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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.

Definition

Let

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

be a C (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

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

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

${\displaystyle (0,0)}$ is the only equilibrium point. The linearization at the equilibrium is

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

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

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

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

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