Hyperbolic equilibrium point

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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]

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

Maps[edit]

If T : RnRn 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[edit]

Let F : RnRn 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[edit]

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[edit]

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[edit]

Notes[edit]

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

References[edit]