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:$F:\; mathbb\{R\}^n\; o\; mathbb\{R\}^n$be a "C"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. [Ralph Abraham and Jerrold E. Marsden, "Foundations of Mechanics", (1978) Benjamin/Cummings Publishing, Reading Mass. ISBN 0-8053-0102-X] [ [http://scholarpedia.org/article/Equilibrium Equilibrium (Scholarpedia)] ]

Example

Consider the nonlinear system:$frac\{\; dx\; \}\{\; dt\; \}\; =\; y,$:$frac\{\; dy\; \}\{\; dt\; \}\; =\; -x-x^3-alpha\; y,~\; alpha\; e\; 0$

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

:.

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

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

* nonhyperbolic equilibrium
* Hyperbolic set
* Anosov flow

References