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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. [14]
In sports biomechanics, dynamical systems theory has emerged in the movement sciences as a viable framework for modeling athletic performance and efficiency. It comes as no surprise, since dynamical systems theory has its roots in Analytical mechanics. From psychophysiological perspective, the human movement system is a highly intricate network ...
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.
A Markov partition in mathematics is a tool used in dynamical systems theory, allowing the methods of symbolic dynamics to be applied to the study of hyperbolic dynamics.By using a Markov partition, the system can be made to resemble a discrete-time Markov process, with the long-term dynamical characteristics of the system represented as a Markov shift.
Conley's decomposition is characterized by a function known as complete Lyapunov function. Unlike traditional Lyapunov functions that are used to assert the stability of an equilibrium point (or a fixed point) and can be defined only on the basin of attraction of the corresponding attractor, complete Lyapunov functions must be defined on the whole phase-portrait.
Sharkovskii's theorem does not immediately apply to dynamical systems on other topological spaces. It is easy to find a circle map with periodic points of period 3 only: take a rotation by 120 degrees, for example. But some generalizations are possible, typically involving the mapping class group of the space minus a periodic orbit.
The theorem states that the behaviour of a dynamical system in a domain near a hyperbolic equilibrium point is qualitatively the same as the behaviour of its linearization near this equilibrium point, where hyperbolicity means that no eigenvalue of the linearization has real part equal to zero. Therefore, when dealing with such dynamical ...
The concept of fixed points is of primary importance in discrete dynamical systems. Another graphical technique that can be used for one-variable mappings is the spider web projection. After determining an initial value x 0 {\displaystyle x_{0}} on the horizontal axis, draw a vertical line from the initial value x 0 {\displaystyle x_{0}} to the ...