Bifurcations and distribution of limit cycles which appear from two nests of periodic orbits

In this paper, we study the distribution and simultaneous bifurcation of limit cycles bifurcated from the two periodic annuli of the holomorphic differential equation ż=iz+z3, after a small polynomial perturbation. We first show that, under small perturbations of the form εP2m−1(z,z̄), where P2m−1(z,z̄) is a polynomial of degree 2m−12m−1 in which the power of zz is odd and the power of z̄ is even, the only possible distribution of limit cycles is (u,u)(u,u) for all values of u=0,1,2,…,m−3u=0,1,2,…,m−3. Hence, the sharp upper bound for the number of limit cycles bifurcated from each two period annuli of ż=iz+z3 is m−3m−3, for m≥4m≥4. Then we consider a perturbation of the form εPm(z,z̄), where Pm(z,z̄) is a polynomial of degree mm in which the power of zz is odd and obtain the upper bound m−5m−5, for m≥6m≥6. Moreover, we show that the distribution (u,v)(u,v) of limit cycles is possible for 0≤u≤m−50≤u≤m−5, 0≤v≤m−50≤v≤m−5 with u+v≤m−2u+v≤m−2 and m≥9m≥9.