Multivalued Dynamics Research Articles

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.

Homogenization of the G-equation: a metric approach

The aim of the paper is to recover some results of Cardaliaguet–Nolen–Souganidis in Cardaliaguet et al. (Arch Rat Mech Anal 199(2): 527–561, 2011) and Xin–Yu in Xin and Yu (Commun Math Sci 8(4): 1067–1078, 2010) about the homogenization of the G-equation, using different and simpler techniques. The main mathematical issue is the lack of coercivity of the Hamiltonians. In our approach we consider a multivalued dynamics without periodic invariants sets, a family of intrinsic distances and perform an approximation by a sequence of coercive Hamiltonians.

Conway topograph, -dynamics and two-valued groups

Conway’s topographic approach to binary quadratic forms and Markov triples is reviewed from the point of view of the theory of two-valued groups. This leads naturally to a new class of commutative two-valued groups, which we call involutive. It is shown that the two-valued group of Conway’s lax vectors plays a special role in this class. The group describing the symmetries of the Conway topograph acts by automorphisms of this two-valued group. Binary quadratic forms are interpreted as primitive elements of the Hopf 2-algebra of functions on the Conway group. This fact is used to construct an explicit embedding of the Conway two-valued group into and thus to introduce a total group ordering on it. The two-valued algebraic involutive groups with symmetric multiplication law are classified, and it is shown that they are all obtained by the coset construction from the addition law on elliptic curves. In particular, this explains the special role of Mordell’s modification of the Markov equation and reveals its connection with two-valued groups in -theory. The survey concludes with a discussion of the role of two-valued groups and the group in the context of integrability in multivalued dynamics. Bibliography: 104 titles.

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.

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.

A metric proof of the converse Lyapunov theorem for semicontinuous multivalued dynamics

We extend the metric proof of the converse Lyapunov Theorem,given in [13] for continuous multivalued dynamics, bymeans of tools issued from weak KAM theory, to the case where theset-valued vector field is just upper semicontinuous. Thisgenerality is justified especially in view of application todiscontinuous ordinary differential equations. The more relevantnew point is that we introduce, to compensate the lack ofcontinuity, a family of perturbed dynamics, obtained throughinternal approximation of the original one, and perform somestability analysis of it.

A metric approach to the converse Lyapunov theorem for continuous multivalued dynamics

We construct a smooth Lyapunov pair for a continuous differential inclusion possessing a compact attractor. Our approach is based on the introduction of an intrinsic length functional, together with the corresponding length distance, defined starting from the polar cone of the multivalued vector field. This method allows a more geometrical insight into the subject and leads to a proof of the result which is, in our opinion, quite simple compared with the others available in the literature.

INVARIANT MEASURES AND THEIR PROJECTIONS IN NONAUTONOMOUS DYNAMICAL SYSTEMS

Invariant measure-type notions are examined for nonautonomous dynamical systems, namely, for flows whose dynamics is affected by an external parameter. The desired measures are obtained as projections of invariant measures of the shift flow on the lifted space of trajectories. The specific cases of multi-valued dynamics and controlled dynamics are examined in detail and other applications are pointed out.

Existence and Relaxation for Finite-Dimensional Optimal Control Problems Driven by Maximal Monotone Operators

We study the optimal control of a class of nonlinear finite-dimensional optimal control problems driven by a maximal monotone operator which is not necessarily everywhere defined. So our model problem incorporates systems monitored by variational inequalities. First we prove an existence theorem using the reduction method of Berkovitz and Cesari. This requires a convexity hypothesis. When this convexity condition is not satisfied, we have to pass to an augmented, convexified problem known as the "relaxed problem". We present four relaxation methods. The first uses Young measures, the second uses multi-valued dynamics, the third is based on Caratheodory's theorem for convex sets in RN and the fourth uses lower semicontinuity arguments and Γ-limits. We show that they are equivalent and admissible, which roughly speaking means that the corresponding relaxed problem is in a sense the "closure" of the original one.

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.

An existence theorem for an optimal economic growth problem with infinite planning horizon

We study a problem of optimal economic growth. Our model is a general one with multivalued dynamics, not necessarily convex technologies, and an infinite planning horizon in continuous time. Using the reduction technique of Cesari and the theory of weighted Sobolev spaces, we prove the existence of an optimal capital accumulation path for an integral utility criterion.