Chaos and stability of Nielsen-Olesen vortex solution with cylindrical symmetry at critical coupling

The chaotic properties of the Nielsen-Olesen vortex solution with cylindrical symmetry in the Abelian-Higgs theories are numerically studied at a critical coupling constant at which the type of the interaction between vortices changes. From the step-wise behaviour of maximal Lyapunov exponents of fields it is shown that the solution exhibits chaos in the case of large perturbation, whose gross structure is similar to the order-to-chaos transition observed in several topological solutions of the SU(2) Yang-Mills-Higgs theories. It is suggested that the time evolution of the system might cause some change in the interaction between the vortices with winding number bigger than one at Bogomolʹnyi limit.