Editing Dynamical system (section)

== Formal definition ==
In the most general sense,<ref>Giunti M. and Mazzola C. (2012), "[https://www.researchgate.net/publication/272943599_Dynamical_Systems_on_Monoids_Toward_a_General_Theory_of_Deterministic_Systems_and_Motion Dynamical systems on monoids: Toward a general theory of deterministic systems and motion]". In Minati G., Abram M., Pessa E. (eds.), ''Methods, models, simulations and approaches towards a general theory of change'', pp. 173–185, Singapore: World Scientific. {{ISBN|978-981-4383-32-5}}
</ref><ref>Mazzola C. and Giunti M. (2012), "[https://www.researchgate.net/publication/281244041_Reversible_dynamics_and_the_directionality_of_time Reversible dynamics and the directionality of time]". In Minati G., Abram M., Pessa E. (eds.), ''Methods, models, simulations and approaches towards a general theory of change'', pp. 161–171, Singapore: World Scientific. {{ISBN|978-981-4383-32-5}}.</ref>
a '''dynamical system''' is a [[tuple]] (''T'', ''X'', Φ) where ''T'' is a [[monoid]], written additively,  ''X'' is a non-empty [[set (mathematics)|set]] and Φ is a [[function (mathematics)|function]]
:<math>\Phi: U \subseteq (T \times X) \to X</math>
with
:<math>\mathrm{proj}_{2}(U) = X</math>  (where <math>\mathrm{proj}_{2}</math> is the 2nd [[Projection (set theory)|projection map]])
and for any ''x'' in ''X'':
:<math>\Phi(0,x) = x</math>
:<math>\Phi(t_2,\Phi(t_1,x)) = \Phi(t_2 + t_1, x),</math> 
for <math>\, t_1,\, t_2 + t_1 \in I(x)</math> and <math>\ t_2 \in I(\Phi(t_1, x)) </math>, where we have defined the set <math> I(x) := \{ t \in T : (t,x) \in U \}</math> for any ''x'' in ''X''.

In particular, in the case that <math> U = T \times X </math> we have for every ''x'' in ''X'' that <math> I(x) = T </math> and thus that Φ defines a [[Semigroup action|monoid action]] of ''T'' on ''X''.

The function Φ(''t'',''x'') is called the '''evolution function''' of the dynamical system: it associates to every point ''x'' in the set ''X'' a unique image, depending on the variable ''t'', called the '''evolution parameter'''. ''X'' is called '''[[phase space]]''' or '''state space''', while the variable ''x'' represents an '''initial state''' of the system.

We often write 
:<math>\Phi_x(t) \equiv \Phi(t,x)</math>
:<math>\Phi^t(x) \equiv \Phi(t,x)</math>
if we take one of the variables as constant. The function 
:<math>\Phi_x:I(x) \to X</math>
is called the '''flow''' through ''x'' and its [[graph (function)|graph]] is called the '''[[trajectory]]''' through ''x''. The set
:<math>\gamma_x \equiv\{\Phi(t,x) : t \in I(x)\}</math>
is called the '''[[orbit (dynamics)|orbit]]''' through ''x''.
The orbit through ''x'' is the [[image (mathematics)|image]] of the flow through ''x''.
A subset ''S'' of the state space ''X'' is called Φ-'''invariant''' if for all ''x'' in ''S'' and all ''t'' in ''T''
:<math>\Phi(t,x) \in S.</math>
Thus, in particular, if ''S'' is Φ-'''invariant''', <math>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''.

More commonly there are two classes of definitions for a dynamical system: one is motivated by [[ordinary differential equation]]s and is geometrical in flavor; and the other is motivated by [[ergodic theory]] and is [[Measure (mathematics)#Measure theory|measure theoretical]] in flavor.

=== Geometrical definition ===
In the geometrical definition, a dynamical system is the tuple <math> \langle  \mathcal{T}, \mathcal{M}, f\rangle </math>. <math>\mathcal{T}</math> is the domain for time – there are many choices, usually the reals or the integers, possibly restricted to be non-negative. <math>\mathcal{M}</math> is a [[manifold]], i.e. locally a Banach space or Euclidean space, or in the discrete case a [[Graph (discrete mathematics)|graph]]. ''f'' is an evolution rule ''t''&nbsp;→&nbsp;''f''<sup>&nbsp;''t''</sup> (with <math>t\in\mathcal{T}</math>) such that ''f<sup>&nbsp;t</sup>'' is a [[diffeomorphism]] of the manifold to itself. So, f is a "smooth" mapping of the time-domain <math> \mathcal{T}</math> into the space of diffeomorphisms of the manifold to itself. In other terms, ''f''(''t'') is a diffeomorphism, for every time ''t'' in the domain <math> \mathcal{T}</math> .

==== Real dynamical system ====
A ''real dynamical system'', ''real-time dynamical system'', ''[[continuous time]] dynamical system'', or ''[[Flow (mathematics)|flow]]'' is a tuple (''T'', ''M'', Φ) with ''T'' an [[open interval]] in the [[real number]]s '''R''', ''M'' a [[manifold]] locally [[diffeomorphic]] to a [[Banach space]], and  Φ a [[continuous function]]. If Φ is [[continuously differentiable]] we say the system is a ''differentiable dynamical system''.  If the manifold ''M'' is locally diffeomorphic to '''R'''<sup>''n''</sup>, the dynamical system is ''finite-dimensional''; if not, the dynamical system is ''infinite-dimensional''. This does not assume a [[symplectic manifold|symplectic structure]]. When ''T'' is taken to be the reals, the dynamical system is called ''global'' or a ''[[Flow (mathematics)|flow]]''; and if ''T'' is restricted to the non-negative reals, then the dynamical system is a ''semi-flow''.

==== Discrete dynamical system ====
A ''discrete dynamical system'', ''[[discrete-time]] dynamical system'' is a tuple (''T'', ''M'', Φ), where ''M'' is a [[manifold]] locally diffeomorphic to a [[Banach space]], and  Φ is a function. When ''T'' is taken to be the integers, it is a ''cascade'' or a ''map''. If ''T'' is restricted to the non-negative integers we call the system a ''semi-cascade''.<ref>{{Cite book|title=Discrete Dynamical Systems|last=Galor|first=Oded|publisher=Springer|year=2010}}</ref>

==== Cellular automaton ====
A ''cellular automaton'' is a tuple (''T'', ''M'', Φ), with ''T'' a [[lattice (group)|lattice]] such as the [[integer]]s or a higher-dimensional [[integer lattice|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.

==== Multidimensional generalization ====
Dynamical systems are usually defined over a single independent variable, thought of as time. A more general class of systems are defined over multiple independent variables and are therefore called [[multidimensional systems]]. Such systems are useful for modeling, for example, [[image processing]].

==== Compactification of a dynamical system ====
Given a global dynamical system ('''R''', ''X'', Φ) on a [[locally compact]] and [[Hausdorff space|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 space|compact]] and [[simply connected]].

===Measure theoretical definition===
{{main|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 (mathematics)|set]], and (''X'', Σ, ''μ'') is a [[measure space|probability space]], meaning that Σ is a [[sigma-algebra]] on ''X'' and μ is a finite [[measure (mathematics)|measure]] on (''X'', Σ). A map Φ: ''X'' → ''X'' is said to be [[measurable function|Σ-measurable]] if and only if, for every σ in Σ, one has <math>\Phi^{-1}\sigma \in \Sigma</math>. A map Φ is said to '''preserve the measure''' if and only if, for every ''σ'' in Σ, one has <math>\mu(\Phi^{-1}\sigma ) = \mu(\sigma)</math>. 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 [[iterated function|iterates]] <math>\Phi^n = \Phi \circ \Phi \circ \dots \circ \Phi</math> for every integer ''n'' are studied. For continuous dynamical systems, the map Φ is understood to be a finite time evolution map and the construction is more complicated.

====Relation to geometric definition====

The measure theoretical definition assumes the existence of a measure-preserving transformation. Many different invariant measures can be associated to any one evolution rule. If the dynamical system is given by a system of differential equations the appropriate measure must be determined. This 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 and assumes the choice has been made. A simple construction (sometimes called the [[Krylov–Bogolyubov theorem]]) shows that for a large class of systems 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.

Some systems have a natural measure, such as the [[Liouville's theorem (Hamiltonian)|Liouville measure]] in [[Hamiltonian system]]s, chosen over other invariant measures, such as the measures supported on periodic orbits of the Hamiltonian system.  For chaotic [[dissipative system]]s 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. A small region of phase space shrinks under time evolution.

For hyperbolic dynamical systems, the [[Sinai–Ruelle–Bowen measure]]s appear to be the natural choice.  They are constructed on the geometrical structure of [[stable manifold|stable and unstable manifold]]s of the dynamical system; they behave physically under small perturbations; and they explain many of the observed statistics of hyperbolic systems.