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 notes that "hyperbolic is an unfortunate name – it sounds like it should mean 'saddle point' – but is has become standard." Several properties hold about a neighborhood of a hyperbolic point, notably

• 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

$T: \mathbb{R}^n \to \mathbb{R}^n$

is a C¹ map and p is a fixed point then p is said to be a hyperbolic fixed point when the differential 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:

$\left[\begin{array}{c} x_{n+1}\\ y_{n+1} \end{array}\right]=\left[\begin{array}{cc} 1 & 1\\ 1 & 2 \end{array}\right]\left[\begin{array}{c} x_{n}\\ y_{n} \end{array}\right]\,\,\text{modulo }1$

Since the eigenvalues are given by

$\lambda_{1}=\left(3+\sqrt{5}\right)/2>1$

and

$\lambda_{2}=\left(3-\sqrt{5}\right)/2<1$.

Flows

Let

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

be a C¹ (that is, continuously differentiable) vector field with a critical 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.

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 $\alpha \ne 0$, 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 $(0,0)$.

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

• Anosov flow
• Hyperbolic set
• Normally hyperbolic invariant manifold

References

• Equilibrium at Scholarpedia, curated by Eugene M. Izhikevich.