On the number of limit cycles in piecewise smooth generalized Abel equations with many separation lines

This paper investigates generalized Abel equations of the form dx/dθ=A(θ)xp+B(θ)xq, where p, q∈Z≥2, p≠q, and A(θ) and B(θ) are piecewise trigonometrical polynomials of degree m with n−1∈N+ separation lines 0<θ1<θ2<⋯<θn−1<2π. The main objective is to obtain the maximum number of non-zero limit cycles (i.e., non-zero isolated periodic solutions) that the equation can have, denoted by Hθ1,θ2,…,θn−1(m), and to analyze how the number and location of separation lines {θi}i=1n−1 affect Hθ1,θ2,…,θn−1(m). By using the theories of Melnikov functions and ECT-systems, we obtain lower bounds for Hθ1,θ2,…,θn−1(m). Our result extend those of Huang et al. who studied the special case of n=2, and reveal that the lower bounds decrease in the presence of pairs of symmetrical separation lines.