Hyperbolic equilibrium point

Hyperbolic equilibrium point

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

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



F: mathbb{R}^n to mathbb{R}^n
be a C1 (that is, continuously differentiable) vector field with fixed 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.


Consider the nonlinear system
frac{ dx }{ dt } = y,
frac{ dy }{ dt } = -x-x^3-alpha y,~ alpha ne 0

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

J(0,0) = begin{pmatrix}
0 & 1 -1 & -alpha end{pmatrix}.

The eigenvalues of this matrix are frac{-alpha pm sqrt{alpha^2-4} }{2}. For all values of alpha ne 0, 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 alpha=0, the system has a nonhyperbolic equilibrium at (0,0).


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.

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