Abstract The present work introduces the problem of simultaneous bifurcation limit cycles and critical periods for a system polynomial differential equations in plane. simultaneity concept is defined, as well idea bi-weakness return map period function. Together with classical methods, we an approach which uses Lie bracket to address some cases. This used find cubic quartic Liénard systems, gen...

The main objective of this paper is to study the number of limit cycles in a family of polynomial systems. Using bifurcation methods, we obtain the maximal number of limit cycles in global bifurcation.

The existence of a heteroclinic bifurcation for the Michaelis–Menten-type ratiodependent predator-prey system is rigorously established. Limit cycles related to the heteroclinic bifurcation are also discussed. It is shown that the heteroclinic bifurcation is characterized by the collision of a stable limit cycle with the origin, and the bifurcation triggers a catastrophic shift from the state o...

This paper concerns with the number of limit cycles for a cubic Hamiltonian system under cubic perturbation. The fact that there exist 9–11 limit cycles is proved. The different distributions of limit cycles are given by using methods of bifurcation theory and qualitative analysis, among which two distributions of eleven limit cycles are new.  2005 Elsevier Inc. All rights reserved.

The existence of periodic solutions for evolution equations is of certain interest for both pure and applied mathematicians. Even for bidimensional systems of differential equations the detection of limit cycles by theoretical means is difficult. The bifurcation theory offers a strong tool for finding limit cycles, namely the theory concerning the Hopf bifurcation (when there is a varying param...

This paper is concerned with bifurcation of limit cycles in a fourth-order near-Hamiltonian system with quartic perturbations. By bifurcation theory, proper perturbations are given to show that the system may have 20, 21 or 23 limit cycles with different distributions. This shows thatH(4) ≥ 20, whereH(n) is the Hilbert number for the second part of Hilbert’s 16th problem. It is well known that ...

In this paper the asymptotic expansion of first-order Melnikov function of a heteroclinic loop connecting a cusp and a nilpotent saddle both of order one for a planar near-Hamiltonian system are given. Next, we consider the bifurcation of limit cycles of a class of hyper-elliptic Liénard system with this kind of heteroclinic loop. It is shown that this system can undergo Poincarè bifurcation fr...

In this paper, we study bifurcation of limit cycles in cubic planar integrable, non-Hamiltonian systems. The systems are assumed to be Z2-equivariant with two symmetric centers. Particular attention is given to bifurcation of limit cycles in the neighborhood of the two centers under cubic perturbations. Such integrable systems can be classified as 11 cases. It is shown that different cases have...