Symbolic dynamics: Difference between revisions

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{{Short description|Modeling a dynamical system's states as infinite sequences of symbols}}

In [[mathematics]], '''symbolic dynamics''' is the practicestudy of modelingdynamical asystems topological or smooth [[dynamical system]]defined byon a discrete space consisting of infinite [[sequence]]s of abstract symbols,. eachThe of which corresponds to a [[Dynamical system|state]]evolution of the system, with the dynamics (evolution) given by the [[shift operator]]. Formally, a [[Markov partition]] is used to provide a [[finite cover]] for the smoothdynamical system; each set of the cover is associateddefined withas a singlesimple symbol, and the sequences of symbols result as a trajectoryshift of the systemsequence. moves from one covering set to another.

Because of their explicit, discrete nature, such systems are often relatively easy to characterize and understand. They form a key tool for studying topological or smooth [[dynamical system]]s, because in many important cases it is possible to reduce the dynamics of a more general dynamical system to a symbolic system. To do so, a [[Markov partition]] is used to provide a [[finite cover]] for the smooth system; each set of the cover is associated with a single symbol, and the sequences of symbols result as a trajectory of the system moves from one covering set to another.

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==Examples==

Consider the set of two-sided infinite sequences on two symbols, 0 and 1. A typical element in this set looks like: (..., 0, 1, 0, 0, 1, 0, 1, ... )

Concepts such as [[heteroclinic orbit]]s and [[homoclinic orbit]]s have a particularly simple representation in symbolic dynamics.

There will be exactly two fixed points under the shift map: the sequence of all zeroes, and the sequence of all ones. A periodic sequence will have a periodic orbit. For instance, the sequence (..., 0, 1, 0, 1, 0, 1, 0, 1, ...) will have period two.

More complex concepts such as [[heteroclinic orbit]]s and [[homoclinic orbit]]s also have simple descriptions in this system. For example, any sequence that has only a finite number of ones will have a homoclinic orbit, tending to the sequence of all zeros in forward and backward iterations.

===Itinerary===