Fractal dimension of a random invariant set

AbstractIn recent years many deterministic parabolic equations have been shown to possess global attractors which, despite being subsets of an infinite-dimensional phase space, are finite-dimensional objects. Debussche showed how to generalize the deterministic theory to show that the random attractors of the corresponding stochastic equations have finite Hausdorff dimension. However, to deduce a parametrization of a ‘finite-dimensional’ set by a finite number of coordinates a bound on the fractal (upper box-counting) dimension is required. There are non-trivial problems in extending Debussche's techniques to this case, which can be overcome by careful use of the Poincaré recurrence theorem. We prove that under the same conditions as in Debussche's paper and an additional concavity assumption, the fractal dimension enjoys the same bound as the Hausdorff dimension. We apply our theorem to the 2d Navier–Stokes equations with additive noise, and give two results that allow different long-time states to be distinguished by a finite number of observations

Determining asymptotic behavior from the dynamics on attracting sets

Two tracking properties for trajectories on attracting sets are studied. We prove that trajectories on the full phase space can be followed arbitrarily closely by skipping from one solution on the global attractor to another. A sufficient condition for asymptotic completeness of invariant exponential attractors is found, obtaining similar results as in the theory of inertial manifolds. Furthermore, such sets are shown to be retracts of the phase space, which implies that they are simply connected.Ministerio de Educación y CienciaDepartamento de Ecuaciones Diferenciales y Análisis Numérico (Universidad de Sevilla

Lower semicontinuity of attractors for non-autonomous dynamical systems

This paper is concerned with the lower semicontinuity of attractors for semilinear non-autonomous differential equations in Banach spaces. We require the unperturbed attractor to be given as the union of unstable manifolds of time-dependent hyperbolic solutions, generalizing previous results valid only for gradient-like systems in which the hyperbolic solutions are equilibria. The tools employed are a study of the continuity of the local unstable manifolds of the hyperbolic solutions and results on the continuity of the exponential dichotomy of the linearization around each of these solutions

Upper Semicontinuity of Attractors for Small Random Perturbations of Dynamical Systems

The relationship between random attractors and global attractors for dynamical systems is studied. If a partial differential equation is perturbed by an ²¡small random term and certain hypotheses are satisfied, the upper semicontinuity of the random attractors is obtained as ² goes to zero. The results are applied to the Navier-Stokes equations and a problem of reaction-diffusion type, both perturbed by an additive white noise

Bifurcations in non-autonomous scalar equations

In a previous paper we introduced various definitions of stability and instability for non-autonomous differential equations, and applied these to investigate the bifurcations in some simple models. In this paper we present a more systematic theory of local bifurcations in scalar non-autonomous equations.Royal Society University Research FellowMinisterio de Educación y Cienci

Bifurcation from zero of a complete trajectory for non-autonomous logistic PDEs

In this paper we extend the well-known bifurcation theory for autonomous logistic equations to the non-autonomous equation ut − ∆u = λu − b(t)u 2 with b(t) ∈ [b0, B0], 0 < b0 < B0 < 2b0. In particular, we prove the existence of a unique uniformly bounded trajectory that bifurcates from zero as λ passes through the first eigenvalue of the Laplacian, which attracts all other trajectories. Although it is this relatively simple equation that we analyse in detail, other more involved models can be treated using similar techniques.Ministerio de Educación y CienciaFondo Europeo de Desarrollo Regiona

Stability, instability, and bifurcation phenomena in non-autonomous differential equations

There is a vast body of literature devoted to the study of bifurcation phenomena in autonomous systems of differential equations. However, there is currently no well-developed theory that treats similar questions for the nonautonomous case. Inspired in part by the theory of pullback attractors, we discuss generalisations of various autonomous concepts of stability, instability, and invariance. Then, by means of relatively simple examples, we illustrate how the idea of a bifurcation as a change in the structure and stability of invariant sets remains a fruitful concept in the non-autonomous case.Comisión Interministerial de Ciencia y TecnologíaRoyal Society University Research Fello

Stability and random attractors for a reaction-diffusion equation with multiplicative noise

We study the asymptotic behaviour of a reaction-diffusion equation, and prove that the addition of multiplicative white noise (in the sense of Itˆo) stabilizes the stationary solution x 0. We show in addition that this stochastic equation has a finite-dimensional random attractor, and from our results conjecture a possible bifurcation scenario

Pullback permanence in a non-autonomous competitive Lotka-Volterra model

The goal of this work is to study in some detail the asymptotic behaviour of a non-autonomous Lotka-Volterra model, both in the conventional sense (as t → ∞) and in the “pullback” sense (starting a fixed initial condition further and further back in time). The non-autonomous terms in our model are chosen such that one species will eventually die out, ruling out any conventional type of permanence. In contrast we introduce the notion of “pullback permanence” and show that this property is enjoyed by our model. This is not just a mathematical artifice, but rather shows that if we come across an ecology that has been evolving for a very long time we still expect that both species are represented (and their numbers are bounded below), even if the final fate of one of them is less happy. The main tools in the paper are the theory of attractors for non-autonomous differential equations, the sub-supersolution method and the spectral theory for linear elliptic equations.Royal Society University Research FellowComisión Interministerial de Ciencia y Tecnologí

Forwards and pullback behaviour of a non-autonomous Lotka-Volterra system

Lotka-Volterra systems have been extensively studied by many authors, both in the autonomous and non-autonomous cases. In previous papers the time asymptotic behaviour as t → ∞ has been considered. In this paper we also consider the “pullback” asymptotic behaviour which roughly corresponds to observing a system “now” that has already been evolving for a long time. For a competitive system that is asymptotically autonomous both as t → −∞ and as t → +∞ we show that these two notions of asymptotic behaviour can be very different but are both important for a full understanding of the dynamics. In particular there are parameter ranges for which, although one species dies out as t → ∞, there is a distinguished time-dependent coexistent state that is attracting in the pullback sense.Ministerio de Ciencia y Tecnología (España). Dirección General de Investigación Científica y TécnicaRoyal Society University Research Fello