SOME BIFURCATION RESULTS IN NEAR-HAMILTONIAN SYSTEMS WITH FINITE-ORDER SMOOTHNESS

Abstract

This paper presents a systematic survey of bifurcation theory for near-Hamiltonian systems with finite-order smoothness. Unlike the classical $C^{\infty}$ or $C^{\omega}$ framework, we collect some known results on limit cycle bifurcations for planar systems with finite-order smoothness, including Poincaré bifurcation, Hopf bifurcation, homoclinic bifurcation, double homoclinic bifurcation, heteroclinic bifurcation, and bifurcations involving nilpotent singularities. For each type, we summarize the relevant bifurcation theorems on properties of the Melnikov functions and the conditions for the existence and the number of limit cycles that can be produced.