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 $T\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}$ 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 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.[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{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)=\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

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

References

Eugene M. Izhikevich (ed.). "Equilibrium". Scholarpedia.

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[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.