Ushiki's theorem

In mathematics, particularly in the study of functions of several complex variables, Ushiki's theorem, named after S. Ushiki, states that certain well-behaved functions cannot have certain kinds of well-behaved invariant manifolds.

Invariant manifolds typically appear as solutions of certain asymptotic problems in dynamical systems.

[1] The theorem appeared in print again several years later, in a certain Russian journal, by an author apparently unaware of Ushiki's work.

The practical consequence is that one cannot show the existence of a Smale's horseshoe in this system by a perturbation method, starting from a homoclinic or heteroclinic connection.

Nevertheless, one can show that Smale's horseshoe exists in the standard map for many parameter values, based on crude rigorous numerical calculations.