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."{{cite book |last=Strogatz |first=Steven |title=Nonlinear Dynamics and Chaos |year=2001 |publisher=Westview Press |isbn=0-7382-0453-6 |url-access=registration |url=https://archive.org/details/nonlineardynamic00stro }} Several properties hold about a neighborhood of a hyperbolic point, notably{{cite book |last=Ott |first=Edward |title=Chaos in Dynamical Systems |url=https://archive.org/details/chaosindynamical0000otte |url-access=registration |year=1994 |publisher=Cambridge University Press |isbn=0-521-43799-7 }}

Image:Phase Portrait Sadle.svg

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.

Maps

If $T \colon \mathbb{R}^{n} \to \mathbb{R}^{n}$ is a C¹ map and p is a fixed point then p is said to be a hyperbolic fixed point when the Jacobian matrix $\operatorname{D} T (p)$ has no eigenvalues on the complex unit circle.

One example of a map whose only fixed point is hyperbolic is Arnold's 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

: $\lambda_1=\frac{3+\sqrt{5}}{2}$

: $\lambda_2=\frac{3-\sqrt{5}}{2}$

We know that the Lyapunov exponents are:

: $\lambda_1=\frac{\ln(3+\sqrt{5})}{2}>1$

: $\lambda_2=\frac{\ln(3-\sqrt{5})}{2}<1$

Therefore it is a saddle point.

Flows

Let $F \colon \mathbb{R}^{n} \to \mathbb{R}^{n}$ 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.{{cite book |first=Ralph |last=Abraham |first2=Jerrold E. |last2=Marsden |title=Foundations of Mechanics |year=1978 |publisher=Benjamin/Cummings |location=Reading Mass. |isbn=0-8053-0102-X }}

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

: $\begin{align}
\frac{dx}{dt} & = y, \\[5pt]
\frac{dy}{dt} & = -x-x^3-\alpha y,~ \alpha \ne 0
\end{align}$

(0, 0) is the only equilibrium point. The Jacobian matrix of the linearization at the equilibrium point is

: $J(0,0) = \left[ \begin{array}{rr}
0 & 1 \\
-1 & -\alpha \end{array} \right].$

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.

Notes

References

{{Scholarpedia|title=Equilibrium|urlname=Equilibrium|curator=Eugene M. Izhikevich}}

Category:Limit sets

Category:Stability theory