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| {{dablink|This article presents the many ways to define a dynamical system. See the main article, [[dynamical system]], for an overview of the topic.}}
| | Andrew Simcox is the title his mothers and fathers gave him and he completely enjoys this title. He works as a bookkeeper. To perform lacross is the factor I love most of all. For many years he's been living in Mississippi and he doesn't plan on changing it.<br><br>Have a look at my weblog ... psychic phone; [http://www.skullrocker.com/blogs/post/10991 her latest blog], |
| {{Refimprove|date=September 2011}}
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| The '''dynamical system''' concept is a [[mathematics|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 (mathematics)|set]], without the need of a [[Differentiability|smooth]] space-time structure defined on it.
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| ==Formal definition==
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| 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 (mathematics)#Measure theory|measure theoretical]] in flavor. The measure theoretical definitions assume the existence of a measure-preserving transformation. This appears to exclude [[dissipative system]]s, 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.
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| 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.
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| == General definition ==
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| 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 (mathematics)|set]] and Φ is a [[function (mathematics)|function]]
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| :<math>\Phi: U \subset T \times M \to M</math>
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| with
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| :<math> I(x) = \{ t \in T : (t,x) \in U \}\,</math>
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| :<math>\Phi(0,x) = x\,</math>
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| :<math>\Phi(t_2,\Phi(t_1,x)) = \Phi(t_1 + t_2, x),\, </math> for <math>\, t_1, t_2, t_1 + t_2 \in I(x)\,</math>
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| 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.
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| We often write
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| :<math>\Phi_x(t) := \Phi(t,x)\,</math>
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| :<math>\Phi^t(x) := \Phi(t,x)\,</math>
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| if we take one of the variables as constant.
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| :<math>\Phi_x:I(x) \to M</math>
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| is called '''flow''' through ''x'' and its [[graph (function)|graph]] '''[[trajectory]]''' through ''x''. The set
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| :<math>\gamma_x:=\{\Phi(t,x) : t \in I(x)\}</math>
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| is called '''[[orbit (dynamics)|orbit]]''' through ''x''.
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| A subset ''S'' of the state space ''M'' is called Φ-'''invariant''' if for all ''x'' in ''S'' and all ''t'' in ''T''
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| :<math>\Phi(t,x) \in S.</math>
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| In particular, for ''S'' to be Φ-'''invariant,''' 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''.
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| == Geometrical cases ==
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| In the following cases, ''M'' is a manifold (or its extreme case a [[Graph (mathematics)|graph]]). Dynamical systems are defined as [[tuple]]s of which one element is a manifold.
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| === Real dynamical system ===
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| 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 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 R<sup>n</sup>, the dynamical system is ''finite-dimensional''; if not, the dynamical system is ''infinite-dimensional''.
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| === Discrete dynamical system ===
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| A ''discrete dynamical system'', ''[[discrete-time]] dynamical system'', ''map'' or ''cascade'' is a tuple (T, M, Φ) with T the [[integer]]s {{clarify|date=August 2013}}, 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''.
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| === Cellular automaton ===
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| 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 a function 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.
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| ==Measure theoretical definition==
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| :''See main article [[measure-preserving dynamical system]].''
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| 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 (mathematics)|set]], and Σ is a [[topology]] on ''X'', so that (''X'', Σ) is a σ-algebra. For every element σ in Σ, μ is its finite [[measure (mathematics)|measure]], so that the triplet (''X'', Σ, μ) is a [[measure space|probability space]]. A map Φ: ''X'' → ''X'' is said to be [[measurable function|Σ-measurable]] if and only if, for every σ in Σ, one has Φ<sup>−1</sup>(σ) ∈ Σ. A map Φ is said to '''preserve the measure''' if and only if, for every σ in Σ, one has μ(Φ<sup>−1</sup>(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'''.
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| The map Φ embodies the time evolution of the dynamical system. Thus, for discrete dynamical systems the [[iterated function|iterates]] <math>\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.
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| ===Relation to geometric definition===
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| 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'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 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.
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| 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 and unstable manifolds of the dynamical system; they behave physically under small perturbations; and they explain many of the observed statistics of hyperbolic systems.
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| == Construction of dynamical systems ==
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| 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 mechanics|classical mechanical systems]], that is the study of the [[initial value problem]]s for their describing systems of [[ordinary differential equation]]s.
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| :<math>\dot{\boldsymbol{x}}=\boldsymbol{v}(t,\boldsymbol{x})</math>
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| :<math>\boldsymbol{x}|_{{t=0}}=\boldsymbol{x}_0</math>
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| where
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| *<math>\scriptstyle{\dot{\boldsymbol{x}}}</math> represents the [[velocity]] of the material point '''x'''
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| *'''v''': ''T'' × ''M'' → ''M'' is a [[vector field]] in '''R'''<sup>''n''</sup> or '''C'''<sup>''n''</sup> and represents the change of [[velocity]] induced by the known [[force]]s acting on the given material point. Depending on the properties of this vector field, the mechanical system is called
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| **'''autonomous''', when '''v'''(''t'', '''x''') = '''v'''('''x''')
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| **'''homogeneous''' when '''v'''(''t'', '''0''') = 0 for all ''t''
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| The solution is the evolution function already introduced in above
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| :<math>\boldsymbol{{x}}(t)=\Phi(t,\boldsymbol{{x}}_0)</math>
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| Some formal manipulation of the system of [[differential equation]]s shown above gives a more general form of equations a dynamical system must satisfy
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| :<math>\dot{\boldsymbol{x}}-\boldsymbol{v}(t,\boldsymbol{x})=0 \qquad\Leftrightarrow\qquad \mathfrak{{G}}\left(t,\Phi(t,\boldsymbol{{x}}_0)\right)=0</math>
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| where <math>\scriptstyle\mathfrak{G}:{{(T\times M)}^M}\to\mathbf{C}</math> is a [[functional (mathematics)|functional]] from the set of evolution functions to the field of the complex numbers.
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| == Compactification of a dynamical system ==
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| 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*).
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| In compact dynamical systems the [[limit set]] of any orbit is [[non-empty]], [[compact space|compact]] and [[simply connected]].
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| == References ==
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| *Vladimir Igorevic Arnol'd "''Ordinary differential equations''", various editions from MIT Press and from Springer Verlag, chapter 1 "''Fundamental concepts''".
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| *I. D. Chueshov "''Introduction to the Theory of Infinite-Dimensional Dissipative Systems''" online version of first edition on the EMIS site [http://www.emis.de/monographs/Chueshov/].
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| *Roger Temam "''Infinite-Dimensional Dynamical Systems in Mechanics and Physics''" Springer Verlag 1988, 1997.
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| [[Category:Dynamical systems|Dynamical systems]]
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| [[it:Sistema dinamico]]
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