Misplaced Pages

Revision as of 18:44, 26 January 2010

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 pieces of hyperbolas centered in that point with respect to a suitable coordinate system.

Maps

If

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

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

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

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

${\displaystyle {\frac {dx}{dt}}=y,}$

${\displaystyle {\frac {dy}{dt}}=-x-x^{3}-\alpha y,~\alpha \neq 0}$

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

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

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

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

• Anosov flow
• Hyperbolic set
• Normally hyperbolic invariant manifold

Notes

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

References

• Eugene M. Izhikevich (ed.). "Equilibrium". Scholarpedia.

• Limit sets
• Stability theory