Higher order Melnikov functions for piecewise smooth integrable non-Hamiltonian systems separated by a nonregular line

This paper deals with the perturbations of piecewise smooth integrable non-Hamiltonian systems separated by a nonregular line. We develop the algorithms of the Melnikov functions of arbitrary order to analyze limit cycles that can bifurcate from the periodic annulus. While previous studies have explored the Melnikov functions of arbitrary order for perturbations of piecewise Hamiltonian systems separated by a straight line, our work extends to piecewise smooth integrable non-Hamiltonian systems with a nonregular line. As applications, lower bounds for the number of limit cycles are provided for two concrete systems.