Cyclicity of periodic annulus and Hopf cyclicity in perturbing a hyper-elliptic Hamiltonian system with a degenerate heteroclinic loop

In this paper, we study the cyclicity of periodic annulus and Hopf cyclicity in perturbing a quintic Hamiltonian system. The undamped system is hyper-elliptic, non-symmetric with a degenerate heteroclinic loop, which connects a hyperbolic saddle to a nilpotent saddle. We rigorously prove that the cyclicity is 3 for periodic annulus when the weak damping term has the same degree as that of the associated Hamiltonian system. This result provides a positive answer to the open question whether the annulus cyclicity is 3 or 4. When the smooth polynomial damping term has degree n, first, a transformation based on the involution of the Hamiltonian is introduced, and then we analyze the coefficients involved in the bifurcation function to show that the Hopf cyclicity is [2n+13]. Further, for piecewise smooth polynomial damping with a switching manifold at the y -axis, we consider the damping terms to have degrees l and n, respectively, and prove that the Hopf cyclicity of the origin is [3l+2n+43] ( [3n+2l+43] ) when l≥n ( n≥l ).