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| [[Image:homoclinic.svg|200px|thumb|right|A homoclinic orbit]]
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| [[Image:oriented.png|200px|thumb|right|An oriented homoclinic orbit]]
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| [[Image:mobius.png|200px|thumb|right|A twisted homoclinic orbit]]
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| In [[mathematics]], a '''homoclinic orbit''' is a trajectory of a [[flow (mathematics)|flow]] of a [[dynamical system]] which joins a saddle equilibrium point to itself. More precisely, a homoclinic orbit lies in the intersection of the [[stable manifold]] and the [[unstable manifold]] of an [[equilibrium point|equilibrium]].
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| Consider the continuous dynamical system described by the ODE
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| :<math>\dot x=f(x)</math>
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| Suppose there is an equilibrium at <math>x=x_0</math>, then a solution <math>\Phi(t)</math> is a homoclinic orbit if
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| :<math>\Phi(t)\rightarrow x_0\quad \mathrm{as}\quad
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| t\rightarrow\pm\infty</math>
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| If the [[phase space]] has three or more dimensions, then it is important to consider the [[topology]] of the unstable manifold of the saddle point. The figures show two cases. First, when the unstable manifold is topologically a cylinder, and secondly, when the unstable manifold is topologically a [[Möbius strip]]; in this case the homoclinic orbit is called ''twisted''.
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| == Discrete dynamical system ==
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| Homoclinic orbits and '''homoclinic points''' are defined in the same way for [[iterated function]]s, as the intersection of the [[stable set]] and [[unstable set]] of some [[Fixed point (mathematics)|fixed point]] or [[periodic point]] of the system.
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| We also have the notion of homoclinic orbit when considering discrete dynamical systems. In such a case, if <math>f:M\rightarrow M</math> is a [[diffeomorphism]] of a [[manifold]] <math>M</math>, we say that <math>x</math> is a homoclinic point if it has the same past and future - more specifically, if it exists a fixed (or periodic) point
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| <math>p</math> such that
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| :<math>\lim_{n\rightarrow \pm\infty}f^n(x)=p.</math>
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| == Properties ==
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| The existence of one homoclinic point implies the existence of infinite number of them.<ref>{{cite book|last=Ott|first=Edward|title=Chaos in Dynamical Systems|year=1994|publisher=Cambridge University Press}}</ref>
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| This comes from its definition: the intersection of a stable and unstable set. Both sets are [[Positive invariant set|invariant]] by definition, which means that the forward iteration of the homoclinic point is both on the stable and unstable set. By iterating N times, the map approaches the equilibrium point by the stable set, but in every iteration it is on the unstable manifold too, which shows this property.
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| This property suggests that complicated dynamics arise by the existence of a homoclinic point. Indeed, Smale (1967)<ref>{{cite book|last=Smale|first=Stephen|title=Differentiable dynamical systems|year=1967|publisher=Bull. Amer. Math. Soc.73, 747-817}}</ref> showed that these points leads to [[horseshoe map]] like dynamics, which is associated with chaos.
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| == Symbolic dynamics ==
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| By using the [[Markov partition]], the long-time behaviour of [[hyperbolic system]] can be studied using the techniques of [[symbolic dynamics]]. In this case, a homoclinic orbit has a particularly simple and clear representation. Suppose that <math>S=\{1,2,\ldots,M\}</math> is a [[finite set]] of ''M'' symbols. The dynamics of a point ''x'' is then represented by a [[bi-infinite string]] of symbols
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| :<math>\sigma =\{(\ldots,s_{-1},s_0,s_1,\ldots) : s_k \in S \; \forall k \in \mathbb{Z} \}</math>
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| A [[periodic point]] of the system is simply a recurring sequence of letters. A [[heteroclinic orbit]] is then the joining of two distinct periodic orbits. It may be written as
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| :<math>p^\omega s_1 s_2 \cdots s_n q^\omega</math>
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| where <math>p= t_1 t_2 \cdots t_k</math> is a sequence of symbols of length ''k'', (of course, <math>t_i\in S</math>), and <math>q = r_1 r_2 \cdots r_m</math> is another sequence of symbols, of length ''m'' (likewise, <math>r_i\in S</math>). The notation <math>p^\omega</math> simply denotes the repetition of ''p'' an infinite number of times. Thus, a heteroclinic orbit can be understood as the transition from one periodic orbit to another. By contrast, a homoclinic orbit can be written as
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| :<math>p^\omega s_1 s_2 \cdots s_n p^\omega</math>
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| with the intermediate sequence <math>s_1 s_2 \cdots s_n</math> being non-empty, and, of course, not being ''p'', as otherwise, the orbit would simply be <math>p^\omega</math>.
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| == See also ==
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| * [[Heteroclinic orbit]]
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| * [[Homoclinic bifurcation]]
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| == References ==
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| {{Reflist}}
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| * [[John Guckenheimer]] and [[Philip Holmes]], Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Applied Mathematical Sciences Vol. 42), Springer
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| == External links ==
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| * [http://www.ibiblio.org/e-notes/Chaos/homoclinic.htm Homoclinic orbits in Henon map] with Java applets and comments
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| [[Category:Dynamical systems]]
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