ON THE LIMIT CYCLES FOR A CLASS OF GENERALIZED KUKLES DIFFERENTIAL SYSTEMS

Amel Boulfoul; Amar Makhlouf; Nawal Mellahi

doi:10.11948/2156-907X.20180083

2019 Volume 9 Issue 3

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Amel Boulfoul, Amar Makhlouf, Nawal Mellahi. ON THE LIMIT CYCLES FOR A CLASS OF GENERALIZED KUKLES DIFFERENTIAL SYSTEMS[J]. Journal of Applied Analysis & Computation, 2019, 9(3): 864-883. doi: 10.11948/2156-907X.20180083

Citation:

Amel Boulfoul, Amar Makhlouf, Nawal Mellahi. ON THE LIMIT CYCLES FOR A CLASS OF GENERALIZED KUKLES DIFFERENTIAL SYSTEMS[J]. Journal of Applied Analysis & Computation, 2019, 9(3): 864-883. doi: 10.11948/2156-907X.20180083

ON THE LIMIT CYCLES FOR A CLASS OF GENERALIZED KUKLES DIFFERENTIAL SYSTEMS

1.
Department of Mathematics, 20 August 1955 University, BP26 El Hadaiek, 21000 Skikda, Algeria
2.
LAMAHIS Laboratory, 20 August 1955 University, BP26 El Hadaiek, 21000 Skikda, Algeria
3.
Departement of Mathematics, LMA Laboratory, Badji-Mokhtar University, BP12 El Hadjar, 23000 Annaba, Algeria

Corresponding author: Email address:a.boulfoul@univ-skikda.dz(A. Boulfoul)

Abstract

Abstract

In this paper, we consider the limit cycles of a class of polynomial differential systems of the form $\dot{x}=-y, \hspace{0.2cm} \dot{y}=x-f(x)-g(x)y-h(x)y^{2}-l(x)y^{3}, $ where $f(x)=\epsilon f_{1}(x)+\epsilon^{2}f_{2}(x), $ $g(x)=\epsilon g_{1}(x)+\epsilon^{2}g_{2}(x), $ $h(x)=\epsilon h_{1}(x)+\epsilon^{2}h_{2}(x)$ and $l(x)=\epsilon l_{1}(x)+\epsilon^{2}l_{2}(x)$ where $f_{k}(x), $ $g_{k}(x), $ $h_{k}(x)$ and $l_{k}(x)$ have degree $n_{1}, $ $n_{2}, $ $n_{3}$ and $n_{4}, $ respectively for each $k=1, 2, $ and $\varepsilon$ is a small parameter. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of the linear center $\dot{x}=-y, $ $\dot{y}=x$ using the averaging theory of first and second order.
- Limit cycle /
- averaging theory /
- Kukles systems
MSC: 34C29, 34C25, 47H11

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References

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