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In mathematics, a **measure-preserving dynamical system** is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular.

A measure-preserving dynamical system is defined as a probability space and a measure-preserving transformation on it. In more detail, it is a system

with the following structure:

This definition can be generalized to the case in which *T* is not a single transformation that is iterated to give the dynamics of the system, but instead is a monoid (or even a group) of transformations *T _{s}* :

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Measure-preserving_dynamical_system

In mathematics, a **dynamical system** is a system in which a function describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a lake.

At any given time a dynamical system has a state given by a set of real numbers (a vector) that can be represented by a point in an appropriate state space (a geometrical manifold). The *evolution rule* of the dynamical system is a function that describes what future states follow from the current state. Often the function is deterministic; in other words, for a given time interval only one future state follows from the current state; however, some systems are stochastic, in that random events also affect the evolution of the state variables.

The concept of a dynamical system has its origins in Newtonian mechanics. 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 relation is either a differential equation, difference equation or other time scale.) To determine the state for all future times requires iterating the relation many times—each advancing time a small step. The iteration procedure is referred to as *solving the system* or *integrating the system*. If the system can be solved, given an initial point it is possible to determine all its future positions, a collection of points known as a *trajectory* or *orbit*.

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Dynamical_system

The **dynamical system** concept is a mathematical formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space. The concept unifies very different types of such "rules" in mathematics: the different choices made for how time is measured and the special properties of the ambient space may give an idea of the vastness of the class of objects described by this concept. 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 ambient space may be simply a set, without the need of a smooth space-time structure defined on it.

There are two classes of definitions for a dynamical system: one is motivated by ordinary differential equations and is geometrical in flavor; and the other is motivated by ergodic theory and is measure theoretical in flavor. The measure theoretical definitions assume the existence of a measure-preserving transformation. This appears to exclude dissipative systems, as in a dissipative system a small region of phase space shrinks under time evolution. A simple construction (sometimes called the Krylov-Bogolyubov theorem) shows that it is always possible to construct a measure so as to make the evolution rule of the dynamical system a measure-preserving transformation. In the construction a given measure of the state space is summed for all future points of a trajectory, assuring the invariance.

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Dynamical_system_(definition)

**Dynamical systems theory** is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called *continuous dynamical systems*. When difference equations are employed, the theory is called *discrete dynamical systems*. When the time variable runs over a set that is discrete over some intervals and continuous over other intervals or is any arbitrary time-set such as a cantor set—one gets dynamic equations on time scales. Some situations may also be modeled by mixed operators, such as differential-difference equations.

This theory deals with the long-term qualitative behavior of dynamical systems, and studies the nature of, and when possible the solutions of, the equations of motion of systems that are often primarily mechanical or otherwise physical in nature, such as planetary orbits and the behaviour of electronic circuits, as well as systems that arise in biology, economics, and elsewhere. Much of modern research is focused on the study of chaotic systems.

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Dynamical_systems_theory

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