Hyperbolic equilibrium point explained

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

T \colon R ^{n} \to 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

λ _{1= (3+ \sqrt{ 5)/()} } λ _{2= (3- \sqrt{ 5)/()} }

We know that the Lyapunov exponents are:

λ _{1= (ln ( 3+ \sqrt{ 5)/()}{ 2 }> 1)} λ _{2= (ln ( 3- \sqrt{ 5)/()}{ 2 }< 1)}

Therefore it is a saddle point.

Flows

Let

F \colon R ^{n} \to 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.^[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

\begin{align} (dx)/(dt) & = y, \\[ 5 pt] (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

(- \alpha \pm \sqrt{\alpha ^{2-4})/() }. 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

Notes

1. Book: Strogatz, Steven . Nonlinear Dynamics and Chaos . 2001 . Westview Press . 0-7382-0453-6 . registration .
2. Book: Ott, Edward . Chaos in Dynamical Systems . registration . 1994 . Cambridge University Press . 0-521-43799-7 .
3. Book: Abraham, Ralph . Jerrold E. . Marsden . Foundations of Mechanics . 1978 . Benjamin/Cummings . Reading Mass. . 0-8053-0102-X .