Local cyclicity in low degree planar piecewise polynomial vector fields

The local cyclicity problem: Melnikov method using Lyapunov constants

“…Finally, § 5 is devoted to piecewise polynomial vector fields of degrees 3, 4, and 5, proving our second main result Theorem 1.2. We also provide a new proof of M c p (3) ≥ 26, different from the one published in [29], using now only first-order developments.…”

“…For n = 2, using averaging theory of fifth-order, and perturbing the linear centre, Llibre and Tang in [44] proved that H c p (2) ≥ 8. Recently, da Cruz, Novaes, and Torregrosa in [18] improve this lower bound and, using local developments of the difference map, Gouveia and Torregrosa in [29] provide the first high lower bounds for piecewise polynomial classes of degrees three, four and five: M c p (3) ≥ 26, M c p (4) ≥ 40, and M c p (5) ≥ 58. Here we update two of these lower bounds using averaging theory in our second main result.…”

“…More details on the pseudo-Hopf bifurcation are collected in [12, 21, 22]. In [14, 27, 29, 30], we can see that a good and simple tool for studying the number of limit cycles of small amplitude bifurcating from the origin is the first-order development of the Lyapunov constants. We notice that, in this case, the Jacobian matrix of the vector field at the origin must be of non-degenerate monodromic type.…”

Bifurcation of limit cycles in piecewise quadratic differential systems with an invariant straight line

“…From the above analysis, in the bifurcation of an analytic planar piecewise vector field, when we have a weak-focus of order k we get (generically) k limit cycles. See more details in [17]. This bifurcation problem with varying parameters and taking into account multiplicities is studied in [19].…”

“…For quadratic vector fields, Bautin showed in [3] that the maximum number of limit cycles of small amplitude near an equilibrium point is three and, moreover, this upper bound is reached. For quadratic discontinuous differential systems, this problem is studied in [17,21]. The work of Bautin is very appreciated because increasing the degree, these problems remain open.…”