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 the ambient space may give an idea of the vastity of the 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. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In logic, mathematics, and computer science, a formal system is a formal grammar used for modelling purposes. ... A pocket watch, a device used to tell time Look up time in Wiktionary, the free dictionary. ... Space has been an interest for philosophers and scientists for much of human history. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ...

Formal definition

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 assumes 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-Bogoliubov 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. NikolaI Sergeevich Krylov (1918–1947) was a Russian theoretical physicist known for his work in statistical physics. ... Nikolai Nikolaevich Bogoliubov (21 August 1909 – 13 February 1992) was a Russian-Ukrainian mathematician and theoretical physicist known for his work in statistical field theory and dynamical systems. ...

The difficulty in constructing the natural measure for a dynamical system makes it difficult to develop ergodic theory starting from differential equations, so it becomes convenient to have a dynamical systems-motivated definition within ergodic theory that side-steps the choice of measure.

General definition

In the most general sense, a dynamical system is a tuple (T, M, Φ) with T is a monoid with the operation of the group indicated by +, M a set and Φ a function In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... This article is about sets in mathematics. ... Partial plot of a function f. ...

with

for

The function Φ(t,x) is called the evolution function of the dynamical system: it associates to every point in the set M a unique image, depending on the variable t, called the evolution parameter. M is called phase space or state space, while the variable x is called initial state of the system.

We often write

if we take one of the variables as constant.

Then

is called flow through x and its graph trajectory through x. The set Mathematically the term trajectory refers to the ordered set of states which are assumed by a dynamical system over time (see e. ...

is called orbit through x. In the study of dynamical systems, an orbit is the sequence generated by iterating a map. ...

A subset S of the state space M is called Φ-invariant if for all x in S and all t in T

Geometrical cases

In these cases M is a manifold (or its extreme case a graph).

Real dynamical system

A real dynamical system, real-time dynamical system or flow is a tuple (T, M, Φ) with T an open interval in the real numbers R, M a manifold locally diffeomorphic to a Banach space, and Φ a continuous function. If T=R we call the system global, if T is restricted to the non-negative reals we call the system a semi-flow. If Φ is continuously differentiable we say the system is a differentiable dynamical system. If the manifold M is locally diffeomorphic to Rⁿ the dynamical system is finite-dimensional and if not, the dynamical system is infinite-dimensional. In mathematics, flow refers to the group action of a one-parameter group on a set. ... In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ...

Discrete dynamical system

A discrete dynamical system, discrete-time dynamical system, map or cascade is a tuple (T, M, Φ) with T the integers, M a manifold locally diffeomorphic to a Banach space, and Φ a function. If T is restricted to the non-negative integers we call the system a semi-cascade. The integers are commonly denoted by the above symbol. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ...

Cellular automaton

A cellular automaton is a tuple (T, M, Φ), with T the integers, M a finite set, and Φ an evolution function. Some cellular automata are reversible dynamical systems, although most are not. A cellular automaton (plural: cellular automata) is a discrete model studied in computability theory and mathematics. ...

Measure theoretical definition

See main article measure-preserving dynamical system.

A dynamical system may be defined formally, as a measure-preserving transformation of a sigma-algebra, the triplet (T,(X,Σ,μ),φ). Here, T is a monoid (usually the non-negative integers), X is a set, and Σ is a topology on X, so that (X,Σ) is a sigma-algebra. For every element , μ is its finite measure, so that the triplet (X,Σ,μ) is a probability space. A map is said to be Σ-measurable if and only if, for every , one has . A map Φ is said to preserve the measure if and only if, for every , one has μ(φ ^{− 1}σ) = μ(σ). Combining the above, a map Φ is said to be a measure-preserving transformation of X , if it is a map from X to itself, it is Σ-measurable, and is measure-preserving. The triplet (T,(X,Σ,μ),φ), for such a Φ, is then defined to be a dynamical system. In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of ergodic theory. ... In mathematics, a σ-algebra (or σ-field) X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ... This article is about sets in mathematics. ... A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... In mathematics, a measure is a function that assigns a number, e. ... In mathematics, a measure is a function that assigns a number, e. ... In mathematics, measurable functions are well-behaved functions between measurable spaces. ...

The map Φ embodies the time evolution of the dynamical system. Thus, for discrete dynamical systems the iterates for every integer n are studied. For continuous dynamical systems, the map Φ is understood to be finite time evolution map and the construction is more complicated. In mathematics, iterated functions are the objects of study in fractals and dynamical systems. ...

Relation to geometric definition

Many different invariant measures can be associated to any one evolution rule. In ergodic theory the choice is assumed made, but if the dynamical system is given by a system of differential equations the appropriate measure must be determined. Some systems have a natural measure, such as the Liouville measure in Hamiltonian systems, chosen over other invariant measures, such as the measures supported on periodic orbits of the Hamiltonian system. For many dissipative chaotic systems the choice of invariant measure is technically more challenging. The measure needs to be supported on the attractor, but attractors have zero Lebesgue measure and the invariant measures must be singular with respect to the Lebesgue measure. In mathematical physics, Liouvilles theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. ... In classical mechanics, a Hamiltonian system is a physical system in which forces are velocity invariant. ... In dynamical systems, an attractor is a set to which the system evolves after a long enough time. ... In mathematics, the Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. ...

For hyperbolic dynamical systems, the Sinai-Ruelle-Bowen measures appear to be the natural choice. They are constructed on the geometrical structure of stable and unstable manifolds of the dynamical system; they behave physically under small perturbations; and they explain many of the observed statistics of hyperbolic systems.

Construction of dynamical systems

The concept of evolution in time is central to the theory of dynamical systems as seen in the previous sections: the basic reason for this fact is that the starting motivation of the theory was the study of time behavior of classical mechanical systems, that is the study of the initial value problems for their describing systems of ordinary differential equations. Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... In mathematics, an initial value problem is a statement of a differential equation together with specified value of the unknown function at a given point in the domain of the solution. ... In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ...

where

• represents the velocity of the material point
• is a vector field in or and represents the change of velocity induced by the known forces acting on the given material point. Depending on the properties of this vector field, the mechanical system is called

• autonomous, when
• homogeneous when for all

The solution is the evolution function already introduced in above The velocity of an object is its speed in a particular direction. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... The velocity of an object is its speed in a particular direction. ... In physics, force is an influence that may cause a body to accelerate. ...

Some formal manipulation of the system of differential equations shown above gives a more general form of equations a dynamical system must satisfy A simulation of airflow into a duct using the Navier-Stokes equations A differential equation is a mathematical equation for an unknown function of one or several variables which relates the values of the function itself and of its derivatives of various orders. ...

where is a functional from the set of evolution functions to the field of the complex numbers. Generally, functional refers to something with and able to fulfill its purpose or function. ...

Compactification of a dynamical system

Given a global dynamical system (R, X, Φ) on a locally compact and Hausdorff topological space X, it is often useful to study the continuous extension Φ* of Φ to the one-point compactification X* of X. Although we loose the differential structure of the original system we can now use compactness arguments to analyze the new system (R, X*, Φ*). In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... Hausdorff may refer to: A Hausdorff space, when used as an adjective, as in the real line is Hausdorff. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, compactification is applied to topological spaces to make them compact spaces. ...

In compact dynamical systems the limit set of any orbit is non-empty, compact and simply connected. In mathematics, a limit set is the set of cluster points of an iterated function. ... In set theory, a set is called non-empty (or nonempty) if it contains at least one element, and is therefore not the empty set. ... Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ...

References

• Vladimir Igorevic Arnol'd "Ordinary differential equations", various editions from MIT Press and from Springer Verlag, chapter 1 "Fundamental concepts".
• I. D. Chuesov "Introduction to the Theory of Infinite-Dimensional Dissipative Systems" online version of first edition on the EMIS site [1].
• Roger Teman "Infinite-dimensional dynamical systems in mechanics and physics" Springer Verlag 1989, 1997.