# Validated numerics for period-tupling and touch-and-go bifurcations of symmetric periodic orbits in reversible systems

## TL;DR

This paper develops a computer-assisted framework to verify symmetry breaking and bifurcations of symmetric periodic orbits in reversible systems, with applications to celestial mechanics.

## Contribution

It introduces a general method for verifying bifurcations in reversible maps and applies it to prove the existence of complex bifurcations of halo orbits in the Three Body Problem.

## Key findings

- Proved existence of wide branches of halo orbits bifurcating from Lyapunov families.
- Demonstrated multiple period doubling and quadrupling bifurcations.
- Validated bifurcations for specific physical parameters.

## Abstract

We propose a general framework for computer-assisted verification of the presence of symmetry breaking, period-tupling and touch-and-go bifurcations of symmetric periodic orbits for reversible maps. The framework is then adopted to Poincar\'e maps in reversible autonomous Hamiltonian systems. In order to justify the applicability of the method, we study bifurcations of halo orbits in the Circular Restricted Three Body Problem. We give a computer-assisted proof of the existence of wide branches of halo orbits bifurcating from $L_{1,2,3}$-Lyapunov families and for wide range of mass parameter. For two physically relevant mass parameters we prove, that halo orbits undergo multiple period doubling, quadrupling and third-order touch-and-go bifurcations.