Necessary conditions for multistationarity in discrete dynamical systems

R. Thomas conjectured, 20 years ago, that the presence of a positive circuit in the interaction graph of a dynamical system is a necessary condition for the presence of several stable states. Recently, E. Remy et al. stated and proved the conjecture for Boolean dynamical systems. Using a similar approach, we generalize the result to discrete dynamical systems, and by focusing on the asynchronous dynamics that R. Thomas used in the course of his analysis of genetic networks, we obtain a more general variant of R. Thomas’ conjecture. In this way, we get a necessary condition for genetic networks to lead to differentiation.     

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.     

Spectral decomposition theorem in equicontinuous nonautonomous discrete dynamical systems

In this paper, we define the notion of weak chain recurrence and study properties of weak chain recurrent sets in a nonautonomous discrete dynamical system induced by a sequence of homeomorphisms on a compact metric space. Our main result is the Smale’s spectral decomposition theorem in an equicontinuous nonautonomous discrete dynamical system.     

A kind of nonnegative matrices and its application on the stability of discrete dynamical systems

In this paper we introduce a new kind of nonnegative matrices which is called (sp) matrices. We show that the zero solutions of a class of linear discrete dynamical systems are asymptotically stable if and only if the coefficient matrices are (sp) matrices. To determine that a matrix is (sp) matrix or not is very simple, we need only to verify that some elements of the coefficient matrices are zero or not. According to the result above, we obtain the conditions for the stability of several classes of discrete dynamical systems.     

Generalized synchronization of two bidirectionally coupled discrete dynamical systems

Based on the modified system approach the generalized synchronization (GS) in two bidirectionally coupled discrete dynamical systems is classified into several types, and under some conditions, the existence, Lipschitz smoothness and Hölder continuity of two kinds of GS therein are derived and theoretically proved. In addition, numerical simulations validate the present theory.     

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.     

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.

     

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.     

On the global periodicity of discrete dynamical systems and application to rational difference equations

A new necessary condition for global periodicity of discrete dynamical systems and of difference equations is obtained here. This condition will be applied to contribute to solving the problem of global periodicity for second order rational difference equations.     

A discrete dynamical system as a model of a neural network with generalized Hebbian synapses

We study the dynamical behavior of a discrete time dynamical system which can serve as a model of a learning process. We determine fixed points of this system and basins of attraction of attracting points. This system was studied by Fernanda Botelho and James J. Jamison in [A learning rule with generalized Hebbian synapses, J. Math. Anal. Appl. 273 (2002) 529-547] but authors used its continuous counterpart to describe basins of attraction.     

Global periodicity and complete integrability of discrete dynamical systems

Consider the discrete dynamical system generated by a map F. It is said that it is globally periodic if there exists a natural number p such that F ^p (x)=x for all x in the phase space. On the other hand, it is called completely integrable if it has as many functionally independent first integrals as the dimension of the phase space. In this paper, we relate both concepts. We also give a large list of globally periodic dynamical systems together with a complete set of their first integrals, emphasizing the ones coming from difference equations.     

Note on the Markus-Yamabe conjecture for gradient dynamical systems

Let be a C¹ vector field which has a singular point O and its linearization is asymptotically stable at every point of Rn. We say that the vector field v satisfies the Markus-Yamabe conjecture if the critical point O is a global attractor of the dynamical system . In this note we prove that if v is a gradient vector field, i.e. v=∇f (f∈C²), then the basin of attraction of the critical point O is the whole Rn, thus implying the Markus-Yamabe conjecture for this class of vector fields. An analogous result for discrete dynamical systems of the form xm+1=∇f(xm) is proved.     

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.