dimension of uniform attractors for non-autonomous dynamical systems with infinite dimensional symbol space

The aim of this paper is to find an upper bound for the box-counting dimension of uniform attractors for non-autonomous dynamical systems. Contrary to the results in literature, we do not ask the symbol space to have finite box-counting dimension. Instead, we ask a condition on the semi-continuity of pullback attractors of the system as time goes to infinity. This semi-continuity can be achieved if we suppose the existence of finite-dimensional exponential uniform attractors for the limit symbols. After showing these new results, we apply them to study the box-counting dimension of the uniform attractor for a reaction-diffusion equation, and we find a specific forcing term such that the symbol space has infinite box-counting dimension but the uniform attractor has finite box-counting dimension anyway.

math.DS↗

arXiv2024

Structural stability of invasion graphs for Lotka--Volterra systems

In this paper, we study in detail the structure of the global attractor for the Lotka--Volterra system with a Volterra--Lyapunov stable structural matrix. We consider the invasion graph as recently introduced in [19] and prove that its edges represent all the heteroclinic connections between the equilibria of the system. We also study the stability of this structure with respect to the perturbation of the problem parameters. This allows us to introduce a definition of structural stability in ecology in coherence with the classical mathematical concept where there exists a detailed geometrical structure, robust under perturbation, that governs the transient and asymptotic dynamics.

math.DS↗

arXiv2024

Forwards attractors for non-autonomous Lotka-Volterra cooperative systems: a detailed geometrical description

Non-autonomous differential equations exhibit a highly intricate dynamics, and various concepts have been introduced to describe their qualitative behavior. In general, it is rare to obtain time dependent invariant compact attracting sets when time goes to plus infinity. Moreover, there are only a few papers in the literature that explore the geometric structure of such sets. In this paper we investigate the long time behaviour of cooperative $n$-dimensional non-autonomous Lotka-Volterra systems is population dynamics. We provide sufficient conditions for the existence of a globally stable (forward in time) entire solution in which one species becomes extinct, or where all species except one become extinct. Furthermore, we obtain the precise geometrical structure of the non-autonomous forward attractor in one, two, and three dimensions by establishing heteroclinic connections between the globally stable solution and the semi-stable solutions in cases of species permanence and extinction. We believe that understanding time-dependent forward attractors paves the way for a comprehensive analysis of both transient and long-term behavior in non-autonomous phenomena.

math.DS↗

arXiv2021

Continuity and topological structural stability for nonautonomous random attractors

In this work, we study continuity and topological structural stability of attractors for nonautonomous random differential equations obtained by small bounded random perturbations of autonomous semilinear problems. First, we study existence and permanence of unstable sets of hyperbolic solutions. Then, we use this to establish lower semicontinuity of nonautonomous random attractors and to show that the gradient structure persists under nonautonomous random perturbations. Finally, we apply the abstract results in a stochastic differential equation and in a damped wave equation with a perturbation on the damping.

math.DS↗

arXiv2020

The effect of a small bounded noise on the hyperbolicity for autonomous semilinear differential equations

In this work we study permanence of hyperbolicity for autonomous differential equations under nonautonomous random/stochastic perturbations. For the linear case, we study robustness and existence of exponential dichotomies for nonautonomous random dynamical systems. Next, we establish a result on the persistence of hyperbolic equilibria for nonlinear differential equations. We show that for each nonautonomous random perturbation of an autonomous semilinear problem with a hyperbolic equilibrium there exists a bounded \textit{random hyperbolic solution} for the associated nonlinear nonautonomous random dynamical systems. Moreover, we show that these random hyperbolic solutions converge to the autonomous equilibrium. As an application, we consider a semilinear differential equation with a small nonautonomous multiplicative white noise, and as an example, we apply the abstract results to a strongly damped wave equation.

math.AP↗