Limit cycles for continuous and discontinuous perturbations of uniform isochronous cubic centers

Abstract

Abstract. Let p be a uniform isochronous cubic polynomial center. We study the maximum number of small or medium limit cycles that bifurcate from p or from the periodic solutions surrounding p respectively, when they are perturbed, either inside the class of all continuous cubic polynomial differential systems, or inside the class of all discontinuous differential systems formed by two cubic differential systems separated by the straight line y = 0.In the case of continuous perturbations using the averaging theory of order 6 we show that the maximum number of small limit cycles that can appear in a Hopf bifurcation at p is 3, and this number can be reached. For a subfamily of these systems using the averaging theory of first order we prove that at most 3 medium limit cycles can bifurcate from the periodic solutions surrounding p, and this number can be reached.In the case of discontinuous perturbations using the averaging theory of order 6 we prove that the maximum number of small limit cycles that can appear in a Hopf bifurcation at p is 5, and this number can be reached. For a subfamily of these systems using the averaging method of first order we show that the maximum number of medium limit cycles that can bifurcate from the periodic solutions surrounding p is 7, and this number can be reached.We also provide all the first integrals and the phase portraits in the Poincaré disc for the uniform isochronous cubic centers.

An Algorithmic Approach to Limit Cycles of Nonlinear Differential Systems

“…. , 5 are consistent with the forms in [26]. However, our averaged function f 6 (r ) looks much simpler than the form given in [26], this is because we rigorously simplify the function f 6 (r ) under the conditions f 1 ≡ f 2 ≡ · · · ≡ f 5 ≡ 0.…”

“…A detailed proof of it can be found in Appendix A. This result is the first work that deals with the bifurcation of limit cycles of system (4) in the general class of perturbations (see [25,26] for a few results on some systems of special form). This theorem tells us that the maximum number of small-amplitude limit cycles of (4), which bifurcate from the center of (3) is always finite ( H k (n 1 , n 2 ) ≤ N k ).…”

“…However, our averaged function f 6 (r ) looks much simpler than the form given in [26], this is because we rigorously simplify the function f 6 (r ) under the conditions f 1 ≡ f 2 ≡ · · · ≡ f 5 ≡ 0. The averaged function f 6 (r ) in [26] is not correct, because the authors do not simplify this expression in a right way (in fact one should note that the isolated parameter β 3, 7 contains the parameter β 1,9 ). As a consequence, the maximum number {3} of limit cycles of system (10) up to the 6th order averaging they obtained can not be reached.…”

On the number of limit cycles for generic Lotka–Volterra system and Bogdanov–Takens system under perturbations of piecewise smooth polynomials

“…Recently, by using Picard-Fuchs equations and Chebyshev criterion, J. Yang and L. Zhao [17] got the exactly 5 and 6 limit cycles for S 1 and S 2 , respectively. For more, one is recommended to see [4,[9][10][11][12].…”

Limit cycles for quadratic and cubic planar differential equations under polynomial perturbations of small degree

“…We also notice that the case δ = 1 with P n , Q n quadratic is studied in [16]. In Theorem 4 of [14], the authors obtained an upper bound of limit cycles when P n and Q n are cubic, but they didn't provide the upper bound for a perturbation of any degree.…”