It is possible to obtain a discrete

dynamical system from a continuous

dynamical system through the sampling of its solution at a regular time interval T, in which the dynamical rule representing the relationship between the consecutive sampled values of the dynamical variables is regarded as a time T map.

Given

dynamical system f : S [right arrow] S, S [subset] R, the iterations of the function f are the composition of a function with itself.

This special issue places its emphasis on the study of the applications of

dynamical system on time scales; such applications include economics models utilizing optimal control theory, fractional calculus, and the development of new population models.

Reconstructed image pixels were obtained by using the initial value problem of differential equations describing the

dynamical system, for example, a continuous-time image reconstruction (CIR) system.

When the states of the entities are updated in a synchronous manner, the system is called a parallel

dynamical system (PDS) [2, 3], while if they are updated in an asynchronous way, the system is named sequential

dynamical system (SDS) [4].

Consider the following three-dimensional

dynamical system:

On the

dynamical system (X , a; w), one has [M.sub.[sigma]] (w) = [M.sub.[tau]].

A

dynamical system {[X.sub.t]} generated by a map [psi](.):

where [OMEGA] is a metric space, E is a finite-dimensional Banach space, ([OMEGA], [Z.sub.+], [sigma]) is a

dynamical system with discrete time [Z.sub.+], [E] is the space of all the linear operators acting on E equipped with operator norm, C([OMEGA], [E]) (respectively, C(E x [OMEGA], E)) is the space of all the continuous functions defined on [OMEGA] (respectively, on E x [OMEGA]) with values in [E] (respectively, E) equipped with compact-open topology and F is a "small" perturbation.

These chapters include discussions of such important issues as the long-run survival of weakly, strictly, and iteratively strictly dominated strategies, and the mapping between stationary states of the

dynamical system on the one hand, and aggregate Nash equilibrium behavior (the static solution concepts discussed in Chapter One) and evolutionary stability criteria (the quasidynamic solution concepts discussed in Chapter Two) on the other.

The simplest chaotic

dynamical system is the Bernoulli shift described by Palmore [8]: