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." Several properties hold about a neighborhood of a hyperbolic point, notably
If T : Rn → Rn is a C1 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
Let F : Rn → Rn be a C1vector 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.
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.
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