en>Mark viking: /* Motivation */ Added wl, period

2014-10-27T08:03:23Z

Motivation: Added wl, period

← Older revision		Revision as of 10:03, 27 October 2014
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	~~In the study of [[dynamical system]]s~~, ~~a '''hyperbolic equilibrium point''' or '''hyperbolic fixed point'~~'' is ~~a [[fixed point (mathematics)\|fixed point]] that~~ does ~~not have any [[center manifold]]s. Near a [[hyperbolic function\|hyperbolic]] point the orbits of~~ a ~~two-dimensional, [[Hamiltonian mechanics\|non-dissipative]] system resemble hyperbolas. This fails to hold in general~~. ~~Strogatz<ref>{{cite book\|last=Strogatz\|first=Steven\|title=Nonlinear Dynamics and Chaos\|year=2001\|publisher=Westview Press}}</ref> notes that "hyperbolic~~ is ~~an unfortunate name – it sounds like it should mean '[[saddle point]]' – but it has become standard~~.~~" Several properties hold about a neighborhood of a hyperbolic point, notably~~<~~ref~~>~~{{cite book\|last=Ott\|first=Edward\|title=Chaos in Dynamical Systems\|year=1994\|publisher=Cambridge University Press}}~~<~~/ref~~>		I would like to introduce myself to you, I am Jayson Simcox but I don't like when individuals use my complete title. I've usually cherished residing in Mississippi. Office supervising is what she does for a living. To play domino is some thing I truly enjoy doing.<br><br>Here is my website online psychics ([http://www.seekavideo.com/playlist/2199/video/ have a peek at this site])

	* A [[stable manifold]] and an unstable manifold exist,
	* [[Shadowing lemma\|Shadowing]] occurs,
	* The dynamics on the invariant set can be represented via [[symbolic dynamics]],
	* A natural measure can be defined,
	* The system is ~~[[Structural stability\|structurally stable]].~~

	~~[[Image:Phase Portrait Sadle.svg\|thumb\|right\|Orbits near a two-dimensional saddle point, an example of a hyperbolic equilibrium.]]~~

	~~== Maps ==~~
	If ''T'' : '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup> is a ''C''<sup>1</sup> map and ''p'' is a [[Fixed point (mathematics)\|fixed point]] then ''p'' is said to be a '''hyperbolic fixed point''' when the [[differential (mathematics)\|differential]]{{disambiguation needed\|date=July 2013}} ''DT''(''p'') has no eigenvalues on the unit circle.

	~~One example of a [[Map~~ (~~mathematics)\|map]] that its only fixed point is hyperbolic is the Arnold Map or~~ [~~[Arnold's cat map\|cat map]]:~~

	~~:<math>\begin{bmatrix} x_{n+1}\\ y_{n+1} \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 1 & 2\end{bmatrix} \begin{bmatrix} x_n\\ y_n\end{bmatrix} \quad \text{modulo }1</math>~~

	~~Since the eigenvalues are given by~~

	~~:<math>\lambda_{1}=\frac{3+\sqrt{5}}{2}>1</math>~~
	~~:<math>\lambda_{2}=\frac{3-\sqrt{5}}{2}<1</math>~~

	~~== Flows ==~~

	~~Let ''F''~~ : ~~'''R'''<sup>''n''<~~/~~sup> → '''R'''<sup>''n''</sup> be a ''C''<sup>1<~~/sup> (that is, continuously differentiable) [[vector field]] with a critical point ''p'' and let ''J'' denote the [[Jacobian matrix]] of ''F'' at ''p''. If the matrix ''J'' has no eigenvalues with zero real parts then ''p'' is called '''hyperbolic'''. Hyperbolic fixed points may also be called '''hyperbolic critical points''' or '''elementary critical points'''.~~<ref>Ralph Abraham and Jerrold E~~. ~~Marsden, ''Foundations of Mechanics'', (1978) Benjamin~~/~~Cummings Publishing, Reading Mass. ISBN 0-8053-0102-X</ref>~~

	The [[Hartman-Grobman theorem]] states that the orbit structure of a dynamical system in a [[neighbourhood (mathematics)\|neighbourhood]] of a hyperbolic equilibrium point is [[topological conjugacy\|topologically equivalent]] to the orbit structure of the [[linearization\|linearized]] dynamical system.

	~~=== Example ===~~
	~~Consider the nonlinear system~~
	~~:<math>\frac{ dx }{ dt } = y,</math>~~
	~~:<math>\frac{ dy }{ dt } = -x-x^3-\alpha y,~ \alpha \ne 0<~~/~~math>~~

	~~(0, 0) is the only equilibrium point. The linearization at the equilibrium is~~

	~~:<math>J(0,0) = \begin{pmatrix}~~
	~~0 & 1 \\~~
	~~-1 & -\alpha \end{pmatrix}<~~/~~math>.~~

	~~The eigenvalues of this matrix are <math>\frac{-\alpha \pm \sqrt{\alpha^2-4}}{2}<~~/~~math>. For all values of α ≠ 0, the eigenvalues~~ have ~~non-zero real part. Thus, this equilibrium point is a hyperbolic equilbrium point. The linearized system will behave similar to the non-linear system near (0, 0). When α = 0, the system has~~ a ~~nonhyperbolic equilibrium~~ at ~~(0, 0~~).

	~~== Comments ==~~
	~~In the case of an infinite dimensional system - for example systems involving a time delay - the notion of the "hyperbolic part of the spectrum" refers to the above property.~~

	~~== See also ==~~

	* [[Anosov flow]]
	* [[Hyperbolic set]]
	* [[Normally hyperbolic invariant manifold]]

	~~== Notes ==~~
	~~<references/>~~

	~~== References ==~~

	* {{Scholarpedia\|title=Equilibrium\|urlname=Equilibrium\|curator=Eugene M. Izhikevich}}

	~~[[Category:Limit sets]]~~
	~~[[Category:Stability theory]]~~

en>TakuyaMurata: /* Definition */ not an exterior derivative per the talk

2013-12-01T00:30:00Z

Definition: not an exterior derivative per the talk

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en>Kreizhn: Fixed hyperlink and article (a->an)

2012-08-24T15:52:26Z

Fixed hyperlink and article (a->an)

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Fundamental vector field - Revision history

en>Mark viking: /* Motivation */ Added wl, period

en>TakuyaMurata: /* Definition */ not an exterior derivative per the talk

en>Kreizhn: Fixed hyperlink and article (a->an)