Analytic continuation of periodic orbits from equilibria of two-degree-of-freedom systems in rotating coordinates

A new theory is formulated for the analytic continuation of periodic (and aperiodic) orbits from equilibrium solutions of a two-degree-of-freedom dynamical system in rotating coordinates:% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% acbiGab8xDayaacaGaa8xlaiaa-jdacaWFUbGaeqyXduNaa8xpaiaa% -zfadaWgaaWcbaGccaWF4baaleqaaOGaaiilaiqbew8a1zaacaGaey% 4kaSIaaGOmaiaad6gacaWG1bGaeyypa0Jaa8NvamaaBaaaleaakiaa% -LhaaSqabaGccaGGSaGabmiEayaacaGaeyypa0JaamyDaiaacYcace% WG5bGbaiaacqGH9aqpcqaHfpqDaaa!54CD!\\dot u - 2n\upsilon = V_x ,\dot \upsilon + 2nu = V_y ,\dot x = u,\dot y = \upsilon \]Away from resonance, a family of nonlinear, normal-mode orbits defines an autonomous velocity field u(x, y), u(x, y) represented by convergent algebraic-series expansions in the two position variables. This approach is useful for determining the global structure of solution curves and nonlinear stability of normal modes using Liapunov's direct method. At resonance, the series coefficients are time dependent because stationary modes are incompatible with the equations of motion. By eliminating small divisors, explicit time dependence provides a natural transition from non-resonance to resonance cases within the same theory.