Hyperbolic equilibrium point

Near a hyperbolic point the orbits of a two-dimensional, non-dissipative system resemble hyperbolas.

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] If

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.

[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.

For all values of α ≠ 0, the eigenvalues have non-zero real part.

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.