Application of renormalization to the dynamics of a particle in a infinite square-well potential driven by an external field

We analyze by a renormalization method, the dynamics of a particle in a infinite square-well potential driven by an external monochromatic field. This method set up for Hamiltonian systems with two degrees of freedom allows us to analyze precisely the stability of the trajectories of the particle as a function of the amplitude $ϵ$ of the external field. We compute numerical values of $ϵ$ for which the motion of the particle with frequency $\omega$ is broken and a transition to a chaotic behavior occurs. We obtain the critical function ${ϵ}_c(\omega )$ associated with this system as a function of the parameters such as the frequency of the field and the width of the potential.