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

In this talk, we discuss  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.    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    $\big[\frac{2n+1}{3}\big]$.    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.