Hyperbolic equilibrium point

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

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

1 Definition
2 Example
3 Comments
4 See also
5 References

[edit] Definition

Let

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

be a C¹ (that is, differentiable) vector field with fixed 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]^[2]

[edit] 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 $(0,0)$ .

[edit] 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.

[edit] See also

nonhyperbolic equilibrium
Hyperbolic set
Anosov flow

[edit] References

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