TENSOR SUM AND DYNAMICAL SYSTEMS

In this paper we introduce the concept of tensor sum semigroups. Also we have given the examples of tensor sum operators which induce dynamical system on weighted locally convex function spaces.     

Hilbertian Jamison sequences and rigid dynamical systems

A strictly increasing sequence (nk)_k?0 of positive integers is said to be a Hilbertian Jamison sequence if for any bounded operator T on a separable Hilbert space such that supk?0‖Tnk‖<+∞, the set of eigenvalues of modulus 1 of T is at most countable. We first give a complete characterization of such sequences. We then turn to the study of rigidity sequences (nk)_k?0 for weakly mixing dynamical systems on measure spaces, and give various conditions, some of which are closely related to the Jamison condition, for a sequence to be a rigidity sequence. We obtain on our way a complete characterization of topological rigidity and uniform rigidity sequences for linear dynamical systems, and we construct in this framework examples of dynamical systems which are both weakly mixing in the measure-theoretic sense and uniformly rigid.     

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.     

Fractional powers of the algebraic sum of normal operators

The main concern in this paper is to give sufficient conditions such that if are unbounded normal operators on a (complex) Hilbert space , then for each , the domain equals . It is then verified that such a result can be applied to characterize the domains of fractional powers of a large class of the Hamiltonians with singular potentials arising in quantum mechanics through the study of the Schrödinger equation.

     

Conley index for random dynamical systems

Conley index theory is a very powerful tool in the study of dynamical systems. In this paper, we generalize Conley index theory to discrete random dynamical systems. Our constructions are basically the random version of Franks and Richeson in [J. Franks, D. Richeson, Shift equivalence and the Conley index, Trans. Amer. Math. Soc. 352 (2000) 3305-3322] for maps, and the relations of isolated invariant sets between time-continuous random dynamical systems and corresponding time-h maps are discussed. Two examples are presented to illustrate results in this paper.     

Stochastic equilibria in von Neumann-Gale dynamical systems

This paper examines a class of random dynamical systems related to the classical von Neumann and Gale models of economic dynamics. Such systems are defined in terms of multivalued operators in spaces of random vectors, possessing certain properties of convexity and homogeneity. We establish a general existence theorem for equilibrium, which holds under conditions analogous to the standard deterministic ones. Our results answer questions that remained open for more than three decades.

     

Continuous random dynamical systems

We study a dynamical system generalizing continuous iterated function systems and stochastic differential equations disturbed by Poisson noise. The main results provide us with sufficient conditions for the existence and uniqueness of an invariant measure for the considered system. Since the dynamical system is defined on an arbitrary Banach space (possibly infinite dimensional), to prove the existence of an invariant measure and its stability we make use of the lower bound technique developed by Lasota and Yorke and extended recently to infinite-dimensional spaces by Szarek. Finally, it is shown that many systems appearing in models of cell division or gene expressions are exactly as those we study. Hence we obtain their stability as well.     

Affine-periodic solutions for discrete dynamical systems

The paper concerns the existence of affine-periodic solutions for discrete dynamicalsystems. This kind of solutions might be periodic, harmonic, quasi-periodic, even non-periodic.We prove the existence of affine-periodic solutions for discrete dynamical systems by using thetheory of Brouwer degree. As applications, another existence theorem is given via Lyapnovfunction.     

Global fractional-order projective dynamical systems

This paper studies a class of global fractional-order projective dynamical systems. First, we show the existence and uniqueness of the solution of this type of system. Then, the existence of the equilibrium point of this class of dynamical systems is obtained. Further more, we obtain the α-exponential stability of the equilibrium point under suitable conditions. In addition, we use a predictor–corrector algorithm to find a solution to this kind of system. Finally a numerical example is provided to illustrate the results obtained in this paper.     

A general method for writing the dynamical equations of nonholonomic systems with ideal constraints

The basic notions of the dynamics of nonholonomic systems are revisited in order to give a general and simple method for writing the dynamical equations for linear as well as non-linear kinematical constraints. The method is based on the representation of the constraints by parametric equations, which are interpreted as dynamical equations, and leads to first-order differential equations in normal form, involving the Lagrangian coordinates and auxiliary variables (the use of Lagrangian multipliers is avoided). Various examples are illustrated.      

On Tensor Products of l-Operators and bo-Operators

It is proved that the tensor product of two linear operators is a cone summing operator (respectively, order bounded operator) if and only if both operators are cone summing (respectively, order bounded).     

Tensor invariants of natural mechanical systems on compact surfaces and the corresponding integrals

The classification problem for tensor invariants of geodesic flows on compact surfaces is considered. Such invariants include first integrals, multivalued integrals, and symmetry fields. The problem is solved for tensor invariants of arbitrary degrees. It is shown that the invariants under consideration can vanish at some point only in the presence of two-valued symmetry fields and multivalued integrals. Translated fromMatematicheskie Zametki, Vol. 66, No. 3, pp. 417–430, September, 1999.     

Linear matrix differential dynamical systems with fuzzy matrices

In this paper, we investigate linear first-order fuzzy matrix differential dynamical systems where the coefficients matrix is described by a fuzzy matrix. We show some properties of the matrix differential dynamical systems, and their phase portraits are described by means of examples.     

Partial symmetries and dynamical systems

We start recalling the characterizing property of the ‘partial symmetries’ of a differential problem, that is, the property of transforming solutions into solutions only in a proper subset of the full solution set. This paper is devoted to analyze the role of partial symmetries in the special context of dynamical systems and also to compare this notion with other notions of ‘weak’ symmetries, namely, the λ‐symmetries and the orbital symmetries. Particular attention is addressed to discuss the relevance of partial symmetries in dynamical systems admitting homoclinic (or heteroclinic) manifolds, which can be ‘broken’ by periodic perturbations, thus giving rise, according to the (suitably rewritten) Mel'nikov theorem, to the appearance of a chaotic behavior of Smale‐horseshoes type. Many examples illustrate all the various aspects and situations. Copyright © 2016 John Wiley & Sons, Ltd.     

Dissipativity of Runge-Kutta methods for dynamical systems with delays

This paper is concerned with the numerical solution of dissipativeinitial value problems with delays by Runge-Kutta methods. Asufficient condition for the dissipativity of the systems isgiven. The concepts of D(l)-dissipativity and GD(l)-dissipativityare introduced. We investigate the dissipativity propertiesof (k,l)-algebraically stable Runge-Kutta methods with piecewiseconstant or linear interpolation procedures for finite-dimensionaland infinite-dimensional dynamical systems with delays.     

The Morse–Witten complex via dynamical systems

Given a smooth closed manifold M, the Morse–Witten complex associated to a Morse function f and a Riemannian metric g on M consists of chain groups generated by the critical points of f and a boundary operator counting isolated flow lines of the negative gradient flow. Its homology reproduces singular homology of M. The geometric approach presented here was developed in Weber [Der Morse–Witten Komplex, Diploma Thesis, TU Berlin, 1993] and is based on tools from hyperbolic dynamical systems. For instance, we apply the Grobman–Hartman theorem and the λ-lemma (Inclination Lemma) to analyze compactness and define gluing for the moduli space of flow lines.     

On the stability of projected dynamical systems

A class of projected dynamical systems (PDS), whose stationary points solve the corresponding variational inequality problem (VIP), was recently studied by Dupuis and Nagurney (Ref. 1). This paper initiates the study of the stability of such PDS around their stationary points and thus gives rise to the study of the dynamical stability of VIP solutions. Examples are constructed showing that such a study can be quite distinct from the classical stability study for dynamical systems (DS). We give the definition of a regular solution to a VIP and introduce the concept of a minimal face flow induced by a PDS, which is a standard DS of a lower dimension. We then show that, at the regular solutions of the VIP, the local stability of the PDS is essentially the same as that of its minimal face flow. Hence, we reduce the problem, in this case, to one of the classical stability study of DS, a more developed discipline. In a more direct way, we then establish a series of local and global stability results of the PDS, under various conditions of monotonicity.This research was supported by the National Science Foundation under Grant DMS-9024071 under the Faculty Awards for Women Program. This support is gratefully acknowledged.     

Random dynamical systems arising from iterated function systems with place-dependent probabilities

We show that if an iterated function system with place-dependent probabilities admits an invariant and attractive measure, then it has the structure of a random dynamical system (in the sense of Ludwig Arnold).     

A class of linear differential dynamical systems with fuzzy matrices

This paper investigates the first order linear fuzzy differential dynamical systems with fuzzy matrices. We use a complex number representation of the α-level sets of the fuzzy system, and obtain the solution by employing such representation. It is applicable to practical computations and has also some implications for the theory of fuzzy differential equations. We then present some properties of the 2-dimensional dynamical systems and their phase portraits. Some examples are considered to show the richness of the theory and we can clearly see that new behaviors appear. We finally present some conclusions and new directions for further research in the area of fuzzy dynamical systems.