Computer assisted proof of the existence of homoclinic tangency for the Henon map and for the forced-damped pendulum

TL;DR

This paper introduces a topological computer-assisted method to verify homoclinic tangencies in planar maps, successfully applied to the Henon map and the forced damped pendulum, advancing rigorous dynamical systems analysis.

Contribution

The paper develops a novel topological approach for computer-assisted proof of homoclinic tangencies in planar systems, applicable to both maps and ODEs.

Findings

Verified homoclinic tangency existence in the Henon map.

Confirmed homoclinic tangency in the forced damped pendulum.

Demonstrated efficiency of the topological method.

Abstract

We present a topological method for the efficient computer assisted verification of the existence of the homoclinic tangency which unfolds generically in a one-parameter family of planar maps. The method has been applied to the Henon map and the forced damped pendulum ODE.