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There are many forms of these maps, [2] many of which are equivalent under a coordinate transformation. For example two of the most common ones are: : +: The second one can be mapped to the first using the fact that . = + (), so : + is the same under the transformation = + ().
A discrete dynamical system, discrete-time dynamical system is a tuple (T, M, Φ), where M is a manifold locally diffeomorphic to a Banach space, and Φ is a function. When T is taken to be the integers, it is a cascade or a map. If T is restricted to the non-negative integers we call the system a semi-cascade. [13]
The ergodic theory of dynamical systems has recently been used to prove combinatorial theorems about number theory which has given rise to the field of arithmetic combinatorics. Also dynamical systems theory is heavily involved in the relatively recent field of combinatorics on words. Also combinatorial aspects of dynamical systems are studied.
From a physical point of view, continuous dynamical systems is a generalization of classical mechanics, a generalization where the equations of motion are postulated directly and are not constrained to be Euler–Lagrange equations of a least action principle. When difference equations are employed, the theory is called discrete dynamical systems.
An update scheme specifying the mechanism by which the mapping of individual vertex states is carried out so as to induce a discrete dynamical system with map F: K n → K n. The phase space associated to a dynamical system with map F: K n → K n is the finite directed graph with vertex set K n and directed edges (x, F(x)).
Maps may be parameterized by a discrete-time or a continuous-time parameter. Discrete maps usually take the form of iterated functions . Chaotic maps often occur in the study of dynamical systems .
In control engineering, a discrete-event dynamic system (DEDS) is a discrete-state, event-driven system of which the state evolution depends entirely on the occurrence of asynchronous discrete events over time.
Sequential dynamical systems (SDSs) are a class of graph dynamical systems. They are discrete dynamical systems which generalize many aspects of for example classical cellular automata, and they provide a framework for studying asynchronous processes over graphs. The analysis of SDSs uses techniques from combinatorics, abstract algebra, graph ...
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