Multivalued Dynamical Systems Research Articles

Fixed points imply chaos for a class of differential inclusions that arise in economic models

We consider multi-valued dynamical systems with continuous time of the form ẋ ∈ F (x), where F (x) is a set-valued function. Such models have been studied recently in mathematical economics. We provide a definition for chaos, ω-chaos and topological entropy for these differential inclusions that is in terms of the natural R-action on the space of all solutions of the model. By considering this more complicated topological space and its R-action we show that chaos is the ‘typical’ behavior in these models by showing that near any hyperbolic fixed point there is a region where the system is chaotic, ω-chaotic, and has infinite topological entropy.

Global attractor and omega-limit sets structure for a phase-field model of thermal alloys

In this paper, the existence of weak solutions is established for a phase-field model of thermal alloys supplemented with Dirichlet boundary conditions. After that, the existence of global attractors for the associated multi-valued dynamical systems is proved, and the relationship among these sets is established. Finally, we provide a more detailed description of the asymptotic behaviour of solutions via the omega-limit sets. Namely, we obtain a characterization–through the natural stationary system associated to the model–of the elements belonging to the omega-limit sets under suitable assumptions.

On explicit a priori estimates of the joint spectral radius by the generalized Gelfand formula

In various problems of control theory, non-autonomous and multivalued dynamical systems, wavelet theory and other fields of mathematics information about the rate of growth of matrix products with factors taken from some matrix set plays a key role. One of the most prominent quantities characterizing the exponential rate of growth of matrix products is the so-called joint or generalized spectral radius. In the work some explicit a priori estimates for the joint spectral radius with the help of the generalized Gelfand formula are obtained. These estimates are based on the notion of the measure of irreducibility (quasi-controllability) of matrix sets proposed previously by A. Pokrovskii and the author.

Conley index over the base morphism for multivalued discrete dynamical systems

We define an index of Conley type for a certain class of uppersemicontinuous multivalued dynamical systems, using techniques introduced by Mrozek, Reineck and Srzednicki [ The Conley index over a base , Trans. Amer. Math. Soc. 352 (2000), no. 9, 4171–4194] for the index over the base. We give the characterisation of the nontrivial index and present an example, proving that our index detects isolated invariant sets that are not detected by Kaczynski and Mrozek's [ Conley index for discrete multi-valued dynamical systems , Topology Appl. 65 (1995), 83–96] index.

Strong centers of attraction for multi-valued dynamical systems on noncompact spaces

In this article, we introduce the notions of strong centers of attraction for multi-valued dynamical systems on arbitrary metric spaces. And we obtain strong centers of attraction for multi-valued dynamical systems.

Stabilities in multi-valued dynamical systems

For the study of the various stabilities in multi-valued dynamical systems, we obtain some necessary and sufficient conditions of the notions of stability and Lyapunov stability, and investigate the connection between the concepts of attractions and the suitable versions of limit sets. Also we consider the notion of characteristic 0 + in multi-valued dynamical systems and obtain necessary and sufficient conditions for the concept of characteristic 0 +.

Homotopy Conley index for discrete multivalued dynamical systems

We define an index of Conley type for a certain class of upper semicontinuous multivalued dynamical systems. We use the Szymczak functor and apply techniques introduced by Reineck, Mrozek and Srzednicki for the index over the base. Moreover we introduce the notion of the homotopy partial functor for the usc maps. We show that the index possesses Ważewski and homotopy properties. We also give four examples that exhibit the benefits of our index over the cohomological index defined by Mrozek and Kaczyński.

Asymptotic behaviour of monotone multi-valued dynamical systems

We introduce the concept of a monotone multi-valued semi-flow as an order-preserving map. This definition is motivated by the applications in the theory of differential equations without uniqueness of solutions. For an order-preserving multi-valued semi-flow we prove several results on the structure of the global attractor. Some applications to models governed by ordinary differential equations and delay equations with continuous right-hand side are presented. In particular, the abstract results are applied to a biochemical control circuit.

Chain recurrence for multi-valued dynamical systems on noncompact spaces

It is well-known that the chain recurrence set of continuous map f is the complement of union of B ( A ) - A , where A is an attractor and B ( A ) is a basin of A. We prove that this result also holds when f is a compact-valued continuous relation.

Attractors for relations in σ-compact spaces

We develop a theory of discrete multi-valued dynamical systems for attractors and Lyapunov functions. In [Topology Appl. 109 (2001) 201–210], Hurley proved that if X is a locally compact and σ-compact space and the zero set Z of a Lyapunov function is closed, then Z is a weak attractor. Here we extend the above result to the case of compact-valued continuous relation on X.

Pullback Attractors of Nonautonomous and Stochastic Multivalued Dynamical Systems

In this paper we study the existence of pullback global attractors for multivalued processes generated by differential inclusions. First, we define multivalued dynamical processes, prove abstract results on the existence of ω-limit sets and global attractors, and study their topological properties (compactness, connectedness). Further, we apply the abstract results to nonautonomous differential inclusions of the reaction–diffusion type in which the forcing term can grow polynomially in time, and to stochastic differential inclusions as well.

Attractors for Nonautonomous Multivalued Evolution Systems Generated by Time‐Dependent Subdifferentials

In a real separable Hilbert space, we consider nonautonomous evolution equations including time‐dependent subdifferentials and their nonmonotone multivalued perturbations. In this paper, we treat the multivalued dynamical systems associated with time‐dependent subdifferentials, in which the solution is not unique for a given initial state. In particular, we discuss the asymptotic behaviour of our multivalued semiflows from the viewpoint of attractors. In fact, assuming that the time‐dependent subdifferential converges asymptotically to a time‐independent one (in a sense) as time goes to infinity, we construct global attractors for nonautonomous multivalued dynamical systems and its limiting autonomous multivalued dynamical system. Moreover, we discuss the relationship between them.

Conley index for set-valued maps: from theory to computation

Recent results on the Conley index theory for discrete multi-valued dynamical systems with their consequences for the computation of the index for representable maps are recapitulated. The terminology is simplified with respect to previous presentations,

Rapid Growth Paths in Multivalued Dynamical Systems Generated by Homogeneous Convex Stochastic Operators

The paper examines dynamical systems generated by convex homogeneous multivalued operators in spaces of random vectors. The primary goal is to investigate the growth rates of random trajectories of these dynamical systems. Existence and characterization theorems for ‘rapid’ trajectories, growing faster in a certain sense than others, are obtained.

On Attractors of Multivalued Semi-Flows and Differential Inclusions

In this paper we study the existence of global attractors for multivalued dynamical systems. These theorems are then applied to dynamical systems generated by differential inclusions for which the solution is not unique for a given initial state. Finally, some boundary-value problems are considered.

Conley index for discrete multi-valued dynamical systems

The definitions of isolating block, index pair, and the Conley index, together with the proof of homotopy and additivity properties of the index are generalized for discrete multi-valued dynamical systems. That generalization provides a theoretical background of numerical computation used by Mischaikow and Mrozek in their computer assisted proof of chaos in the Lorenz equations, where finitely represented multi-valued mappings appear as a tool for discretization.

Fuzzy dynamical systems

A fuzzy dynamical system on an underlying complete, locally compact metric state space X is defined axiomatically in terms of a fuzzy attainability set mapping on X. This definition includes as special cases crisp single and multivalued dynamical systems on X. It is shown that the support of such a fuzzy dynamical system on X is a crisp multivalued dynamical system on X, and that such a fuzzy dynamical system can be considered as a crisp dynamical system on a state space of nonempty compact fuzzy subsets of X. In addition fuzzy trajectories are defined, their existence established and various properties investigated.

The funnel boundary of multivalued dynamical systems

AbstractProperties of the funnel boundary are investigated for multivalued dynamical systems defined axiomatically in terms of attainability set mappings on complete, locally compact metric state spaces. The set of regular boundary events is shown to be dense in the funnel boundary and theorems of Fukuhara and Zaremba on peripheral attainability are generalized to the systems considered here.

Stability of multivalued discrete dynamical systems

Stability of multivalued discrete dynamical systems