1. 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 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.

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.

2. General Definition

In the most general sense,^{[1] [2]} a dynamical system is a tuple (T, M, Φ) where T is a monoid, written additively, M is a non-empty set and Φ is a function

[math]\displaystyle{ \Phi: U \subseteq (T \times M) \to M }[/math]

with

[math]\displaystyle{ \mathrm{proj}_{2}(U) = M }[/math] (where [math]\displaystyle{ \mathrm{proj}_{2} }[/math] is the 2nd projection map)

[math]\displaystyle{ I(x) = \{ t \in T : (t,x) \in U \} }[/math]

[math]\displaystyle{ \Phi(0,x) = x }[/math]

[math]\displaystyle{ \Phi(t_2,\Phi(t_1,x)) = \Phi(t_2 + t_1, x), }[/math] for [math]\displaystyle{ \, t_1,\, t_2 + t_1 \in I(x) }[/math] and [math]\displaystyle{ \ t_2 \in I(\Phi(t_1, x)) }[/math]

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

[math]\displaystyle{ \Phi_x(t) \equiv \Phi(t,x) }[/math]

[math]\displaystyle{ \Phi^t(x) \equiv \Phi(t,x) }[/math]

if we take one of the variables as constant.

[math]\displaystyle{ \Phi_x:I(x) \to M }[/math]

is called the flow through x and its graph trajectory through x. The set

[math]\displaystyle{ \gamma_x \equiv\{\Phi(t,x) : t \in I(x)\} }[/math]

is called the orbit through x. Note that the orbit through x is the image of the flow through x. A subset S of the state space M is called Φ-invariant if for all x in S and all t in T

[math]\displaystyle{ \Phi(t,x) \in S. }[/math]

Thus, in particular, if S is Φ-invariant, [math]\displaystyle{ I(x) = T }[/math] for all x in S. That is, the flow through x must be defined for all time for every element of S.

3. Geometrical Cases

In the following cases, M is a manifold (or its extreme case a graph). Dynamical systems are defined as tuples of which one element is a manifold.

3.1. Real dDynamical sSystem

A real dynamical system, real-time dynamical system, continuous 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; if not, the dynamical system is infinite-dimensional. Note that this does not assume a symplectic structure.

3.2. Discrete dynamical sSystem

A discrete dynamical system, discrete-time dynamical system, map or cascade is a tuple (T, M, Φ) where T is the set of integers, M is a manifold locally diffeomorphic to a Banach space, and Φ is a function. If T is restricted to the non-negative integers we call the system a semi-cascade.^[3]

3.3. Cellular aAutomaton

A cellular automaton is a tuple (T, M, Φ), with T a lattice such as the integers or a higher-dimensional integer grid, M is a set of functions from an integer lattice (again, with one or more dimensions) to a finite set, and Φ a (locally defined) evolution function. As such cellular automata are dynamical systems. The lattice in M represents the "space" lattice, while the one in T represents the "time" lattice.

4. Measure Theoretical Definition

Main page: Measure-preserving dynamical system

A dynamical system may be defined formally, as a measure-preserving transformation of a measure space, the triplet (T, (X, Σ, μ), Φ). Here, T is a monoid (usually the non-negative integers), X is a set, and (X, Σ, μ) is a probability space. A map Φ: X → X is said to be Σ-measurable if and only if, for every σ in Σ, one has Φ⁻¹(σ) ∈ Σ. A map Φ is said to preserve the measure if and only if, for every σ in Σ, 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.

The map Φ embodies the time evolution of the dynamical system. Thus, for discrete dynamical systems the iterates [math]\displaystyle{ \scriptstyle\phi^n=\phi\circ\phi\circ\ldots\circ\phi }[/math] 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.

4.1. Relation to gGeometric dDefinition

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 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.

5. 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.

[math]\displaystyle{ \dot{\boldsymbol{x}}=\boldsymbol{v}(t,\boldsymbol{x}) }[/math]

[math]\displaystyle{ \boldsymbol{x}|_{{t=0}}=\boldsymbol{x}_0 }[/math]

where

• [math]\displaystyle{ \scriptstyle{\dot{\boldsymbol{x}}} }[/math] represents the velocity of the material point x
• v: T × M → TM is a vector field in Rⁿ or Cⁿ 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 v(t, x) = v(x)
  • homogeneous when v(t, 0) = 0 for all t

The solution is the evolution function already introduced in above

[math]\displaystyle{ \boldsymbol{{x}}(t)=\Phi(t,\boldsymbol{{x}}_0) }[/math]

Some formal manipulation of the system of differential equations shown above gives a more general form of equations a dynamical system must satisfy

[math]\displaystyle{ \dot{\boldsymbol{x}}-\boldsymbol{v}(t,\boldsymbol{x})=0 \qquad\Leftrightarrow\qquad \mathfrak{{G}}\left(t,\Phi(t,\boldsymbol{{x}}_0)\right)=0 }[/math]

where [math]\displaystyle{ \mathfrak{G}:{{(T\times M)}^M}\to\mathbf{C} }[/math] is a functional from the set of evolution functions to the field of the complex numbers.

6. 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 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 non-empty, compact and simply connected.