Global Saddles for Planar Maps

TL;DR

This paper investigates conditions under which a hyperbolic saddle fixed point in planar maps becomes a global saddle, with specific results for symmetric maps and applications to differential equations.

Contribution

It provides new sufficient conditions for global saddles in planar diffeomorphisms, including symmetric cases, extending understanding of global dynamics in such systems.

Findings

Sufficient conditions for a fixed point to be a global saddle

Results for $D_2$-symmetric maps and $C^1$ homeomorphisms

Applications to differential equations

Abstract

We study the dynamics of planar diffeomorphisms having a unique fixed point that is a hyperbolic local saddle. We obtain sufficient conditions under which the fixed point is a global saddle. We also address the special case of $D_2$-symmetric maps, for which we obtain a similar result for $C^1$ homeomorphisms. Some applications to differential equations are also given.