A dynamical system
The same statement holds if the vector field v is defined on the unit disk and is transversal to the boundary.
Orbital topological stability of a dynamical system means that for any sufficiently small perturbation (in the C1-metric), there exists a homeomorphism close to the identity map which transforms the orbits of the original dynamical system to the orbits of the perturbed system (cf structural stability).
The first condition of the theorem is known as global hyperbolicity. A zero of a vector field v, i.e. a point x0 where v(x0)=0, is said to be hyperbolic if none of the eigenvalues of the linearization of v at x0 is purely imaginary. A periodic orbit of a flow is said to be hyperbolic if none of the eigenvalues of the Poincaré return map at a point on the orbit has absolute value one.
Finally, saddle connection refers to a situation where an orbit from one saddle point enters the same or another saddle point, i.e. the unstable and stable separatrices are connected (cf homoclinic orbit and heteroclinic orbit).
- Andronov, Aleksandr A.; Lev S. Pontryagin (1937). "Грубые системы" [Coarse systems]. Doklady Akademii Nauk SSSR. 14 (5): 247–250. Cited in Kuznetsov (2004).
- Kuznetsov, Yuri A. (2004). Elements of Applied Bifurcation Theory. Springer. ISBN 978-0-387-21906-6.. See Theorem 2.5.