Generalized quadrangle - Revision history

en>Anurag Bishnoi: /* References */

2014-12-06T16:25:30Z

References

← Older revision		Revision as of 18:25, 6 December 2014
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	~~[[Image:homoclinic.svg\|200px\|thumb\|right\|A homoclinic orbit]]~~		Hi there, I am Alyson Pomerleau and I believe it sounds quite great when you say it. Alaska is exactly where I've always been living. He functions as a bookkeeper. What I love doing is football but I don't have the time recently.<br><br>Visit my site - [http://jplusfn.gaplus.kr/xe/qna/78647 best psychic]
	~~[[Image:oriented.png\|200px\|thumb\|right\|An oriented homoclinic orbit]]~~
	~~[[Image:mobius.png\|200px\|thumb\|right\|A twisted homoclinic orbit]]~~

	~~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]].~~

	~~Consider the continuous dynamical system described by the ODE~~

	~~:<math>\dot x=f(x)</math>~~

	~~Suppose there is an equilibrium at <math>x=x_0</math>, then a solution <math>\Phi(t)</math> is a homoclinic orbit if~~

	~~:<math>\Phi(t)\rightarrow x_0\quad \mathrm{as}\quad~~
	~~t\rightarrow\pm\infty</math>~~

	~~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''.~~

	~~== Discrete dynamical system ==~~
	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.

	~~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~~
	~~<math>p</math> such that~~

	~~:<math>\lim_{n\rightarrow \pm\infty}f^n(x)=p~~.~~</math>~~

	~~== Properties ==~~
	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>
	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.

	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.

	~~== Symbolic dynamics ==~~
	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~~

	~~:<math>\sigma =\{(\ldots,s_{-1},s_0,s_1,\ldots) : s_k \in S \; \forall k \in \mathbb{Z} \}</math>~~

	~~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~~

	~~:<math>p^\omega s_1 s_2 \cdots s_n q^\omega</math>~~

	~~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~~

	:<~~math~~>~~p^\omega s_1 s_2 \cdots s_n p^\omega~~<~~/math~~>

	~~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>.~~

	~~== See also ==~~
	* [[Heteroclinic orbit]]
	* [[Homoclinic bifurcation]]

	~~== References ==~~
	~~{{Reflist}}~~
	* [[John Guckenheimer]] and [[Philip Holmes]], Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Applied Mathematical Sciences Vol. 42), Springer

	~~== External links ==~~
	* [http://~~www~~.~~ibiblio~~.~~org~~/~~e-notes~~/~~Chaos~~/~~homoclinic.htm Homoclinic orbits in Henon map] with Java applets and comments~~

	~~[[Category:Dynamical systems]~~]

en>Addbot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q2754558

2013-03-15T04:58:19Z

Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q2754558

← Older revision		Revision as of 06:58, 15 March 2013
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	Hi there. My title is ~~Sophia Meagher although~~ it is ~~not~~ the title ~~on my beginning certificate~~. ~~To play lacross~~ is 1 of the ~~issues she enjoys most~~. ~~Her family life~~ in ~~Alaska but her spouse desires them to transfer~~. ~~He is~~ an ~~information officer~~.<br><br>~~Feel free to surf to my web~~-~~site;~~ [http://~~conniecolin~~.~~com~~/xe/~~community~~/~~24580 real psychic readings~~]		[[Image:homoclinic.svg\|200px\|thumb\|right\|A homoclinic orbit]]
			[[Image:oriented.png\|200px\|thumb\|right\|An oriented homoclinic orbit]]
			[[Image:mobius.png\|200px\|thumb\|right\|A twisted homoclinic orbit]]

			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]].

			Consider the continuous dynamical system described by the ODE

			:<math>\dot x=f(x)</math>

			Suppose there is an equilibrium at <math>x=x_0</math>, then a solution <math>\Phi(t)</math> is a homoclinic orbit if

			:<math>\Phi(t)\rightarrow x_0\quad \mathrm{as}\quad
			t\rightarrow\pm\infty</math>

			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''.

			== Discrete dynamical system ==
			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.

			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
			<math>p</math> such that

			:<math>\lim_{n\rightarrow \pm\infty}f^n(x)=p.</math>

			== Properties ==
			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>
			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.

			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.

			== Symbolic dynamics ==
			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

			:<math>\sigma =\{(\ldots,s_{-1},s_0,s_1,\ldots) : s_k \in S \; \forall k \in \mathbb{Z} \}</math>

			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

			:<math>p^\omega s_1 s_2 \cdots s_n q^\omega</math>

			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

			:<math>p^\omega s_1 s_2 \cdots s_n p^\omega</math>

			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>.

			== See also ==
			* [[Heteroclinic orbit]]
			* [[Homoclinic bifurcation]]

			== References ==
			{{Reflist}}
			* [[John Guckenheimer]] and [[Philip Holmes]], Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Applied Mathematical Sciences Vol. 42), Springer

			== External links ==
			* [http://www.ibiblio.org/e-notes/Chaos/homoclinic.htm Homoclinic orbits in Henon map] with Java applets and comments

			[[Category:Dynamical systems]]

en>Wcherowi: fixed caption

2012-08-25T03:01:06Z

fixed caption

New page

Hi there. My title is Sophia Meagher although it is not the title on my beginning certificate. To play lacross is 1 of the issues she enjoys most. Her family life in Alaska but her spouse desires them to transfer. He is an information officer.<br><br>Feel free to surf to my web-site; [http://conniecolin.com/xe/community/24580 real psychic readings]