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.     

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.     

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.     

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.     

Weak shadowing for discrete dynamical systems on nonsmooth manifolds

In J. Math. Anal. Appl. 189 (1995) 409-423, Corless and Pilyugin proved that weak shadowing is a C⁰ generic property in the space of discrete dynamical systems on a compact smooth manifold M. In our paper we give another proof of this theorem which does not assume that M has a differential structure. Moreover, our method also works for systems on some compact metric spaces that are not manifolds, such as a Hilbert cube (or generally, a countably infinite Cartesian product of manifolds with boundary) and a Cantor set.     

A new class of generalized f-projected dynamical system in Banach spaces

In this article, a new class of generalized f-projected dynamical systems is introduced and studied in Banach spaces. A global existence and uniqueness result of generalized f-projected dynamical system is proved, which generalizes the existence result of Xia and Vincent [Y.S. Xia and T.L. Vincent, On the stability of global projected dynamical systems, J. Optim. Theory Appl. 106 (2009), pp. 129–150]. The global convergence stability of the generalized f-projected dynamical system and the sensitivity result of solutions set with perturbations of the constraint sets are also obtained under some suitable conditions.     

Dynamic range in a neuron network with electrical and chemical synapses

The dynamic range is the logarithmic difference between maximum and minimum levels of sensation produced by known stimuli. In the human sensory systems the dynamic ranges are typically larger than for single neurons, this amplification being essentially a collective effect of the neural network. We investigated the dynamic range exhibited by a cellular automaton network with electrical and chemical synapses, when the stimuli are modelled by a Poisson process of suprathreshold events of stereotyped unit amplitude and the neuron response is its average firing rate.     

Stability and bifurcation of a Cohen-Grossberg neural network with discrete delays

A Cohen-Grossberg neural network with discrete delays is investigated in this paper. The qualitative analysis is given for the system and it is found that the system undergoes a sequence of Hopf bifurcations by choosing the discrete time delay as a bifurcation parameter. Moreover, by applying the normal form theory and the center manifold theorem, the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions are obtained. Numerical simulations are given to illustrate the obtained results.     

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.     

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.

     

Stability and bifurcation analysis for a neural network model with discrete and distributed delays

In this paper, a two‐neuron network with both discrete and distributed delays is considered. With the corresponding characteristic equation analyzed, the local stability of the trivial equilibrium is investigated. With the discrete time delay taken as a bifurcation parameter, the existence of Hopf bifurcation is established. Moreover, formulae for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are derived. Finally, numerical simulations are carried out to illustrate the main results and further to exhibit that there is a characteristic sequence of bifurcations leading to a chaotic dynamics, which implies that the system admits rich and complex dynamics. Copyright © 2012 John Wiley & Sons, Ltd.     

Rest points of generalized dynamical systems

In this paper we consider generalized dynamical systems whose integral vortex (that is, the set of all trajectories of the system starting at a given point) is an acyclic set in the corresponding space of curves. For such systems we apply the theory of fixed points for multi-valued maps in order to prove the existence of rest points. In this way we obtain new existence theorems for rest points of generalized dynamical systems. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 28–36, January, 1999.     

Stabilizing Hopfield neural networks via inhibitory self-connections

The theory of monotone dynamical systems is employed to establish some sufficient conditions for the global attractivity of the Hopfield neural networks with finite distributed delays. The results show that self-inhibitory connections can be used to stabilize a delayed network provided the diagonal delays corresponding to the inhibitory self-connections are small enough.     

A discrete model of a large polling system

For a network with Poisson incoming flow of customers (particles) and unit time of the motion of servers (annihilators), we obtain the limit distribution of the number of customers at the node for a fixed general number of nodes.     

Uniform persistence for a discrete predator-prey system with delays

In this paper, we deal with a discrete predator-prey model with monotonic functional responses and give a sufficient condition of uniform persistence for the system.     

Bifurcation of a three-unit neural network

Considered is a system of delay differential equations modeling a time-delayed connecting network of three neurons without self-feedback. Discussing the change of the number of eigenvalues with zero real part, we locate the boundary of the stability region and finally determine the largest stability region of trivial solution. We investigate the existence of bifurcation phenomena of codimension one/two of the trivial equilibrium by considering the intersections of some parameter curves, which, in the aτ-half parameter plane, correspond to zero root or pure imaginary roots. In particular, the equivariant bifurcation is studied because of the equivariance of the system. We also present numerical simulations to demonstrate the rich dynamical behavior near the equivariant Pitchfork-Hopf bifurcation points, Hopf-Hopf bifurcation points, and some higher codimension bifurcation points.