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In ], especially in the study of ]s, a '''hyperbolic equilibrium point''' or '''hyperbolic fixed point''' is a special type of an ''equilibrium'', or a ]. In ], especially in the study of ]s, a '''hyperbolic equilibrium point''' or '''hyperbolic fixed point''' is a special type of an ], or a ].

The word ''hyperbolic'' is due to the fact that in the 2 dimensional case the orbits near the hyperbolic point lay on ]s centered in that point.

== Maps ==
If
:<math>T: \mathbb{R}^n \to \mathbb{R}^n</math>
is a ''C''<sup>1</sup> map and ''p'' is a ] then ''p'' is said to be a '''hyperbolic fixed point''' when the ] ''DT''(''p'') has no eigenvalues with zero real parts.


The ] states that the orbit structure of a dynamical system in a ] of a hyperbolic fixed point is ] to the orbit structure of the ] dynamical system. The ] states that the orbit structure of a dynamical system in a ] of a hyperbolic fixed point is ] to the orbit structure of the ] dynamical system.


== Definition == == Flows ==


Let Let
:<math>F: \mathbb{R}^n \to \mathbb{R}^n</math> :<math>F: \mathbb{R}^n \to \mathbb{R}^n</math>
be a ''C''<sup>1</sup> (that is, continuously differentiable) ] with fixed point ''p'' and let ''J'' denote the ] 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> be a ''C''<sup>1</sup> (that is, continuously differentiable) ] with a critical point ''p'' and let ''J'' denote the ] 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>


== Example == == Example ==

Revision as of 12:31, 29 April 2009

In mathematics, especially in the study of dynamical systems, a hyperbolic equilibrium point or hyperbolic fixed point is a special type of an equilibrium point, or a fixed point.

The word hyperbolic is due to the fact that in the 2 dimensional case the orbits near the hyperbolic point lay on hyperbolas centered in that point.

Maps

If

T : R n → R n {\displaystyle T:\mathbb {R} ^{n}\to \mathbb {R} ^{n}}

is a C map and p is a fixed point then p is said to be a hyperbolic fixed point when the differential DT(p) has no eigenvalues with zero real parts.

The Hartman-Grobman theorem states that the orbit structure of a dynamical system in a neighbourhood of a hyperbolic fixed point is topologically equivalent to the orbit structure of the linearized dynamical system.

Flows

Let

F : R n → R n {\displaystyle F:\mathbb {R} ^{n}\to \mathbb {R} ^{n}}

be a C (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.

Example

Consider the nonlinear system

d x d t = y , {\displaystyle {\frac {dx}{dt}}=y,}
d y d t = − x − x 3 − α y ,   α ≠ 0 {\displaystyle {\frac {dy}{dt}}=-x-x^{3}-\alpha y,~\alpha \neq 0}

( 0 , 0 ) {\displaystyle (0,0)} is the only equilibrium point. The linearization at the equilibrium is

J ( 0 , 0 ) = ( 0 1 − 1 − α ) {\displaystyle J(0,0)={\begin{pmatrix}0&1\\-1&-\alpha \end{pmatrix}}} .

The eigenvalues of this matrix are − α ± α 2 − 4 2 {\displaystyle {\frac {-\alpha \pm {\sqrt {\alpha ^{2}-4}}}{2}}} . For all values of α ≠ 0 {\displaystyle \alpha \neq 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 ) {\displaystyle (0,0)} . When α = 0 {\displaystyle \alpha =0} , the system has a nonhyperbolic equilibrium at ( 0 , 0 ) {\displaystyle (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

Notes

  1. Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin/Cummings Publishing, Reading Mass. ISBN 0-8053-0102-X

References

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