Skew-product Semiflows Research Articles

Skew-product attractors of non-autonomous Caputo fractional differential equations.

A non-autonomous Caputo fractional differential equation (FDE) of order α∈(0,1) in Rd with a driving system on a compact base space P is shown to generate a skew-product semi-flow on Cα×P, where Cα is the space of continuous functions f:R+→Rd with a weighted norm giving uniform convergence on compact time subsets. This skew-product semi-flow is then shown to have a bounded and closed attractor when the vector field of the Caputo FDE satisfies a uniform dissipativity condition. It attracts bounded sets of constant initial functions f in here Cα. The properties and structure of this attractor in Cα×P are also discussed.

Zabczyk Type Criteria for Asymptotic Behavior of Dynamical Systems and Applications

AbstractWe obtain new characterizations for nonuniform and uniform asymptotic behaviors of variational systems in infinite dimensional spaces, following the line of studies recently developed in Dragičević et al. (J Differ Equ 268:4786–4829, 2020, J Dyn Differ Equ 34:1107–1137, 2022, Appl Math Comput 414:1–22, 2022, J Math Anal Appl 515:1–37, 2022). First, we give complete descriptions for both nonuniform and uniform stability by means of some nonuniform conditions of convergence for series of suitable nonlinear trajectories. We apply our criteria to present a novel method of exploring the robustness of the nonuniform exponential stability under additive and multiplicative perturbations and we also deduce some consequences for the robustness of the uniform exponential stability. After that, we obtain characterizations for nonuniform and uniform instability in terms of certain nonuniform conditions imposed to the sums of some well-chosen series of nonlinear trajectories. We apply our results to provide a new technique of exploring the robustness of the nonuniform exponential instability under additive and multiplicative perturbations and discuss the consequences for uniform exponential instability. Hence, for both stability and instability we generalize the previous Zabczyk type results and extend their applicability. Next, as another class of applications, we provide nonuniform criteria of Rolewicz type for stability and instability of skew-product semiflows. Thus, we extend the Zabczyk–Rolewicz type methods in two directions: by giving local conditions for global behaviors and by employing specific nonuniform boundedness conditions. Our criteria can be applied to broad classes of discrete and continuous variational dynamical systems.

Attractors in almost periodic Nicholson systems and some numerical simulations

The existence of a global attractor is proved for the skew-product semiflow induced by almost periodic Nicholson systems and new conditions are given for the existence of a unique almost periodic positive solution which exponentially attracts every other positive solution. Besides, some numerical simulations are included to illustrate our results in some concrete Nicholson systems.

The exponential ordering for nonautonomous delay systems with applications to compartmental Nicholson systems

The exponential ordering is exploited in the context of nonautonomous delay systems, inducing monotone skew-product semiflows under less restrictive conditions than usual. Some dynamical concepts linked to the order, such as semiequilibria, are considered for the exponential ordering, with implications for the determination of the presence of uniform persistence or the existence of global attractors. Also, some important conclusions on the long-term dynamics and attraction are obtained for monotone and sublinear delay systems for this ordering. The results are then applied to almost periodic Nicholson systems and new conditions are given for the existence of a unique almost periodic positive solution which asymptotically attracts every other positive solution.

Dynamical behavior of almost periodically forced neutral delayed equation and its applications

In this paper, we consider a class of almost periodically forced neutral delayed equation, which arises from population model with delays. A threshold parameter in terms of basic reproduction ratio R 0 R_0 is introduced into this neutral system. We derive the strongly subhomonogenous property of skew-product semiflow generated by the linearized neutral system under the assumptions of non-neutral case. We show that the positive almost periodic solution is globally stable by applying the approach of monotone skew-product semiflow. Finally, as a classical example, we illustrate the asymptotic behavior of Nicholson model with neutral type delays by using of the new theoretical results. The dynamical behaviors of neutral delayed equation forced by almost periods in this paper cover automatically some known ones of the non-neutral cases.

Nonuniform input-output criteria for exponential expansiveness of discrete dynamical systems and applications

The aim of this paper is to present a new and complete study based on input-output methods for exponential expansiveness of discrete variational systems, treating both the nonuniform behavior and the uniform one. We continue the line of studies initiated in Dragičević et al. (2020) [19] and Dragičević et al. (2022) [20], providing a new approach in exploring the expansiveness properties. We associate to a discrete variational system a control type system that consists of a family of input-output discrete-time equations defined between two sequence spaces whose norms are translations invariant and we introduce new concepts of exact admissibility that rely on nonuniform local conditions. First, by imposing some minimal hypotheses on the input space or respectively on the output space, we establish connections between the exact admissibilities and the nonuniform exponential expansiveness. Moreover, we give for the first time a complete description of the structure of exactly admissible pairs that can be considered to explore nonuniform expansive behaviors in the discrete variational case. In addition, all the hypotheses on the classes from which we choose the input or the output spaces are motivated by illustrative examples. Next, we present characterizations for uniform exponential expansiveness via certain nonuniform exact admissibilities and we generalize the recent criteria for expansiveness of dynamical systems from Dragičević et al. (2020) [19]. Furthermore, we show via relevant examples that the results are the most general and the working hypotheses cannot be weakened or removed. As application, we give general criteria for uniform exponential expansiveness of skew-product semiflows in terms of nonuniform admissibility properties with sequence spaces invariant under translations.

Almost automorphy of minimal sets for C1${C}^{1}$‐smooth strongly monotone skew‐product semiflows on Banach spaces

We focus on the presence of almost automorphy in strongly monotone skew-product semiflows on Banach spaces. Under the $C^1$-smoothness assumption, it is shown that any linearly stable minimal set must be almost automorphic. This extends the celebrated result of Shen and Yi [Mem. Amer. Math. Soc. 136(1998), No. 647] for the classical $C^{1,\alpha}$-smooth systems. Based on this, one can reduce the regularity of the almost periodically forced differential equations and obtain the almost automorphic phenomena in a wider range.

Input-output criteria for stability and expansiveness of dynamical systems

The aim of the present paper is to provide new criteria for the asymptotic behaviors of dynamical systems, approaching simultaneously four directions: nonuniform, uniform stability and respectively nonuniform, uniform expansiveness, treating both discrete and continuous-time case and using computational type methods. In order to characterize the asymptotic properties of discrete variational systems, we consider two classes of weighted ℓ∞-spaces and we define two types of nonuniform input-output properties, by employing spaces from these classes in the admissible pair. Our study continues the line initiated in Dragičević, Sasu and Sasu [J. Differential Equations 268 (2020), 4786-4829] and [J. Dynam. Differential Equations, doi.org/10.1007/s10884-020-09918-4], giving a completely new perspective regarding the connections between admissibility techniques for stability and expansiveness. We obtain necessary and sufficient conditions for nonuniform, uniform exponential stability and also for nonuniform, uniform exponential expansiveness, providing a unified approach and emphasizing the key differences when passing from nonuniform to uniform behavior. Next, we present two categories of applications of the discrete-time results. First, we deduce criteria for stability and expansiveness in terms of input-output type properties along period orbits. After that, we obtain characterizations for uniform exponential stability and uniform exponential expansiveness of skew-product semiflows in terms of the solvability of an associated integral control system between well-chosen weighted spaces of continuous functions, pointing out new computational methods of recovering the continuous information from the discrete-time behavior in the framework of nonuniform input-output techniques. Our results are applicable to general classes of variational systems.

Morse decompositions of uniform random attractors

In this paper, we study the Morse theory on uniform random attractors for non-autonomous random dynamical systems. To handle the non-invariance of uniform random attractors, we construct the Morse sets of uniform random attractor as the projections of the Morse sets on pullback attractor of the skew product semiflow. First, we obtain the Morse decomposition of uniform random attractors in probability. Second, we describe a decaying energy level on such attractor by Lyapunov function with probability one. Furthermore, we study the stability of Morse decompositions of deterministic uniform attractor under a small random disturbance. Finally, we discuss and collect the results on the Morse decomposition under random setting.

Stage-structured hematopoiesis model with delays in an almost periodic environment

In this article, we focus on stage-structured hematopoiesis model with time-dependent delays in an almost periodic environment. A threshold dynamic is characterized by basic reproduction ratio R0. By making use of skew-product semiflow approach, we show that the population is extinct if R01. The existence of a unique positive almost periodic solution is derived when R0>1, meanwhile, the global stability of the solution is verified under weaker conditions.

Non-autonomous scalar linear-dissipative and purely dissipative parabolic PDEs over a compact base flow

In this paper a family of non-autonomous scalar parabolic PDEs over a general compact and connected flow is considered. The existence or not of a neighbourhood of zero where the problems are linear has an influence on the methods used and on the dynamics of the induced skew-product semiflow. That is why two cases are distinguished: linear-dissipative and purely dissipative problems. In both cases, the structure of the global and pullback attractors is studied using principal spectral theory. Besides, in the purely dissipative setting, a simple condition is given, involving both the underlying linear dynamics and some properties of the nonlinear term, to determine the nontrivial sections of the attractor.

Monotone skew-Product Semiflows for Carathéodory Differential Equations and Applications

The first part of the paper is devoted to studying the continuous dependence of the solutions of Carath\'eodory constant delay differential equations where the vector fields satisfy classical cooperative conditions. As a consequence, when the set of considered vector fields is invariant with respect to the time-translation map, the continuity of the respective induced skew-product semiflows is obtained. These results are important for the study of the long-term behavior of the trajectories. In particular, the construction of semicontinuous semiequilibria and equilibria is extended to the context of ordinary and delay Carath\'eodory differential equations. Under appropriate assumptions of sublinearity, the existence of a unique continuous equilibrium, whose graph coincides with the pullback attractor for the evolution processes, is shown. The conditions under which such a solution is the forward attractor of the considered problem are outlined. Two examples of application of the developed tools are also provided.

Forwards attraction properties in scalar non-autonomous linear–dissipative parabolic PDEs. The case of null upper Lyapunov exponent

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 neighbourhood of zero and have null upper Lyapunov exponent. Besides, the notion of Li–Yorke chaotic pullback attractor for a process is introduced, and we prove that this chaotic behaviour might occur for almost all the processes. When the problems are additionally sublinear, more cases of forwards attraction are found, which had not been previously considered even in the case of linear–dissipative ODEs.

Lyapunov Type Theorems for Exponential Stability of Linear Skew-Product Three-Parameter Semiflows with Discrete Time

For linear skew-product three-parameter semiflows with discrete time acting on an arbitrary Hilbert space, we obtain a complete characterization of exponential stability in terms of the existence of appropriate Lyapunov functions. As a nontrivial application of our work, we prove that the notion of an exponential stability persists under sufficiently small linear perturbations.

Local limit theorems for suspended semiflows

We prove local limit theorems for a cocycle over a semiflow by establishing topological, mixing properties of the associated skew product semiflow. We also establish conditional rational weak mixing of certain skew product semiflows and various mixing properties including order 2 rational weak mixing of hyperbolic geodesic flows on the tangent spaces of cyclic covers.

Almost Automorphically and Almost Periodically Forced Circle Flows of Almost Periodic Parabolic Equations on $$S^1$$

We consider the skew-product semiflow which is generated by a scalar reaction–diffusion equation where f is uniformly almost periodic in t. The structure of the minimal set M is thoroughly investigated under the assumption that the center space $$V^c(M)$$ associated with M is no more than 2-dimensional. Such situation naturally occurs while, for instance, M is hyperbolic or uniquely ergodic. It is shown in this paper that M is a 1-cover of the hull H(f) provided that M is hyperbolic (equivalently, $$\mathrm{dim}V^c(M)=0$$ ). If $$\mathrm{dim}V^c(M)=1$$ (resp. $$\mathrm{dim}V^c(M)=2$$ with $$\mathrm{dim}V^u(M)$$ being odd), then either M is an almost 1-cover of H(f) and topologically conjugate to a minimal flow in $${\mathbb {R}}\times H(f)$$ ; or M can be (resp. residually) embedded into an almost periodically (resp. almost automorphically) forced circle-flow $$S^1\times H(f)$$ . When $$f(t,u,u_x)=f(t,u,-u_x)$$ (which includes the case $$f=f(t,u)$$ ), it is proved that any minimal set M is an almost 1-cover of H(f). In particular, any hyperbolic minimal set M is a 1-cover of H(f). Furthermore, if $$\mathrm{dim}V^c(M)=1$$ , then M is either a 1-cover of H(f) or is topologically conjugate to a minimal flow in $${\mathbb {R}}\times H(f)$$ . For the general spatially-dependent nonlinearity $$f=f(t,x,u,u_{x})$$ , we show that any stable or linearly stable minimal invariant set M is residually embedded into $${\mathbb {R}}^2\times H(f)$$ .

Equilibrium-Like Solutions of Asymptotically Autonomous Differential Equations

We analyze the chain recurrent set of skew product semiflows obtained from nonautonomous differential equations—ordinary differential equations or semilinear parabolic differential equations. For many gradient-like dynamical systems, Morse–Smale dynamical systems e.g., the chain recurrent set contains only isolated equilibria. The structure in the asymptotically autonomous setting is richer but still close to the structure of a Morse–Smale dynamical system. The main tool used in this paper is a nonautonomous flavour of Conley index theory developed by the author. We will see that for a class of good equations, the Conley index can be understood in terms of equilibria (in a generalized meaning) and their connections. This allows us to find specific solutions of asymptotically autonomous equations and generalizes properties of Morse–Smale dynamical systems to the asymptotically autonomous setting.

Individual exponential stability for linear skew-products semiflows over a semiflow

In this paper we consider individual stability for linear skew-product semiflows over semiflows by using the standard Perron’s method. To our knowledge, in the study of this problem, the constant from the boundedness theorem has been so far always assumed to be independent of the variable from the metric space. The novelty of our approach consists in being able to choose this constant for each value of the metric space variable in a separate manner.

Topologies of continuity for Carathéodory delay differential equations with applications in non-autonomous dynamics

We study some already introduced and some new strong and weak topologies of integral type to provide continuous dependence on continuous initial data for the solutions of non-autonomous Carath\'eodory delay differential equations. As a consequence, we obtain new families of continuous skew-product semiflows generated by delay differential equations whose vector fields belong to such metric topological vector spaces of Lipschitz Carath\'eodory functions. Sufficient conditions for the equivalence of all or some of the considered strong or weak topologies are also given. Finally, we also provide results of continuous dependence of the solutions as well as of continuity of the skew-product semiflows generated by Carath\'eodory delay differential equations when the considered phase space is a Sobolev space.

On uniform exponential trisplitting for cocycles of linear operators in Banach spaces

Abstract The aim of this paper is to study the concept of uniform exponential trisplitting for skew-product semiflow in Banach spaces. This concept is a generalisation of the well-known concept of uniform exponential trichotomy. We obtain necessary and sufficient conditions for this concept of Datko’s type. a character-isation in terms of Lyapunov functions is provided. The results are obtained from the point of view of the projector families, i.e. invariant and strongly invariant.