Crystalline undulator radiation and sub-harmonic bifurcation of system

Looking for a new light source, especially short wavelength laser light source has attracted widespread attention. In the paper analytically described is the radiation of crystalline undulator field by the sine-squared potential. In the classical mechanics and the dipole approximation, the motion equation of particle is reduced to a generalized pendulum equation with damping term and forcing term. The bifurcation behavior of periodic orbits is analyzed by using the Melnikov method and numerical method, and the stability of the system is discussed. The results show that the stability of the system relates to its parameters, and only by adjusting appropriately these parameters, in principle can the occurrence of bifurcation be avoided or suppressed.