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 it 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 : 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:

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

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.

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 equilibrium 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.

See also

• Anosov flow
• Hyperbolic set
• Normally hyperbolic invariant manifold

References

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