Hyperbolicity and Averaging for the Srzednicki-W\'ojcik equation

TL;DR

This paper investigates how the hyperbolic region of the Srzednicki-Wójcik equation expands with increasing parameter magnitude using averaging techniques, providing bounds on bounded orbit sizes.

Contribution

It introduces an averaging method to analyze hyperbolicity growth in the equation and derives bounds on bounded orbit sizes based on the parameter.

Findings

Hyperbolic region enlarges with increasing |6|

Bounds on sizes of bounded orbits are established

Averaging technique effectively analyzes nonautonomous ODEs

Abstract

For the Srzednicki-W\'ojcik equation, the planar nonautonomous ODE parameterized by $\kappa \in \mathbb{R}$, $$ z'=\overline{z}(1+ |z|^2 \exp(i \kappa t)), \qquad z(t) \in \mathbb{C} $$ using averaging we show how the region of hyperbolicty grows with $|\kappa|$. Based on this we give bounds on the sizes of bounded orbits.