dynamical system


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dynamical system

[dī¦nam·ə·kəl ′sis·təm]
(mathematics)
An abstraction of the concept of a family of solutions to an ordinary differential equation; namely, an action of the real numbers on a topological space satisfying certain “flow” properties.
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The dimensionality of the dynamical system decides the number of Lyapunov exponents, that is, if the system is defined in [R.sup.m], then it possesses m Lyapunov exponents ([[lambda].sub.1] [greater than or equal to] [[lambda].sub.2] [greater than or equal to], ..., [[lambda].sub.m]).
A dynamical system (X, F) is topologically transitive if and only if, for all nonempty subsets U and V of X, there exist n positive such that the iteration of F on U intersects V at least at a point.
If the initial value [x.sup.0] of the switched dynamical system in (4) is chosen in [R.sup.j.sub.++], its solution [phi](t, [x.sup.0]) behaves in [R.sup.j.sub.++] for all t [greater than or equal to] 0.
where [y.sub.i] is the updated state of the entity i by applying a local function [f.sub.i] over the states of the entities in {<} [union] [A.sub.G](i), constitutes a discrete dynamical system called parallel dynamical system (PDS) over [{0,1}.sup.n].
Then we get [sigma]([[summation].sub.m]) = [[summation].sub.m] and [sigma] is continuous, and the m-shift map a as a dynamical system defined on [[summation].sub.m] has the following properties: (1) [sigma] has a countable infinity of periodic orbits consisting of orbits of all periods; (2) [sigma] has an uncountable infinity of nonperiodic orbits; and (3) [sigma] has a dense orbit [23].
It is, however, not trivial that the quantities [N.sub.m] have an interpretation in terms of dynamical systems, as in the case of Artin-Mazur zeta function.
Recall [8, 19] that a triple <W, [phi], ([OMEGA], [Z.sub.+], [sigma])> (in brief [phi]) is called a cocycle over the semigroup dynamical system ([OMEGA], [Z.sub.+],a) with fiber W.
The author's focus on continuous evolutionary models of the deterministic variety means that, with the exception of some cursory discussion of the consequences of switching from continuous to discrete formulations of dynamical systems, these modelling issues do not get addressed.
Gleick [3] in his popular book on chaos, gives the views of several researchers: the complicated, aperiodic attracting orbits of certain dynamical systems, apparently random recurrent behavior in a simple deterministic system, the irregular, unpredictable behavior of deterministic nonlinear dynamical systems.
In general, a nonlinear dynamical system may posses several coexistent attractors; thus it will have many basins of attraction, i.e.
It is induced on a Banach space by a skew-evolution cocycle (see [13]) defined over a semiflow associated with a generalized dynamical system. Both a nonautonomous equation and a variational equation can be addressed in terms of skew-evolution semiflows.
An equity market could be considered as complex dynamical system, with different agents as well as institutions having different time horizons in mind, carrying out transactions that result in complex patterns that are reflected in the data.

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