On the problem of periodic solution's stability under Hopf bifurcation

This work is devoted to the problem of stability of a small periodic solution of a normal Autonomous system of ordinary differential equations. It is natural to analyze the local dynamics of intersections of perturbed trajectories with orthogonal sections of the corresponding cycle when studying the stability of the periodic solution of an Autonomous system. The problem of orbital stability of the periodic solution is reduced to the problem of Lyapunov stability of the zero solution of an auxiliary system with a periodic $t$ right-hand side by introducing a special coordinate system in which one of the axes is directed tangentially to the trajectory of the periodic solution. For an auxiliary system whose dimension is one less than the dimension of the original system, in a linear approximation, the question of the stability of the zero solution is reduced to an estimate of the multipliers of the monodromy matrix. Thus, according to the Andronov — Witt theorem, the classical approach to the study of the orbital stability of the periodic solution is realized. There is a non-critical case of orbital stability. This approach is traditionally used in Hopf-type bifurcation for systems with a parameter. In this paper, for an autonomous system with a parameter, the bifurcation conditions of a small solution whose period is close to the solution period of the corresponding linear homogeneous system are obtained. The determination of the orbital stability property by the parameter is formulated. According to this condition, the perturbed right half-vectors are arbitrarily close to the studied cycle not only due to the smallness of the initial values perturbations, but also due to the smallness of the parameter. In this case, the idea of weakening the requirements for determining the stability of the Lyapunov type, proposed by M. M. Khapaev, is used. The property of orbital stability with respect to the parameter can also take place in the presence of orbital instability of the studied cycle in the classical sense. A nonlinear approximation of the above-mentioned auxiliary system of perturbed motions is used to study the orbital stability of a small periodic solution with respect to the parameter.