AMBROSETTI-PRODI PERIODIC PROBLEM OF SINGULAR φ-LAPLACIAN RAYLEIGH EQUATIONS: THEORY AND NUMERICAL BIFURCATION ANALYSIS

This work focuses on positive periodic solutions for a class of phi Laplacian Rayleigh equations with indefinite repulsive singularities, studying their existence, multiplicity and dynamics. We obtain an Ambrosetti-Proditype result without anticoercivity condition, following the continuation theorem and the properties of the Leray-Schauder degree, and further reveal a special dynamic behavior of periodic solutions as the parameter s approaches negative infinity. Moreover, utilizing the numerical bifurcation analysis, we examine an exact phi-Laplacian Rayleigh equation with two singular terms by investigating its Poincare<acute accent> map. The results show the validity of the main theorem and reveal rich dynamic behaviors of the equation including saddle-node bifurcation, period-doubling bifurcation and torus bifurcation of periodic solutions. The combination action of these bifurcations leads to the multiple periodic solutions and multiple stable states associated with the harmonic solutions and second-order subharmonic solutions.