Multivalued Dynamical System Research Articles

Dynamics of multi-valued retarded p-Laplace equations driven by nonlinear colored noise

This paper mainly considers the long-term behavior of p-Laplace equations with infinite delays driven by nonlinear colored noise. We firstly prove the existence of weak solutions to the equation, but the uniqueness of solutions cannot be guaranteed due to the lack of Lipschitz continuity conditions, and thus generate a multi-valued dynamical system. Moreover, the regularity of solutions is also proved. Then we prove the existence of a pullback attractor. Subsequently, the measurability of the pullback attractor and the multi-valued dynamical system are also proved.

Random attractors for multi-valued multi-stochastic delayed p-Laplace lattice equations

ABSTRACT The dynamical behaviour of the stochastic non-autonomous p-Laplace lattice equation driven by variable delays, random viscosity, multiplicative noise and non-Lipschitz nonlinearity is concerned. The existence of global solutions for the equation is proved, while the solutions are possibly multi-valued and generate a multi-valued dynamical system. The existence of a pullback attractor is shown. Moreover, the measurability of the pullback attractor as well as the multi-valued cocycle is proved by using the weak continuity of the discrete p-Laplace operator and a countable decomposition of the Wiener probability space.

Random dynamics for non-autonomous stochastic evolution equations without uniqueness on unbounded narrow domains

This paper deals with the limiting behavior of non-autonomous stochastic reaction-diffusion equations without uniqueness on unbounded narrow domains. We prove the existence and upper semicontinuity of random attractors for the equations on a family of unbounded (n + 1)-dimensional narrow domains, which collapses onto an n-dimensional domain. Since the solutions are non-uniqueness, which leads to a multivalued random dynamical system with the solution operators of the equation, we will prove the existence and upper semicontinuity of attractors by multivalued random dynamical system theory.

Linking Combinatorial and Classical Dynamics: Conley Index and Morse Decompositions

We prove that every combinatorial dynamical system in the sense of Forman, defined on a family of simplices of a simplicial complex, gives rise to a multivalued dynamical system F on the geometric realization of the simplicial complex. Moreover, F may be chosen in such a way that the isolated invariant sets, Conley indices, Morse decompositions and Conley–Morse graphs of the combinatorial vector field give rise to isomorphic objects in the multivalued map case.

Complex Dynamics of Bouncing Motions at Boundaries and Corners in a Discontinuous Dynamical System

In this paper, from the local theory of flow at the corner in discontinuous dynamical systems, obtained are analytical conditions for switching impact-alike chatter at corners. The objective of this investigation is to find the dynamics mechanism of border-collision bifurcations in discontinuous dynamical systems. Multivalued linear vector fields are employed, and generic mappings are defined among boundaries and corners. From mapping structures, periodic motions switching at the boundaries and corners are determined, and the corresponding stability and bifurcations of periodic motions are investigated by eigenvalue analysis. However, the grazing and sliding bifurcations are determined by the local singularity theory of discontinuous dynamical systems. From such analytical conditions, the corresponding parameter map is developed for periodic motions in such a multivalued dynamical system in the single domain with corners. Numerical simulations of periodic motions are presented for illustrations of motions complexity and catastrophe in such a discontinuous dynamical system.

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.

Attractors for a random evolution equation with infinite memory: Theoretical results

The long-time behavior of solutions (more precisely, the existence of random pullback attractors) for an integro-differential parabolic equation of diffusion type with memory terms, more particularly with terms containing both finite and infinite delays, as well as some kind of randomness, is analyzed in this paper. We impose general assumptions not ensuring uniqueness of solutions, which implies that the theory of multivalued dynamical system has to be used. Furthermore, the emphasis is put on the existence of random pullback attractors by exploiting the techniques of the theory of multivalued nonautonomous/random dynamical systems.

Global attractors of impulsive parabolic inclusions

In this work we consider an impulsive multi-valued dynamical system generated by a parabolic inclusion with upper semicontinuous right-hand side \begin{document}$\varepsilon F(y)$\end{document} and with impulsive multi-valued perturbations. Moments of impulses are not fixed and defined by moments of intersection of solutions with some subset of the phase space. We prove that for sufficiently small value of the parameter \begin{document}$\varepsilon>0$\end{document} this system has a global attractor.

Regularity and structure of pullback attractors for reaction–diffusion type systems without uniqueness

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 backwards bounded complete trajectories. As an application, a general reaction–diffusion system is proved to have an invariant (H,V)-pullback attractor A={A(τ)}τ∈R. This attractor is composed of all the backwards compact complete trajectories of the system, pullback attracts bounded subsets of H in the topology of V, and moreover⋃s⩽τA(s)is precompact in V,∀τ∈R.A non-autonomous Fitz-Hugh–Nagumo equation is studied as a specific example of the reaction–diffusion system.

On One 2-Valued Transformation: Its Invariant Measure and Application to Masked Dynamical Systems

We consider one familySof 2-valued transformations on the interval [0, 1] with measureμ, endowed with a set of weight functions. We construct invariant measureμS=μfor this multivalued dynamical system with weights and show the interplay between such systems and masked dynamical systems, which leads to image processing.

Pullback attractors for three-dimensional Navier-Stokes-Voigt equations with delays

Our aim in this paper is to study the existence of pullback attractors for the 3D Navier-Stokes-Voigt equations with delays. The forcing term containing the delay is sub-linear and continuous with respect to u. Since the solution of the model is not unique, which is caused by the continuity assumption, we establish the existence of pullback attractors for our problem by using the theory of multi-valued dynamical system.

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.

Global attractor of a parabolic inclusion with nonautonomous main part

On solutions of an evolution inclusion with nonautonomous parabolic operator, we construct a nonautonomous multivalued dynamical system. For this system, we prove the existence of an invariant, connected, stable global attractor in the phase space.

Cascade search of the coincidence set of collections of multivalued mappings

The present paper is a continuation of the previous papers of the author dealing with this subject. We study the cascade search of a given subset A, i.e., the construction on the metric space X of a multicascade with given limit subset A in X. A multicascade is a multivalued dynamical system with translation semigroup equal to the additive semigroup of nonnegative integers. We propose a finer (than in the author’s previous papers) version of cascade search for cases in which (1) A is the complete preimage of a closed subspace under the multivalued mapping of metric spaces; (2) A is the set of coincidence points n, n > 1, of the multivalued mappings. An estimate of the distance from the initial to any corresponding limit point is given. In particular, in case (2), a new generalization of a recent theorem due to Arutyunov is obtained for n = 2.

Non-autonomous attractors for integro-differential evolution equations

We show that infinite-dimensional integro-differential equations which involve an integral of the solution over the time interval since starting can be formulated as non-autonomous delay differential equations with an infinite delay. Moreover, when conditions guaranteeing uniqueness of solutions do not hold, they generate a non-autonomous (possibly) multi-valued dynamical system (MNDS). The pullback attractors here are defined with respect to a universe of subsets of the state space with sub-exponetial growth, rather than restricted to bounded sets. The theory of non-autonomous pullback attractors is extended to such MNDS in a general setting and then applied to the original integro-differential equations. Examples based on the logistic equations with and without a diffusion term are considered.

Conley type index applied to Hamiltonian inclusions

In the paper we develop the theory of a cohomological index of the Conley type detecting invariant sets of a multivalued dynamical system generated by semilinear differential inclusion in an infinite dimensional Hilbert space. An application to the existence of periodic orbits to asymptotically linear Hamiltonian inclusions is presented.

Discrete Dynamical System Framework for Construction of Connections between Critical Regions in Lattice Height Data

We define a new mathematical model for the topological study of lattice height data. A discrete multivalued dynamical system framework is used to establish discrete analogies of a Morse function, its gradient field, and its stable and unstable manifolds in order to interpret functions numerically given on finite sets of pixels. We present efficient algorithms detecting critical components of a height function f and displaying connections between them by means of a graph, called the Morse connections graph whose nodes represent the critical components of f and edges show the existence of connecting trajectories between nodes. This graph encodes efficiently the topological structure of the data and makes it easy to manipulate for subsequent processing.

Global Attractor for Impulsive Reaction-Diffusion Equation

In this paper, we consider a reaction-diffusion equation with nonsmooth nonlinearity whose solutions have impulse effects at fixed moments of time. We show how this object generates a nonautonomous multivalued dynamical system and prove the existence of a compact semiinvariant global attractor in the phase space.

Random attractors for ambiguously solvable systems dissipative with respect to probability

We prove a theorem on the existence of a random attractor for a multivalued random dynamical system dissipative with respect to probability. Abstract results are used for the analysis of the qualitative behavior of solutions of a system of ordinary differential equations with continuous right-hand side perturbed by a stationary random process. In terms of the Lyapunov function, for an unperturbed system, we give sufficient conditions for the existence of a random attractor.

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.