Hyperbolic equilibrium point

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In mathematics, especially in the study of dynamical systems, a hyperbolic equilibrium point or hyperbolic fixed point is a special type of an equilibrium point, or a fixed point.

The word hyperbolic is due to the fact that in the 2 dimensional case the orbits near the hyperbolic point lay on pieces of hyperbolas centered in that point with respect to a suitable coordinate system.

Contents 1 Maps 2 Flows 3 Example 4 Comments 5 See also 6 Notes 7 References

[edit] Maps

If

is a C¹ map and p is a fixed point then p is said to be a hyperbolic fixed point when the differential DT(p) has no eigenvalues with zero real parts.

The Hartman-Grobman theorem states that the orbit structure of a dynamical system in a neighbourhood of a hyperbolic fixed point is topologically equivalent to the orbit structure of the linearized dynamical system.

[edit] Flows

Let

be a C¹ (that is, continuously differentiable) vector field with a critical point p 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.^[1]

[edit] 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 , 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).

[edit] 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.

[edit] See also

• Anosov flow
• Hyperbolic set

[edit] Notes

1. ^ Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin/Cummings Publishing, Reading Mass. ISBN 0-8053-0102-X

[edit] References

• Equilibrium at Scholarpedia, curated by Eugene M. Izhikevich.