  The unexpected emerging stability of a time-modulated magnetic guide,
realized without external offset fields, is demonstrated. We found a steady
periodic solution around which the nonlinear dynamics is linearized. To
investigate the orbital stability of the guiding, a formal criterion based on
the analysis of the eigenvalues of the monodromy matrix of the system dynamics
is used. To circumvent the difficulty in finding an analytical expression for
these eigenvalues, a Lyapunov transformation of the system variables is
proposed. From this transformation, an equation of state for the system
parameters is derived, and it allows one to estimate an upper bound for the
computed stability domains. In particular, we found a general expression for
the threshold modulation frequency below which the guiding is unstable. Using
experimentally accessible parameters, the stable guiding of 87Rb atoms is
investigated.
