Local cyclicity and criticality in FF-type piecewise smooth cubic and quartic Kukles systems

We investigate small-amplitude crossing limit cycles and local critical periods for some classes of piecewise smooth Kukles systems with a focus–focus type (abbr.FF-type) equilibrium point, which is formed by two regions separated by a straight line. In this class, we acquire new lower bounds for the local cyclicity Mpff(n) and the criticality Cp(n) with n=3,4. More precisely, Mpff(3)≥10, Mpff(4)≥12 and Cp(3)≥6 for the discontinuous case without a sliding segment, and Mpff(3)≥9, Mpff(4)≥13 and Cp(3)≥6 for the continuous case. Furthermore, one more limit cycle is obtained for the discontinuous case with a sliding segment in comparison with the one without a sliding segment, namely 11 for cubic system and 13 for quartic one. These lower bounds are also new, that are known up to now, for the general FF-type piecewise smooth cubic and quartic systems around an equilibrium. Besides, we give the Finite Order Bifurcation Lemma of local critical periods in FF-type piecewise smooth systems, which is the generalization of the smooth ones.