Fourteen limit cycles in a cubic Hamiltonian system with nine-order perturbed term

On some open problems in planar differential systems and Hilbert’s 16th problem

“…In [125] it was shown that H(3) P 12 and is the best result, so far, on the number of limit cycles for cubic systems. In the last few years different results have been obtained, for instance H(4) P 15, H(5) P 23 and H(6) P 35, see [115,120,121,128] and references therein.…”

Limit cycle analysis on a cubic Hamiltonian system with quintic perturbed terms

“…Conversely, If Γ h is constrained inside as h increases, the stability of the limit cycle is opposite to the results above. The proof of this proposition can be found elsewhere [12]. For the sake of completeness, we briefly present the proof as below:…”

“…Cao et als study [3] indicated that this system has 13 limit cycles when R(x, y, λ) = S(x, y, λ) = mx 6 + ny 6 − λ. Further, Tang and Hong [12] found that the system (1.3) has 14 limit cycles when R(x, y, λ) = S(x, y, λ) = mx 8 + ny 8 − λ. Zhang et al [13] also explored the number of limit cycles for the Hamiltonian system of (1.3) under quartic perturbations. Wu et al [6] also explored the number of limit cycles for the Hamiltonian system of (1.3) under quintic perturbations.…”

“…m, n, k and λ here, rather than m, n and λ only in the existing work, are all variable. Using the same method of detection function and method of numerical exploration as in [10,12], the study firstly shows that the system…”

The same distribution of limit cycles in a Hamiltonian system with nine seven-order perturbed terms

“…see [1,7,8,9,14,15]), most of which mainly focused on a Hamiltonian system with one perturbed term [1,7,8,15]. It was shown that five perturbed Hamiltonian systems have the same distribution of limit cycles [9,14].…”