Multiple Hopf bifurcation in R-3 and inverse Jacobi multipliers

作者：Buica Adriana; Garcia Isaac A^*; Maza Susanna

来源：Journal of Differential Equations, 2014, 256(1): 310-325.

DOI：10.1016/j.jde.2013.09.006

• Jacobi last multipliers
• Multiple Hopf bifurcation
• Poincare map
• Limit cycle

In this paper we study the maximum number of limit cycles that can bifurcate from a singular point of saddle-focus type of an analytic, autonomous differential system in R-3 under any analytic perturbation that keeps the location and nature of the singularity. We only consider those foci on center manifolds having associated two nonzero purely imaginary and one nonzero real eigenvalues. Our approach is different from the classical one in the sense that we do not use any center manifold reduction to compute Poincare-Lyapunov constants. Instead, we study the multiple Hopf bifurcation first doing a Lyapunov-Schmidt reduction to the associated Poincare map, obtaining in this way an analytic reduced displacement map. Next we prove that the order of this displacement map coincides with the vanishing multiplicity (denoted in) of any locally smooth and non-flat inverse Jacobi multiplier. Finally the cyclicity of the focus is given in terms of m.

• 出版日期2014-1-1