The cyclicity of a class of global nilpotent center under perturbations of piecewise smooth polynomials with four zones

In this paper, we study the bifurcation of limit cycles of near-Hamilton system  with four zones separated by nonlinear switching curves. We derive the expression of the first order Melnikov function. As an application, we consider the cyclicity of the system
x˙ = y, y˙ = -x^{2m-1}, where (0, 0) is a global nilpotent center and 2 ≤ m ∈ N+, under the  perturbations of piecewise smooth polynomials with four zones separated by y = ±kx^m  with k > 0. By analyzing the first order Melnikov function, we obtain the exact bound of the number of limit cycles bifurcating from the period annulus if the first order Melnikov  function is not identically zero. We also give some examples to illustrate our results.