Definitions

# Hyperbolic equilibrium point

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.

## Definition

Let

$F: mathbb\left\{R\right\}^n to mathbb\left\{R\right\}^n$
be a C1 (that is, continuously 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.

## Example

Consider the nonlinear system
$frac\left\{ dx \right\}\left\{ dt \right\} = y,$
$frac\left\{ dy \right\}\left\{ dt \right\} = -x-x^3-alpha y,~ alpha ne 0$

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

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

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

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.