Hyperbolic equilibrium point: Information from Answers.com

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.

1 Maps
2 Flows
3 Example
4 Comments
5 See also
6 Notes
7 References

Maps

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

is a C¹ 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

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

be a C¹ (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.^[1]

Example

Consider the nonlinear system

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

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

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

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

The eigenvalues of this matrix are $\frac{-\alpha \pm \sqrt{\alpha^2-4} }{2}$ . For all values of $\alpha \ne 0$ , 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.

Notes

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

References

Equilibrium at Scholarpedia, curated by Eugene M. Izhikevich.

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Hyperbolic equilibrium point

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