# J. A. LangaUniversidad de Sevilla | US · Differential Equations and Numerical Analysis

J. A. Langa

Professor

## About

149

Publications

13,834

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3,431

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Citations since 2016

Introduction

José A. Langa received a degree in mathematics in 1994, and a European Doctorate in mathematics in 1998, from Seville University. He received a prize for best Ph.D. thesis in mathematics 1998, and a prize of the Academia Sevillana de Ciencias for best young researcher 2003. He is Full Professor at the Department of Differential Equations and Numerical Analysis, Seville University. Spain.

Additional affiliations

September 1994 - January 2017

Education

September 1989 - September 1994

**Seville University**

Field of study

- Mathematics

## Publications

Publications (149)

In this paper we study in detail the structure of the global attractor for a generalized Lotka-Volterra system with Volterra--Lyapunov stable structural matrix. We provide the full characterization of this structure and we show that it coincides with the invasion graph as recently introduced in [15]. We also study the stability of the structure wit...

The self-organising global dynamics underlying brain states emerge from complex recursive nonlinear interactions between interconnected brain regions. Until now, most efforts of capturing the causal mechanistic generating principles have supposed underlying stationarity, being unable to describe the non-stationarity of brain dynamics, i.e. time-dep...

In this work, we study the 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 the existence and permanence of unstable sets of hyperbolic solutions. Then, we use this to establish the lower sem...

The model transform fits exactly the parameters of a suitable model to empirical or simulated data in each point in time and/or space. We describe several examples of concrete model transforms and their applications. The model transform allows simple theoretical models to be applied to complex empirical systems in each short interval of time or/and...

Dynamical systems on graphs allow to describe multiple phenomena from different areas of Science. In particular, many complex systems in Ecology are studied by this approach. In this paper we analize the mathematical framework for the study of the structural stability of each stationary point, feasible or not, introducing a generalization for this...

Finite-dimensional attractors play an important role in finite-dimensional reduction of PDEs in mathematical modelization and numerical simulations. For non-autonomous random dynamical systems, Cui and Langa (J Differ Equ, 263:1225–1268, 2017) developed a random uniform attractor as a minimal compact random set which provides a certain description...

In this work we will study the structure of the skew-product attractor for a planar diffusively coupled ordinary differential equation, given by \begin{document}$ \dot{x} = k(y-x)+x-\beta(t)x^3 $\end{document} and \begin{document}$ \dot{y} = k(x-y)+y-\beta(t)y^3 $\end{document}, \begin{document}$ t\geq 0 $\end{document}. We identify the non-autonom...

In this work we study nonuniform exponential dichotomies and existence of pullback and forward attractors for evolution processes associated to nonautonomous differential equations. We define a new concept of nonuniform exponential dichotomy, for which we provide several examples, study the relation with the standard notion, and establish a robustn...

We give a simple proof of a result due to Mañé (1981) [17] that a compact subset A of a Banach space that is negatively invariant for a map S is finite-dimensional if DS(x)=C(x)+L(x), where C is compact and L is a contraction (and both are linear). In particular, we show that if S is compact and differentiable then A is finite-dimensional. We also...

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...

The self-organising global dynamics underlying brain states emerge from complex recursive nonlinear interactions between interconnected brain regions. Until now, all efforts of capturing the causal mechanistic generating principles have proven elusive, since they have been unable to describe the non-stationarity of brain dynamics, i.e. time-depende...

The self-organising global dynamics underlying brain states emerge from complex recursive nonlinear interactions between interconnected brain regions. Until now, all efforts of capturing the causal mechanistic generating principles have proven elusive, since they have been unable to describe the non-stationarity of brain dynamics, i.e. time-depende...

The self-organising global dynamics underlying brain states emerge from complex recursive nonlinear interactions between interconnected brain regions. Until now, all efforts of capturing the causal mechanistic generating principles have proven elusive, since they have been unable to describe the non-stationarity of brain dynamics, i.e. time-depende...

In this work we study Morse–Smale semigroups under nonautonomous perturbations, which leads us to introduce the concept of Morse–Smale evolution processes of hyperbolic type, associated to nonautonomous evolutionary equations. They are amongst the dynamically gradient evolution processes (in the sense of Carvalho et al., in: Applied Mathematical Sc...

Brain dynamics depicts an extremely complex energy landscape that changes over time, and its characterisation is a central unsolved problem in neuroscience. We approximate the non-stationary landscape sustained by the human brain through a novel mathematical formalism that allows us characterise the attractor structure, i.e. the stationary points a...

In this paper, we study stability properties of nonuniform hyperbolicity for evolution processes associated with differential equations in Banach spaces. We prove a robustness result of nonuniform hyperbolicity for linear evolution processes, that is, we show that the property of admitting a nonuniform exponential dichotomy is stable under perturba...

The aim of this paper is to find an upper bound for the fractal dimension of uniform attractors in Banach spaces. The main technique we employ is essentially based on a compact embedding of some auxiliary Banach space into the phase space and a corresponding smoothing effect between these spaces. Our bounds on the fractal dimension of uniform attra...

Dynamical systems on graphs allow to describe multiple phenomena from different areas of Science. In particular, many complex systems in Ecology are studied by this approach. In this paper we analize the mathematical framework for the study of the structural stability of each stationary point, feasible or not, introducing a generalization for this...

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 nonlin...

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 nonline...

In this paper, we study stability properties of nonuniform hyperbolicity for evolution processes associated with differential equations in Banach spaces. We prove a robustness result of nonuniform hyperbolicity for linear evolution processes, that is, we show that the property of admitting a nonuniform exponential dichotomy is stable under perturba...

This book provides a comprehensive study of how attractors behave under perturbations for both autonomous and non-autonomous problems. Furthermore, the forward asymptotics of non-autonomous dynamical systems is presented here for the first time in a unified manner.
When modelling real world phenomena imprecisions are unavoidable. On the other hand...

The book treats the theory of attractors for non-autonomous dynamical systems. The aim of the book is to give a coherent account of the current state of the theory, using the framework of processes to impose the minimum of restrictions on the nature of the non-autonomous dependence.
The book is intended as an up-to-date summary of the field, but m...

The dynamical activity of the human brain describes an extremely complex energy landscape changing over time and its characterisation is central unsolved problem in neuroscience. We propose a novel mathematical formalism for characterizing how the landscape of attractors sustained by a dynamical system evolves in time. This mathematical formalism i...

As it is well-known, the forwards and pullback dynamics are in general unrelated. In this paper we present an in-depth study of whether the pullback attractor is also a forwards attractor for the processes involved with the skew-product semiflow induced by a family of scalar non-autonomous reaction-diffusion equations which are linear in a neighbou...

In this paper we establish a strong comparison principle for a nonautonomous differential inclusion with a forcing term of Heaviside type. Using this principle, we study the structure of the global attractor in both the autonomous and nonautonomous cases. In particular, in the last case we prove that the pullback attractor is confined between two s...

Informational Structures (IS) and Informational Fields (IF) have been recently introduced to deal with a continuous dynamical systems-based approach to Integrated Information Theory (IIT). IS and IF contain all the geometrical and topological constraints in the phase space. This allows one to characterize all the past and future dynamical scenarios...

Background and objective:
Knowing whether a subject is conscious or not is a current challenge with a deep potential clinical impact. Recent theoretical considerations suggest that consciousness is linked to the complexity of distributed interactions within the corticothalamic system. The fractal dimension (FD) is a quantitative parameter that has...

In order to obtain the measurability of a random attractor, the RDS is usually required to be continuous which, however, is hard to verify in many applications. In this paper, we introduce a quasi strong-to-weak (abbrev. quasi-S2W) continuity and establish a new existence theorem for random attractors. It is shown that such continuity is equivalent...

We consider the Rayleigh–Bénard problem for the two-dimensional Boussinesq system for the micropolar fluid. Our main goal is to compare the value of the critical Rayleigh number, and estimates of the Nusselt number and the fractal dimension of the global attractor with those values for the same problem for the classical Navier–Stokes system. Our es...

Author summary
In this paper we introduce a space-time continuous version for the level of integrated information of a network on which a dynamics is defined. The concept of integrated information comes from the IIT of consciousness. By a strict mathematical formulation, we complement the existing IIT theoretical framework from a dynamical systems...

In this paper we study cocycle attractors, pullback attractors and uniform attractors for multi-valued non-autonomous dynamical systems. We first consider the relationship between the three attractors and find that, under suitable conditions, they imply each other. Then, for generalized dynamical systems, we find that these attractors can be charac...

In this paper we study topological structural stability for a family of nonlinear semigroups \(T_h(\cdot )\) on Banach space \(X_h\) depending on the parameter h. Our results shows the robustness of the internal dynamics and characterization of global attractors for projected Banach spaces, generalizing previous results for small perturbations of p...

In this paper, we study the squeezing property and finite dimensionality of cocycle attractors for non-autonomous dynamical systems (NRDS). We show that the generalized random cocycle squeezing property (RCSP) is a sufficient condition to prove a determining modes result and the finite dimensionality of invariant non-autonomous random sets, where t...

We study the long-time behavior of porous-elastic system, focusing on the interplay between nonlinear damping and source terms. The sources may represent restoring forces, but may also be focusing thus potentially amplifying the total energy which is the primary scenario of interest. By employing nonlinear semigroups and the theory of monotone oper...

In this work we prove the lower and upper semicontinuity of pullback, uniform, and cocycle attractors for the non-autonomous dynamical system given by hyperbolic equation on a bounded domain Ω⊂R3ϵutt+ut−Δu=fϵ(t,u).
For each ϵ>0, this equation has uniform, pullback, and cocycle attractors in H01(Ω)×L2(Ω) and for ϵ=0 the limit parabolic equationut−Δu...

A mathematical system of differential equations for the modelization of mutualistic networks in Ecology has been proposed in Bastolla et al. (2007). Basically, it is studied how the complex structure of cooperation interactions between groups of plants and pollinators or seed dispersals affects to the whole network. In this paper we prove existence...

In this paper, for non-autonomous RDS we study cocycle attractors with autonomous attraction universes, i.e. pullback attracting some autonomous random sets, instead of non-autonomous ones. We first compare cocycle attractors with autonomous and non-autonomous attraction universes, and then for autonomous ones we establish some existence criteria a...

This paper is devoted to establishing a (random) uniform attractor theory for non-autonomous random dynamical systems (NRDS). The uniform attractor is defined as the minimal compact uniformly pullback attracting random set. Nevertheless, the uniform pullback attraction in fact implies a uniform forward attraction in probability, and implies also an...

The global attractor of a skew product semiflow for a non-autonomous differential equation describes the asymptotic behaviour of the model. This attractor is usually characterized as the union, for all the parameters in the base space, of the associated cocycle attractors in the product space. The continuity of the cocycle attractor in the paramete...

In this paper we prove the equivalence between equi-attraction and continuity of attractors for skew-product semi-flows, and equi-attraction and continuity of uniform and cocycle attractors associated to non-autonomous dynamical systems. To this aim proper notions of equi-attraction have to be introduced in phase spaces where the driving systems de...

Real phenomena from different areas of Life Sciences can be described by complex networks, whose structure is usually determining their intrinsic dynamics. On the other hand, Dynamical Systems Theory is a powerful tool for the study of evolution processes in real situations. The concept of global attractor is the central one in this theory. In the...

The structure of attractors for differential equations is one of the main topics in the qualitative theory of dynamical systems. However, the theory is still in its infancy in the case of multivalued dynamical systems. In this paper we study in detail the structure and internal dynamics of a scalar differential equation, both in the autonomous and...

In this paper, we study the pullback attractor for a general reaction–diffusion system for which the uniqueness of solutions is not assumed. We first establish some general results for a multi-valued dynamical system to have a bi-spatial pullback attractor, and then we find that the attractor can be backwards compact and composed of all the backwar...

In this paper, we describe in detail the global and cocycle attractors related to nonautonomous scalar differential equations with diffusion. In particular, we investigate reaction-diffusion equations with almost-periodic coefficients. The associated semiflows are strongly monotone which allow us to give a full characterization of the cocycle attra...

In this work we study the topological structural stability for a family of nonlinear semigroups Th(·) on Banach spaces Xh which dependent on a parameter h.

In this paper we describe some dynamical properties of a Morse decomposition with a countable number of sets. In particular, we are able to prove that the gradient dynamics on Morse sets together with a separation assumption is equivalent to the existence of an ordered Lyapunov function associated to the Morse sets and also to the existence of a Mo...

This review paper treats the dynamics of non-autonomous dynamical systems. To study the forwards asymptotic behaviour of a non-autonomous differential equation we need to analyse the asymptotic configurations of the non-autonomous terms present in the equations. This fact leads to the definition of concepts such as skew-products and cocycles and th...

This review paper treats the dynamics of non-autonomous dynam-ical systems. To study the forwards asymptotic behaviour of a non-autonomous differential equation we need to analyse the asymptotic configurations of the non-autonomous terms present in the equations. This fact leads to the definition of concepts such as skew-products and cocycles and t...

In this work we study the continuity and structural stability of the uniform attractor associated with non-autonomous perturbations of differential equations. By a careful study of the different definitions of attractor in the non-autonomous framework, we introduce the notion of lifted-invariance on the uniform attractor, which becomes compatible w...

In this paper we study a three dimensional mutualistic model of two plants in competition and a pollinator with cooperative relation with plants. We compare the dynamical properties of this system with the associated one under absence of the pollinator. We observe how cooperation is a common fact to increase biodiversity, which it is known that, ge...

This paper is devoted to the investigation of the dynamics of non-autonomous differential equations. The description of the asymptotic dynamics of non-autonomous equations lies on dynamical structures of some associated limiting non-autonomous – and autonomous – differential equations (one for each global solution in the attractor of the driving se...

We define (time dependent) Morse-decompositions for non-autonomous evolution processes (non-autonomous dynamical systems) and prove that a non-autonomous gradient-like evolution process possesses a Morsedecomposition on the associated pullback attractor. We also prove the existence of an associated Lyapunov function which describes the gradient beh...

In this appendix we discuss another approach to the asymptotic dynamics of non-autonomous equations, the uniform attractor, which was developed by Chepyzhov and Vishik (2002) [see also the appendix in the book by Vishik (1992)]. Reinterpreted in the language of processes, the uniform attractor is the minimal fixed (time-independent) compact subset...

As our final example we take the non-autonomous damped wave equation $${u}_{tt} + \beta (t){u}_{t} = \Delta u + f(u).$$ Such an equation provides, in the autonomous case, a fairly canonical example in which the semigroup is not compact, and the use of asymptotic compactness is required [see Temam (1988), for example]. There are, of course, similar...

We have already seen that the structure of the attractor of an autonomous gradient semigroup can be completely described: it is given by the union of the unstable sets of the equilibria (Theorem 2.43). However, key to the definition of a gradient semigroup (Definition 2.38) is the existence of a Lyapunov function, and this is a very delicate matter...

In this chapter we develop the existence theory for pullback attractors in a way that recovers well known results for the global attractors of autonomous systems as a particular case (see, for example, Babin and Vishik 1992; Chepyzhov and Vishik 2002;Cholewa and Dlotko 2000; Chueshov 1999; Hale 1988; Ladyzhenskaya 1991; Robinson 2001; Temam 1988).

The global attractor, whose well established definition we recall below, is an object that captures the asymptotic behaviour of autonomous systems. The aim of this chapter is to introduce the ‘pullback attractor’, which seems to be the correct generalisation of this concept for use with non-autonomous processes. We pay particular attention to how t...

A Morse decomposition of a global attractor describes its internal dynamics, i.e., the dynamics on invariant compact sets in the attractor and the connections between them. When we deal with non-autonomous dynamical systems, the concept of pullback attractor for the associated skew-product flow appears as a powerful tool to analyze the asymptotic b...

In this chapter we consider a parabolic problem in which the diffusion coefficient depends on a parameter, $${u}_{t} - {({a}_{\epsilon }(x){u}_{x})}_{x} = f(u).$$ The final goal in this direction would be to compare the asymptotic dynamics of systems with different ‘parameter values’ by comparing their attractors and the flow on them. We assume tha...

In this chapter we study general non-autonomous delay differential equations of the form $$\dot{x}(t) = F(t,x(t),x(t - \rho (t))).$$ Our intention is to demonstrate how pullback attractors can be used to investigate the behaviour of such models. In particular, following the ideas in the preceding chapters we are able to compare the dynamics of syst...

In this chapter we consider the asymptotic dynamics of parabolic problems of the form $$\begin{array}{rcl}{ u}_{t} -\mbox{ div}(a(x)\nabla u) + c(x)u& =& f(x,t,u),\quad \mbox{ in}\quad \Omega, \\ u& =& 0,\quad \mbox{ on}\quad \partial \Omega, \end{array}$$ (12.1) where N is a positive integer, \(\Omega \subset {\mathbb{R}}^{N}\) is a bounded domain...

For the majority of this chapter we study the continuity under perturbation of hyperbolic global solutions and their stable and unstable manifolds, for an abstract process S( ⋅, ⋅) on a Banach space X. Such results are the main ingredient required to apply the lower semicontinuity results for global and pullback attractors like Theorems 3.8 and 3.1...

The two-dimensional incompressible Navier–Stokes equations provide one of the canonical examples of an infinite-dimensional dynamical system. In this chapter we illustrate the results of Chaps. 2 and 4 by driving the dynamics with a non-autonomous forcing term. With such an equation, which has no clear underlying structure (like a Lyapunov function...

In this chapter we study the continuity of attractors under perturbation. The problems of upper semicontinuity (‘no explosion’) and lower semicontinuity (‘no implosion’) are distinct, and we will treat them separately. Broadly speaking, one expects upper semicontinuity to hold widely, but lower semicontinuity requires structural assumptions on the...

In this chapter we study the existence and characterisation of pullback attractors for a non-autonomous version of the Chafee–Infante equation on the domain (0, π), $${u}_{t} - {u}_{xx} = \lambda u - b(t){u}^{3},\qquad u(0,t) = u(\pi ,t) = 0,$$ (13.1) when there exist 0 < b0 < B0 such that $$0 < {b}_{0} \leq b(t) \leq {B}_{0}.$$ Theorem 12.1 guaran...

As our first extended example we will consider a non-autonomous Lotka–Volterra model, $$\begin{array}{rcl} \dot{u}& = u(\lambda (t) - au - bv)& \\ \dot{v}& = v(\mu - cu - dv), &\end{array}$$ (9.1) where the parameters a, b,c,d, and μ are positive, ad>bc, and 0t)≤Λ. In line with the interpretation of this model in terms of the numbers of two competi...

The notion of hyperbolicity plays a fundamental role in the study of autonomous dynamical systems and provides the main assumption in some of the most significant results on their fine structure. The equivalent notion for non-autonomous dynamical systems is that of an exponential dichotomy, and we spend this chapter analysing this important concept...

In this chapter we investigate the dimension of attractors for autonomous and non-autonomous problems. The treatment is necessarily abstract since application of the results generally makes use of certain differentiability properties that need to be checked carefully in each particular application. We will apply the results of this chapter to an ab...

In this chapter we consider the local and global well-posedness of abstract non-autonomous semilinear problems of the form $$\dot{x} = -Ax + f(t,x)\qquad x({t}_{0}) = {x}_{0}

The pullback attractor.- Existence results for pullback attractors.- Continuity of attractors.- Finite-dimensional attractors.- Gradient semigroups and their dynamical properties.- Semilinear Differential Equations.- Exponential dichotomies.- Hyperbolic solutions and their stable and unstable manifolds.- A non-autonomous competitive Lotka-Volterra...

This paper is dedicated to estimate the fractal dimension of exponential global attractors of some generalized gradient-like semigroups in a general Banach space in terms of the maximum of the dimension of the local unstable manifolds of the isolated invariant sets, Lipschitz properties of the semigroup and the rate of exponential attraction. We al...

In this paper we study the continuity of invariant sets for nonautonomous infinite-dimensional dynamical systems under singular perturbations. We extend the existing results on lower-semicontinuity of attractors of autonomous and nonautonomous dynamical systems. This is accomplished through a detailed analysis of the structure of the invariant sets...