Local singularities of dynamical systems with shock interactions

Local (qualitative) singularities of a special class of dynamical systems with shock interactions are classified. For the first four of the specified types of local singularities, certain properties of the qualitative structure are described and used to establish the topological equivalence of the corresponding singularities.Translated fromMatematicheskie Zametki, Vol. 64, No. 4, pp. 531–542, October, 1998.     

Positive circuits and maximal number of fixed points in discrete dynamical systems

We consider a product X of n finite intervals of integers, a map F from X to itself, the asynchronous state transition graph Γ(F) on X that Thomas proposed as a model for the dynamics of a network of n genes, and the interaction graph G(F) that describes the topology of the system in terms of positive and negative interactions between its n components. Then, we establish an upper bound on the number of fixed points for F, and more generally on the number of attractors in Γ(F), which only depends on X and on the topology of the positive circuits of G(F). This result generalizes the following discrete version of Thomas’ conjecture recently proved by Richard and Comet: If G(F) has no positive circuit, then Γ(F) has a unique attractor. This result also generalizes a result on the maximal number of fixed points in Boolean networks obtained by Aracena, Demongeot and Goles. The interest of this work in the context of gene network modeling is briefly discussed.     

On generalized equilibrium points

Generalizations of the usual definition of saddle point and equilibrium point are introduced in this paper. The existence of these points is shown to be related to a class of functions that we call perturbed convex functions. First and second order conditions regarding the existence of these points are also proved.     

关于非自治动力系统中的h-极小覆盖

设（X,d1,f1∞）与（Y ,d2,g1,∞）为两个非自治动力系统,h是从（X,d1,f．∞）到（Y,d2,g1∞）的拓扑半共轭．通过对自治动力系统中的h一极小覆盖的研究,本文得到了以下结论：1）对于任意的Y∈Y及X∈h-1（y）,orb（x,f1∞）被h映射为orb（y,g1∞）,w（x,f1∞）被h映射为w（y,g1∞）;2）在（X,d1,f1∞）中引入关于拓扑半共轭的h-极小覆盖的定义,证明了h一极小覆盖的存在性;3）对于任意的XEX和Y∈Y,在（w（z,f1∞）,f1∞。（x,f1,∞）与（w（y,g∞）,g1,∞（y,g1∞））均构成原系统的子系统的前提下,R（f1∞）被h映射为R（g1∞）．这些结论丰富了非自治动力系统的内容．     

Combinatorics of sequential dynamical systems

In this paper we study sequential dynamical systems (SDS) over words. Our main result is the classification of SDS over words for fixed graph Y and family of local maps (F_vi) by means of a novel notion of SDS equivalence. This equivalence arises from a natural group action on acyclic orientations. An SDS consists of: (a) a graph Y, (b) a family of vertex indexed Y-local maps F_vi:Kⁿ→Kⁿ, where K is a finite field and (c) a word w, i.e. a family (w₁,…,w_k), where w_j is a Y-vertex. A map F_vi(x_v1,…,x_vn) is called Y-local iff it fixes all variables x_vj≠x_vi and depends exclusively on the variables x_vj, for v_j∈B₁(v_i). The SDS-map is obtained by composing the local maps F_vi according to the word w: . Mutual dependencies of the local maps arising from their sequential application are expressed in the graph G(w,Y) having vertex set {1,…,k} (the indices of the word w) and in which r,s are adjacent iff w_s,w_r are adjacent in Y. We prove a bijection from equivalence classes of SDS-words into equivalence classes of acyclic orientations of G(w,Y). We show that within these equivalence classes the induced SDS are equivalent in the sense that their respective phase spaces are isomorphic as digraphs.     

Monotonic dynamical systems under spatial discretization

We estimate the probability of replicating the asymptotic behaviour of a dynamical system generated by a monotonic mapping for randomly centered roundoff lattices.

     

Structural properties of universal minimal dynamical systems for discrete semigroups

We show that for a discrete semigroup there exists a uniquely determined complete Boolean algebra - the algebra of clopen subsets of . is the phase space of the universal minimal dynamical system for and it is an extremally disconnected compact Hausdorff space. We deal with this connection of semigroups and complete Boolean algebras focusing on structural properties of these algebras. We show that is either atomic or atomless; that is weakly homogenous provided has a minimal left ideal; and that for countable semigroups is semi-Cohen. We also present a class of what we call group-like semigroups that includes commutative semigroups, inverse semigroups, and right groups. The group reflection of a group-like semigroup can be constructed via universal minimal dynamical system for and, moreover, and are the same.

     

Heteroclinic cycles in lattice dynamical systems

A criterion of spatial chaos occurring in lattice dynamical systems—heteroclinic cycle—is discussed It is proved that if the system has asymptotically stable heteroclinic cycle, then it has asymptotically stable homoclinic point which implies spatial chaos. Project supported by the National Natural Science Foundation of China.     

Some problems on fractal geometry and topological dynamical systems

This paper is a continuation of [14], some new problems on fractal geometry and topological dynamical systems have been posed.     

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.     

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.     

Exponential attractors for first-order lattice dynamical systems

In l², we investigate the existence of an exponential attractor for the solution semigroup of a first-order lattice dynamical system acting on a closed bounded positively invariant set which needs not to be compact since l² is infinite dimensional. Up to our knowledge, this is the first time to examine the existence of exponential attractors for lattice dynamical systems.     

Dimension of invariant measures for continuous random dynamical systems

We consider continuous random dynamical systems with jumps. We estimate the dimension of the invariant measures and apply the results to a model of stochastic gene expression. Copyright © 2016 John Wiley & Sons, Ltd.     

Some classes of projected dynamical systems in Banach spaces and variational inequalities

We introduce some Projected Dynamical Systems based on metric and generalized Projection Operator in a strictly convex and smooth Banach Space. Then we prove that critical points of these systems coincide with the solution of a Variational Inequality.     

Two notes on measure-theoretic entropy of random dynamical systems

In this paper, Brin-Katok local entropy formula and Katok＇s definition of the measuretheoretic entropy using spanning set are established for the random dynamical system over an invertible ergodic system.