# 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^{[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]}

- 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} is a *C*^{1} 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:

Since the eigenvalues are given by

## Flows

Let *F* : **R**^{n} → **R**^{n} be a *C*^{1} 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

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

- .

The eigenvalues of this matrix are . 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

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

## References

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