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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.## Definition

^{1} (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
## 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

## References

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.

Let

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

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

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

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

- $J(0,0)\; =\; begin\{pmatrix\}$

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 $alpha=0$, the system has a nonhyperbolic equilibrium at $(0,0)$.

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Last updated on Wednesday July 30, 2008 at 08:38:04 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Wednesday July 30, 2008 at 08:38:04 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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