The local cyclicity problem: Melnikov method using Lyapunov constants

Abstract

In 1991, Chicone and Jacobs showed the equivalence between the computation of the first-order Taylor developments of the Lyapunov constants and the developments of the first Melnikov function near a non-degenerate monodromic equilibrium point, in the study of limit cycles of small-amplitude bifurcating from a quadratic centre. We show that their proof is also valid for polynomial vector fields of any degree. This equivalence is used to provide a new lower bound for the local cyclicity of degree six polynomial vector fields, so $\mathcal {M}(6) \geq 44$ . Moreover, we extend this equivalence to the piecewise polynomial class. Finally, we prove that $\mathcal {M}^{c}_{p}(4) \geq 43$ and $\mathcal {M}^{c}_{p}(5) \geq 65.$

Lower bounds for the local cyclicity for families of centers

“…After Figure 1. Phaseportraits in the Poincaré disk of the center (8) for a = −1/12, a = 1/2, and a = 2 using the Implicit Function Theorem to vanish the first nine Lyapunov constants, the simplified expressions of the next two, are, except non zero mult'iplicative constants, L 10 = (3a + 1) 13 (6a + 1) 18 a 3 (9a 2 − 3a + 1) 10 (a − 1) 9 f 0 (a) g(a)…”

“…Drawing the zero level sets of f 0 , f 1 , and f 2 in (13) in red, green, and blue, respectively Remark 5.4. Following the same procedure as for the Lyapunov constants given in (13) we compute also the next Lyapunov constant that can be written as…”

“…This is a direct application of the Implicit Function Theorem to prove that M (n) ≥ k. Usually we use this technique to provide lower bounds for the local cyclicity problem in the class of polynomial vector fields of degree n. In [10,11], Giné presents a conjecture that the number M (n) is bounded below by n 2 + 3n − 7 and study the cyclicity of different families of centers presented in [9]. In [13,14] new lower bounds for M (n) and n small has been obtained. The new values are M (4) ≥ 20, M (5) ≥ 33, M (6) ≥ 44, M (7) ≥ 61, M (8) ≥ 76, and M (9) ≥ 88.…”

Criticality via first order development of the period constants

“…Theorem 1 presented in Section 1 states that the first order coefficients in T 1 (ρ, λ) from ( 25), these are functions θ j (λ), are exactly the first order truncation of the Taylor series of the period constants in (21) with respect to λ. This is inspired by [18], where the authors prove the equivalence between the first order truncation of the Lyapunov constants and the first Melnikov function for limit cycles. Before the proof of Theorem 1 and its Corollary 2, we will start by illustrating the equivalence between both methods with a particular example.…”

“…The method we propose to obtain lower bounds on the number of critical periods is based on the equivalence of the first Melnikov function for the period of the perturbation of an isochronous system and the linear developments with respect to the perturbation parameters of the period constants also near the same isochronous system, an idea already introduced in [18] for cyclicity and Lyapunov constants. This is our main technique and is presented in the following result.…”