There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is an implicit relation that gives the state of the system for only a short time into the future. The concept of a dynamical system has its origins in Newtonian mechanics. Dynamical systems are a fundamental part of chaos theory, logistic map dynamics, bifurcation theory, the self-assembly and self-organization processes, and the edge of chaos concept. The study of dynamical systems is the focus of dynamical systems theory, which has applications to a wide variety of fields such as mathematics, physics, biology, chemistry, engineering, economics, history, and medicine. In order to make a prediction about the system's future behavior, an analytical solution of such equations or their integration over time through computer simulation is realized. In physics, a dynamical system is described as a "particle or ensemble of particles whose state varies over time and thus obeys differential equations involving time derivatives".
However, some systems are stochastic, in that random events also affect the evolution of the state variables. Often the function is deterministic, that is, for a given time interval only one future state follows from the current state. The evolution rule of the dynamical system is a function that describes what future states follow from the current state. This state is often given by a tuple of real numbers or by a vector in a geometrical manifold. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it.Īt any given time, a dynamical system has a state representing a point in an appropriate state space. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. The Lorenz attractor arises in the study of the Lorenz oscillator, a dynamical system.
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