Local Bifurcation Cyclicity for a Non-Polynomial System

Abstract

In this paper, we propose a class of general non-polynomial analytic oscillator models, and study the limit cycle bifurcation at the nilpotent singularity or elementary center-focus. By Taylor expansion, two specific systems from the original model are transformed into two equivalent infinite polynomial systems, and the highest order of fine focus as the nilpotent Hopf bifurcation or Hopf bifurcation point is determined respectively. At the same time, the local bifurcation cyclicities and center problems for two systems are solved respectively. To our knowledge, such dynamic properties are rarely analyzed in many non-polynomial models.