Limit cycles of multi-parameter polynomial dynamical systems

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This is an overview of our recent works on the global bifurcation analysis of multi-parameter polynomial dynamical systems. In particular, using our bifurcation-geometric approach, we study the global dynamics and solve the problem on the maximum number and the distribution of limit cycles in a polynomial Euler–Lagrange–Liénard-type mechanical system. We also consider a rational endocrine system by carrying out the global bifurcation analysis of a reduced planar quartic Topp system, which models the dynamics of diabetes. By analyzing global bifurcations and applying the Wintner–Perko termination principle, we prove that such a system can have at most two limit cycles.