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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>
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| * A [[stable manifold]] and an unstable manifold exist,
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| * [[Shadowing lemma|Shadowing]] occurs,
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| * The dynamics on the invariant set can be represented via [[symbolic dynamics]],
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| * A natural measure can be defined,
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| * The system is [[Structural stability|structurally stable]].
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| [[Image:Phase Portrait Sadle.svg|thumb|right|Orbits near a two-dimensional saddle point, an example of a hyperbolic equilibrium.]]
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| == Maps ==
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| 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.
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| 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]]:
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| :<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>
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| Since the eigenvalues are given by
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| :<math>\lambda_{1}=\frac{3+\sqrt{5}}{2}>1</math>
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| :<math>\lambda_{2}=\frac{3-\sqrt{5}}{2}<1</math>
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| == Flows ==
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| 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>
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| 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.
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| === Example ===
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| Consider the nonlinear system
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| :<math>\frac{ dx }{ dt } = y,</math>
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| :<math>\frac{ dy }{ dt } = -x-x^3-\alpha y,~ \alpha \ne 0</math>
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| (0, 0) is the only equilibrium point. The linearization at the equilibrium is
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| :<math>J(0,0) = \begin{pmatrix}
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| 0 & 1 \\
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| -1 & -\alpha \end{pmatrix}</math>.
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| 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).
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| == Comments ==
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| 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.
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| == See also ==
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| * [[Anosov flow]]
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| * [[Hyperbolic set]]
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| * [[Normally hyperbolic invariant manifold]]
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| == Notes ==
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| <references/>
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| == References ==
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| * {{Scholarpedia|title=Equilibrium|urlname=Equilibrium|curator=Eugene M. Izhikevich}}
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| [[Category:Limit sets]]
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| [[Category:Stability theory]]
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