Английская Википедия: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."[1] 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.
Maps
If <math>T \colon \mathbb{R}^{n} \to \mathbb{R}^{n}</math> is a C1 map and p is a fixed point then p is said to be a hyperbolic fixed point when the Jacobian matrix <math>\operatorname{D} T (p)</math> has no eigenvalues on the complex unit circle.
One example of a map whose only fixed point is hyperbolic is Arnold's cat map:
- <math>\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} </math>
Since the eigenvalues are given by
- <math>\lambda_1=\frac{3+\sqrt{5}}{2}</math>
- <math>\lambda_2=\frac{3-\sqrt{5}}{2}</math>
We know that the Lyapunov exponents are:
- <math>\lambda_1=\frac{\ln(3+\sqrt{5})}{2}>1</math>
- <math>\lambda_2=\frac{\ln(3-\sqrt{5})}{2}<1</math>
Therefore it is a saddle point.
Flows
Let <math>F \colon \mathbb{R}^{n} \to \mathbb{R}^{n}</math> 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
- <math>
\begin{align} \frac{dx}{dt} & = y, \\[5pt] \frac{dy}{dt} & = -x-x^3-\alpha y,~ \alpha \ne 0 \end{align} </math>
(0, 0) is the only equilibrium point. The Jacobian matrix of the linearization at the equilibrium point is
- <math>J(0,0) = \left[ \begin{array}{rr}
0 & 1 \\ -1 & -\alpha \end{array} \right].</math>
The eigenvalues of this matrix are <math>\frac{-\alpha \pm \sqrt{\alpha^2-4}}{2}</math>. 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
Notes
References
