Up to the first two order Melnikov analysis for the exact cyclicity of planar piecewise linear vector fields with nonlinear switching curve.

In this paper, we focus on providing the exact bounds for the maximum number of limit cycles Z (3 , n) that planar piecewise linear differential systems with two zones separated by the curve y = x 3 under perturbation of arbitrary polynomials of x , y with degree n can have, where n ∈ N. By the first two order Melnikov functions, we achieve that Z (3 , 2) = 12 , Z (3 , n) = 2 n + 1 for 3 ≤ n ≤ 88 and Z (3 , n) ≥ 2 n + 1 for any n. The results are novel and improve the previous results in the literature. • We achieved a result about the coexistence of small amplitude and big amplitude limit cycles in Lemma 3.6 (see P 18), which yields system (1.1) ϵ can have 2 n small amplitude limit cycles simultaneously a large one for n odd. • We got a good way to avoid finding the zero points of the Wronskians in applying the useful algebraic method developed in [20]. Specifically, although [26] proved that (F 3) (which is the generating functions of M 1 (u)) is an ECT-system with accuracy 1 for n = 1 , 3 , 5 , it is impossible to find the zero points of the Wronskians for general n. We constructed an auxiliary ordered set of functions (F 2) ⊃ (F 3) (see (3.38) on P 19 and Lemma 3.7) so that we avoided these troubles. In fact, we have checked each Wronskians of (F 2) is non-vanishing since they have the form of M × P (v) v m / 2 (v > 0) , where M , m ≠ 0 are some constants and P (v) are some polynomials of v 2 with positive coefficients. Thus, we achieved Z (3 , n) ≤ 2 n + 1 for n ≤ 88 odd. • We obtained the expression of M 2 (u) without any restriction on perturbation parameters in Lemma 4.1, see P 25. So we can obtain the exact value of Z (3 , 2) = 12 , which improves the result Z (3 , 2) ≥ 5 in [9]. • We performed a transformation in Lemma 3.7 on P 19 so that the calculations of Wronskians are reduced greatly, and we chose clearly the perturbation parameters in Lemma 3.8(ii) (see P 20) such that the corresponding system (1.1) ϵ has at least 2 n + 1 limit cycles. [ABSTRACT FROM AUTHOR]