Limit cycles in a quartic system with a third-order nilpotent singular point

A quartic system with 26 limit cycles

Eight Limit cycles around a Center in quadratic Hamiltonian System with Third-Order perturbation

In this paper, we show that generic planar quadratic Hamiltonian systems with third degree polynomial perturbation can have eight small-amplitude limit cycles around a center. We use higher-order focus value computation to prove this result, which is equivalent to the computation of higher-order Melnikov functions. Previous results have shown, based on first-order and higher-order Melnikov func...

Twelve limit cycles around a singular point in a planar cubic-degree polynomial system

In this paper, we prove the existence of 12 small-amplitude limit cycles around a singular point in a planar cubic-degree polynomial system. Based on two previously developed cubic systems in the literature, which have been proved to exhibit 11 small-amplitude limit cycles, we applied a different method to show 11 limit cycles. Moreover, we show that one of the systems can actually have 12 smal...

Bifurcation of limit cycles at degenerate singular point and in“nity in a septic system

In this paper, the problem of bifurcation of limit cycles from degenerate singular point and infinity in a class of septic polynomial differential systems is investigated. Using the computer algebra system Mathematica, the limit cycle configurations of {(8), 3} and {(3), 6} are obtained under synchronous perturbation at degenerate singular point and infinity. To our knowledge, up to now, this i...

Limit Cycles and Integrability in a Class of Systems with High-Order Nilpotent Critical Points

and Applied Analysis 3 Thus, the origin of system (13) is an element critical point. It could be investigated using the classical integral factor method. Now, we consider the following system: dx dt = y + A30x3n + A21x2ny + A12xny2 + A03y3, dy dt = −x2n−1 + xn−1 (B30x3n + B21x2ny + B12xny2 + B03y3) . (14) When n = 2k + 1, by those transformations, system (14) is changed into dx dt = −y − √2k + ...