Multivalued Dynamical Systems Research Articles

Non-Newtonian incompressible fluids with nonlinear shear tensor and hereditary conditions

We consider a mathematical model with delay for non-Newtonian incompressible fluids in a bounded domain. Existence of global weak solutions is proved under suitable regularity on the initial data and the forces. Conditions for uniqueness are also given, but in general the results are stated in a multi-valued framework. Suitable multi-valued dynamical systems are well-posed, using basically L2(Ω)n×L2(−h,0;L2(Ω)n) or C([−h,0];L2(Ω)n) norms. Then the existence of pullback attractors is ensured acting on several universes, some of them of fixed bounded sets and others of tempered type, depending on parameters related to an integrability condition of the force and the delay term. Finally, relationships between these families of attractors are also provided, improving the characterization of attraction with respect to previous results.

Multi-valued dynamical systems on time-dependent metric spaces with applications to Navier–Stokes equations

Multi-valued dynamical systems on time-dependent metric spaces with applications to Navier–Stokes equations

The index theory for multivalued dynamical systems with applications to reaction-diffusion equations with discontinuous nonlinearity

The index theory for multivalued dynamical systems with applications to reaction-diffusion equations with discontinuous nonlinearity

The Morse Equation in the Conley Index Theory for Discrete Multivalued Dynamical Systems

A recent generalization of the Conley index to discrete multivalued dynamical systems without a continuous selector is motivated by applications to data–driven dynamics. In the present paper we continue the program by studying attractor–repeller pairs and Morse decompositions in this setting. In particular, we prove Morse equation and Morse inequalities.

Attractors for multi-valued lattice dynamical systems with nonlinear diffusion terms

This paper is devoted to the long-term behavior of nonautonomous random lattice dynamical systems with nonlinear diffusion terms. The nonlinear drift and diffusion terms are not expected to be Lipschitz continuous but satisfy the continuity and growth conditions. We first prove the existence of solutions, and establish the existence of a multi-valued nonautonomous cocycle. We then show the existence and uniqueness of pullback attractors parameterized by sample parameters. Finally, we establish the measurability of this pullback attractor by the method based on the weak upper semicontinuity of the solutions.

Conley Index Approach to Sampled Dynamics.

The topological method for the reconstruction of dynamics from time series [K. Mischaikow et al., Phys. Rev. Lett., 82 (1999), pp. 1144-1147] is reshaped to improve its range of applicability, particularly in the presence of sparse data and strong expansion. The improvement is based on a multivalued map representation of the data. However, unlike the previous approach, it is not required that the representation has a continuous selector. Instead of a selector, a recently developed new version of Conley index theory for multivalued maps [B. Batko, SIAM J. Appl. Dyn. Syst., 16 (2017), pp. 1587-1617; B. Batko and M. Mrozek, SIAM J. Appl. Dyn. Syst., 15 (2016), pp. 1143-1162] is used in computations. The existence of a continuous, single valued generator of the relevant dynamics is guaranteed in the vicinity of the graph of the multivalued map constructed from data. Some numerical examples based on time series derived from the iteration of Hénon-type maps are presented.

Tail convergences of pullback attractors for asymptotically converging multi-valued dynamical systems

In this paper we study pullback attractors of multi-valued dynamical systems that are asymptotically convergent. It is shown that, under certain conditions, the components of the pullback attractor of a dynamical system can converge in time to those of the pullback attractor of the limiting dynamic al system. Particular examples are asymptotically autonomous and asymptotically periodic pullback attractors. Different criteria theorems requiring different conditions are established and their applicability and advantages are highlighted.

Strong Pullback Attractors of the Nonautonomous Suspension Bridge Equation with Variable Delay

In this paper, we consider the suspension bridge equation with variable delay. The long-time dynamics of the solutions for the suspension equations without delay effects have been investigated by many authors. But there are few works on suspension equations with delay. Moreover there are not many studies on attractors for other systems with delay. Thus, we study the existence of pullback attractor for the suspension equation with variable delay by using the theory of attractors for multivalued dynamical systems.

Invariant measures on multi-valued functions

In this paper we consider the question of under which conditions multi-valued dynamical systems admit invariant measures. We give results on the existence of invariant measures with full support on orbit spaces of multi-valued dynamical systems. We use these measures on the orbit space to induce measures on the original dynamical system. We focus on the question of when a non-atomic invariant measure on the orbit space induces an atomic invariant measure on the multi-valued dynamical system. This phenomenon is an indicator of complicated multi-periodic behaviour.

Preface to the special issue "Finite and infinite dimensional multivalued dynamical systems"

Preface to the special issue "Finite and infinite dimensional multivalued dynamical systems"

Weak Index Pairs and the Conley Index for Discrete Multivalued Dynamical Systems. Part II: Properties of the Index

Motivation to revisit the Conley index theory for discrete multivalued dynamical systems stems from the needs of broader real applications, in particular in sampled dynamics or in combinatorial dynamics. The new construction of the index in [B. Batko and M. Mrozek, SIAM J. Appl. Dyn. Syst., 15 (2016), pp. 1143--1162] based on weak index pairs, under the circumstances of the absence of index pairs caused by relaxing the isolation property, seems to be a promising step toward this direction. The present paper is a direct continuation of [B. Batko and M. Mrozek, SIAM J. Appl. Dyn. Syst., 15 (2016), pp. 1143--1162] and concerns properties of the index defined therin, namely, the Ważewski property, the additivity property, the homotopy (continuation) property, and the commutativity property. We also present the construction of weak index pairs in an isolating block.

Weak Index Pairs and the Conley Index for Discrete Multivalued Dynamical Systems

Motivated by the problem of reconstructing dynamics from samples, we revisit the Conley index theory for discrete multivalued dynamical systems [T. Kaczynski and M. Mrozek, Topology Appl., 65 (1995), pp. 83--96]. We introduce a new, less restrictive definition of the isolating neighborhood. It turns out that then the main tool for the construction of the index, i.e., the index pair, is no longer useful. In order to overcome this obstacle we use the concept of weak index pairs.

Structure of the pullback attractor for a non-autonomous scalar differential inclusion

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 non-autonomous cases. To this aim, we will also show a general result on the characterization of a pullback attractor for a multivalued process by the union of all the complete bounded trajectories of the system.

Minimality Properties of Set-Valued Processes and their Pullback Attractors

We discuss the existence of pullback attractors for multivalued dynamical systems on metric spaces. Such attractors are shown to exist without any assumptions in terms of continuity of the solution maps, based only on minimality properties with respect to the notion of pullback attraction. When invariance is required, a very weak closed graph condition on the solving operators is assumed. The presentation is complemented with examples and counterexamples to test the sharpness of the hypotheses involved, including a reaction-diffusion equation, a discontinuous ordinary differential equation, and an irregular form of the heat equation.

Discrete dynamics on noncommutative CW complexes

The concept of discrete multivalued dynamical systems for noncommutative CW complexes is developed. Stable and unstable manifolds are introduced and their role in geometric and topological configurations of noncommutative CW complexes is studied. Our technique is illustrated by an example on the noncommutative CW complex decomposition of the algebra of continuous functions on two dimensional torus.

Pullback attractors of a multi-valued process generated by parabolic differential equations with unbounded delays

The theory of nonautonomous multi-valued dynamical systems and a method of asymptotic compactness based on the concept of the Kuratowski measure of the noncompactness of a bounded set are used to prove the existence of pullback attractors in the weighted space Cγ,V for the multi-valued process associated with the nonautonomous parabolic differential equation with infinite delay for which the uniqueness of solutions need not hold.

The Envelope Attractor of Non-strict Multivalued Dynamical Systems with Application to the 3D Navier–Stokes and Reaction–Diffusion Equations

Multivalued semiflows generated by evolution equations without uniqueness sometimes satisfy a semigroup set inclusion rather than equality because, for example, the concatentation of solutions satisfying an energy inequality almost everywhere may not satisfy the energy inequality at the joining time. Such multivalued semiflows are said to be non-strict and their attractors need only be negatively semi-invariant. In this paper the problem of enveloping a non-strict multivalued dynamical system in a strict one is analyzed and their attactors are compared. Two constructions are proposed. In the first, the attainability set mapping is extending successively to be strict at the dyadic numbers, which essentially means (in the case of the Navier–Stokes system) that the energy inequality is satisfied piecewise on successively finer dyadic subintervals. The other deals directly with trajectories and their concatenations, which are then used to define a strict multivalued dynamical system. The first is shown to be applicable to the three-dimensional Navier–Stokes equations and the second to a reaction–diffusion problem without unique solutions.

GENERALIZED SEMIFLOWS AND CHAOS IN MULTIVALUED DYNAMICAL SYSTEMS

This contribution addresses a possible description of the chaotic behavior in multivalued dynamical systems. For the multivalued system formulated via differential inclusion the practical conditions on the right-hand side are derived to guarantee existence of a solution, which leads to the chaotic behavior. Our approach uses the notion of the generalized semiflow but it does not require construction of a selector on the set of solutions. Several applications are provided by concrete examples of multivalued dynamical systems including the one having a clear physical motivation.

Random attractor of stochastic partly dissipative systems perturbed by Lévy noise

The current paper is devoted to random dynamics of stochastic partly dissipative systems perturbed by Levy noise. By the technique of dissipative in probability and multivalued random dynamical systems (MRDS), the existences of random attractor for MRDS generated by the stochastically perturbed partly dissipative systems are provided, both the weaker restrictions and stronger restrictions on the coefficients of Levy noise respectively. Mathematics Subject Classification: 35R60; 60H15; 35K57.

Multivalued attractors and their approximation: applications to the Navier–Stokes equations

This article is devoted to the study of multivalued semigroups and their asymptotic behavior, with particular attention to iterations of set-valued mappings. After developing a general abstract framework, we present an application to a time discretization of the two-dimensional Navier---Stokes equations. More precisely, we prove that the fully implicit Euler scheme generates a family of discrete multivalued dynamical systems, whose global attractors converge to the global attractor of the continuous system as the time-step parameter approaches zero.