Hyperbolic equilibrium point

In the study of dynamical systems, a hyperbolic equilibrium point or hyperbolic fixed point is a fixed point that does not have any center manifolds. Near a hyperbolic point the orbits of a two-dimensional, non-dissipative system resemble hyperbolas. This fails to hold in general. Strogatz^[1] notes that "hyperbolic is an unfortunate name – it sounds like it should mean 'saddle point' – but it has become standard." Several properties hold about a neighborhood of a hyperbolic point, notably^[2]

A stable manifold and an unstable manifold exist,
Shadowing occurs,
The dynamics on the invariant set can be represented via symbolic dynamics,
A natural measure can be defined,
The system is structurally stable.

Orbits near a two-dimensional saddle point, an example of a hyperbolic equilibrium.

Maps

If T : Rⁿ → Rⁿ is a C¹ map and p is a fixed point then p is said to be a hyperbolic fixed point when the Jacobian matrix DT(p) has no eigenvalues on the unit circle.

One example of a map that its only fixed point is hyperbolic is the Arnold Map or cat map:

\begin{bmatrix} x_{n+1}\\ y_{n+1} \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 1 & 2\end{bmatrix} \begin{bmatrix} x_n\\ y_n\end{bmatrix} \quad \text{modulo }1

Since the eigenvalues are given by

\lambda_{1}=\frac{3+\sqrt{5}}{2}>1

\lambda_{2}=\frac{3-\sqrt{5}}{2}<1

Flows

Let F : Rⁿ → Rⁿ be a C¹ vector field with a critical point p, i.e., F(p) = 0, 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.^[3]

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

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 α ≠ 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).

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.

Notes

↑ Strogatz, Steven (2001). Nonlinear Dynamics and Chaos. Westview Press.
↑ Ott, Edward (1994). Chaos in Dynamical Systems. Cambridge University Press.
↑ Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin/Cummings Publishing, Reading Mass. ISBN 0-8053-0102-X

References

Equilibrium at Scholarpedia, curated by Eugene M. Izhikevich.