Dynamical system (definition)

This article presents the many ways to define a dynamical system. See the main article, dynamical system, for an overview of the topic.
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 spacetime structure defined on it.
Contents
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 assume the existence of a measurepreserving 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 KrylovBogolyubov theorem) shows that it is always possible to construct a measure so as to make the evolution rule of the dynamical system a measurepreserving transformation. In the construction a given measure of the state space is summed for all future points of a trajectory, assuring the invariance.
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 systemsmotivated definition within ergodic theory that sidesteps the choice of measure.
General definition
In the most general sense, a dynamical system is a tuple (T, M, Φ) where T is a monoid, written additively, M is a set and Φ is a function
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 represents an initial state of the system.
We often write
if we take one of the variables as constant.
is called flow through x and its graph trajectory through x. The set
is called orbit through x.
A subset S of the state space M is called Φinvariant if for all x in S and all t in T
In particular, for S to be Φinvariant, we require that I(x) = T for all x in S. That is, the flow through x should be defined for all time for every element of S.
Geometrical cases
In these cases M is a manifold (or its extreme case a graph).
Real dynamical system
A real dynamical system, realtime 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 nonnegative reals we call the system a semiflow. If Φ is continuously differentiable we say the system is a differentiable dynamical system. If the manifold M is locally diffeomorphic to R^{n} the dynamical system is finitedimensional and if not, the dynamical system is infinitedimensional.
Discrete dynamical system
A discrete dynamical system, discretetime 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 nonnegative integers we call the system a semicascade.
Cellular automaton
A cellular automaton is a tuple (T, M, Φ), with T a lattice such as the integers or a higher dimensional integer grid, M a finite set, and Φ an evolution function. Some cellular automata are reversible dynamical systems, although most are not.
Measure theoretical definition
 See main article measurepreserving dynamical system.
A dynamical system may be defined formally, as a measurepreserving transformation of a sigmaalgebra, the triplet (T,(X,Σ,μ),ϕ). Here, T is a monoid (usually the nonnegative integers), X is a set, and Σ is a topology on X, so that (X,Σ) is a sigmaalgebra. 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 Σmeasurable 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 measurepreserving transformation of X , if it is a map from X to itself, it is Σmeasurable, and is measurepreserving. The triplet (T,(X,Σ,μ),ϕ), for such a Φ, is then defined to be a dynamical system.
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.
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.
For hyperbolic dynamical systems, the SinaiRuelleBowen 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.
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
Some formal manipulation of the system of differential equations shown above gives a more general form of equations a dynamical system must satisfy
where is a functional from the set of evolution functions to the field of the complex numbers.
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 onepoint compactification X* of X. Although we lose the differential structure of the original system we can now use compactness arguments to analyze the new system (R, X*, Φ*).
In compact dynamical systems the limit set of any orbit is nonempty, compact and simply connected.
References
 Vladimir Igorevic Arnol'd "Ordinary differential equations", various editions from MIT Press and from Springer Verlag, chapter 1 "Fundamental concepts".
 I. D. Chueshov "Introduction to the Theory of InfiniteDimensional Dissipative Systems" online version of first edition on the EMIS site [1].
 Roger Temam "InfiniteDimensional Dynamical Systems in Mechanics and Physics" Springer Verlag 1988, 1997.
Categories: Dynamical systems
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