Bifurcation of critical periods for plane vector fields

Abstract:

A bifurcation problem in families of plane analytic vector fields which have a nondegenerate center at the origin for all values of a parameter $\lambda \in {{\mathbf {R}}^N}$ is studied. In particular, for such a family, the period function $(\xi ,\lambda ) \mapsto P(\xi ,\lambda )$ is defined; it assigns the minimum period to each member of the continuous band of periodic orbits (parametrized by $\xi \in {\mathbf {R}}$) surrounding the origin. The bifurcation problem is to determine the critical points of this function near the center with $\lambda$ as bifurcation parameter. Generally, if the function $\rho$, given by $\xi \mapsto P(\xi ,{\lambda _\ast }) - P(0,{\lambda _\ast })$, vanishes to order $2k$ at the origin, then it is shown that the period function, after a perturbation of ${\lambda _\ast }$, has at most $k$ critical points near the origin. If $\rho$ vanishes to infinite order, i.e., the center is isochronous, it is shown that the number of critical points of $P$ for perturbations of ${\lambda _\ast }$ depends on the number of generators of the ideal of all Taylor coefficients of $\rho (\xi ,\lambda )$, where the coefficients are considered elements of the ring of convergent power series in $\lambda$. Specifically, if the ideal is generated by the first $2k$ Taylor coefficients, then a perturbation of ${\lambda _\ast }$ produces at most $k$ critical points of $P$ near the origin. These theorems are applied to the quadratic systems with Bautin centers and to one degree of freedom "kinetic+potential" Hamiltonian systems with polynomial potentials. For the quadratic systems a complete solution of the bifurcation problem is obtained. For the Hamiltonian systems a number of results are proved independent of the degree of the potential and a complete solution is obtained for potentials of degree less than seven. Aside from their intrinsic interest, monotonicity properties of the period function are important in the question of existence and uniqueness of autonomous boundary value problems, in the study of subharmonic bifurcation of periodic oscillations, and in the analysis of the problem of linearization. In this regard it is shown that a Hamiltonian system with a polynomial potential of degree larger than two cannot be linearized. However, while these topics are the subject of a large literature, the spirit of this paper is more akin to that of N. Bautin’s treatment of the multiple Hopf bifurcation for quadratic systems and the work on various forms of the weakened Hilbert’s 16th problem first posed by V. Arnold.