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]

Orbits near a two-dimensional saddle point, an example of a hyperbolic equilibrium.
  • 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 : R n → R n {\displaystyle 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 D ⁡ T ( p ) {\displaystyle \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:

[ x n + 1 y n + 1 ] = [ 1 1 1 2 ] [ x n y n ] {\displaystyle {\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 + 5 2 {\displaystyle \lambda _{1}={\frac {3+{\sqrt {5}}}{2}}}
λ 2 = 3 − 5 2 {\displaystyle \lambda _{2}={\frac {3-{\sqrt {5}}}{2}}}

We know that the Lyapunov exponents are:

λ 1 = ln ⁡ ( 3 + 5 ) 2 > 1 {\displaystyle \lambda _{1}={\frac {\ln(3+{\sqrt {5}})}{2}}>1}
λ 2 = ln ⁡ ( 3 − 5 ) 2 < 1 {\displaystyle \lambda _{2}={\frac {\ln(3-{\sqrt {5}})}{2}}<1}

Therefore it is a saddle point.

Flows

Let F : R n → R n {\displaystyle 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.[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

d x d t = y , d y d t = − x − x 3 − α y ,   α ≠ 0 {\displaystyle {\begin{aligned}{\frac {dx}{dt}}&=y,\\[5pt]{\frac {dy}{dt}}&=-x-x^{3}-\alpha y,~\alpha \neq 0\end{aligned}}}

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

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

The eigenvalues of this matrix are − α ± α 2 − 4 2 {\displaystyle {\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

Notes

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

References