Maximum number of limit cycles for generalized Kukles differential system

Houdeifa Melki; Amar Makhlouf

doi:10.1515/jaa-2021-2070

Published by De Gruyter April 8, 2022

Maximum number of limit cycles for generalized Kukles differential system

Houdeifa Melki and Amar Makhlouf

From the journal Journal of Applied Analysis

https://doi.org/10.1515/jaa-2021-2070

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Abstract

We apply the averaging theory of first and second order to a class of generalized polynomial Kukles differential systems, which can bifurcate from the periodic orbits of the linear center x ˙ = y , y ˙ = - x , in order to study the maximum number of limit cycles of these systems.

Keywords: Limit cycle; averaging theory; Kukles systems

MSC 2010: 34C29; 34C25; 47H11

A Appendix

We have

∫ 0 2 ⁢ π cos i ⁡ θ ⁢ sin j + 1 ⁡ θ ⁢ cos ⁡ ( ( 2 ⁢ h ) ⁢ θ ) ⁢ 𝑑 θ = { π ⁢ A i ⁢ j 2 ⁢ h if ⁢ i ⁢ even, ⁢ j ⁢ odd , h = 0 , 1 , … , n 1 , π ⁢ A ~ i ⁢ j 2 ⁢ h if ⁢ i ⁢ even, ⁢ j ⁢ odd , h = 0 , 1 , … , n 2 , π ⁢ A ^ i ⁢ j 2 ⁢ h if ⁢ i ⁢ even, ⁢ j ⁢ odd , h = 0 , 1 , … , n 3 , π ⁢ A ¯ i ⁢ j 2 ⁢ h if ⁢ i ⁢ even, ⁢ j ⁢ odd , h = 0 , 1 , … , n 4 , 0 otherwise ,

∫ 0 2 ⁢ π cos i ⁡ θ ⁢ sin j + 1 ⁡ θ ⁢ sin ⁡ ( ( 2 ⁢ h + 1 ) ⁢ θ ) ⁢ 𝑑 θ = { π ⁢ B i ⁢ j 2 ⁢ h + 1 if ⁢ i ⁢ even, ⁢ j ⁢ even , h = 0 , 1 , … , n 1 , π ⁢ B ~ i ⁢ j 2 ⁢ h + 1 if ⁢ i ⁢ even, ⁢ j ⁢ even , h = 0 , 1 , … , n 2 , π ⁢ B ^ i ⁢ j 2 ⁢ h + 1 if ⁢ i ⁢ even, ⁢ j ⁢ even , h = 0 , 1 , … , n 3 , π ⁢ B ¯ i ⁢ j 2 ⁢ h + 1 if ⁢ i ⁢ even, ⁢ j ⁢ even , h = 0 , 1 , … , n 4 , 0 otherwise ,

∫ 0 2 ⁢ π cos i ⁡ θ ⁢ sin j + 1 ⁡ θ ⁢ cos ⁡ ( ( 2 ⁢ h + 1 ) ⁢ θ ) ⁢ 𝑑 θ = { π ⁢ C i ⁢ j 2 ⁢ h + 1 if ⁢ i ⁢ odd, ⁢ j ⁢ odd , h = 0 , 1 , … , n 1 , π ⁢ C ~ i ⁢ j 2 ⁢ h + 1 if ⁢ i ⁢ odd, ⁢ j ⁢ odd , h = 0 , 1 , … , n 2 , π ⁢ C ^ i ⁢ j 2 ⁢ h + 1 if ⁢ i ⁢ odd, ⁢ j ⁢ odd , h = 0 , 1 , … , n 3 , π ⁢ C ¯ i ⁢ j 2 ⁢ h + 1 if ⁢ i ⁢ odd, ⁢ j ⁢ odd , h = 0 , 1 , … , n 4 , 0 otherwise ,

∫ 0 2 ⁢ π cos i ⁡ θ ⁢ sin j + 2 ⁡ θ ⁢ sin ⁡ ( ( 2 ⁢ h + 1 ) ⁢ θ ) ⁢ 𝑑 θ = { π ⁢ D i ⁢ j 2 ⁢ h + 1 if ⁢ i ⁢ even, ⁢ j ⁢ odd , h = 0 , 1 , … , n 1 , π ⁢ D ~ i ⁢ j 2 ⁢ h + 1 if ⁢ i ⁢ even, ⁢ j ⁢ odd , h = 0 , 1 , … , n 2 , π ⁢ D ^ i ⁢ j 2 ⁢ h + 1 if ⁢ i ⁢ even, ⁢ j ⁢ odd , h = 0 , 1 , … , n 3 , π ⁢ D ¯ i ⁢ j 2 ⁢ h + 1 if ⁢ i ⁢ even, ⁢ j ⁢ odd , h = 0 , 1 , … , n 4 , 0 otherwise ,

∫ 0 2 ⁢ π cos i ⁡ θ ⁢ sin j + 2 ⁡ θ ⁢ sin ⁡ ( ( 2 ⁢ h ) ⁢ θ ) ⁢ 𝑑 θ = { π ⁢ E i ⁢ j 2 ⁢ h if ⁢ i ⁢ even, ⁢ j ⁢ even , h = 0 , 1 , … , n 1 , π ⁢ E ~ i ⁢ j 2 ⁢ h if ⁢ i ⁢ even, ⁢ j ⁢ even , h = 0 , 1 , … , n 2 , π ⁢ E ^ i ⁢ j 2 ⁢ h if ⁢ i ⁢ even, ⁢ j ⁢ even , h = 0 , 1 , … , n 3 , π ⁢ E ¯ i ⁢ j 2 ⁢ h if ⁢ i ⁢ even, ⁢ j ⁢ even , h = 0 , 1 , … , n 4 , 0 otherwise ,

∫ 0 2 ⁢ π cos i ⁡ θ ⁢ sin j + 2 ⁡ θ ⁢ cos ⁡ ( ( 2 ⁢ h + 1 ) ⁢ θ ) ⁢ 𝑑 θ = { π ⁢ F i ⁢ j 2 ⁢ h + 1 if ⁢ i ⁢ odd, ⁢ j ⁢ even , h = 0 , 1 , … , n 1 , π ⁢ F ~ i ⁢ j 2 ⁢ h + 1 if ⁢ i ⁢ odd, ⁢ j ⁢ even , h = 0 , 1 , … , n 2 , π ⁢ F ^ i ⁢ j 2 ⁢ h + 1 if ⁢ i ⁢ odd, ⁢ j ⁢ even , h = 0 , 1 , … , n 3 , π ⁢ F ¯ i ⁢ j 2 ⁢ h + 1 if ⁢ i ⁢ odd, ⁢ j ⁢ even , h = 0 , 1 , … , n 4 , 0 otherwise ,

∫ 0 2 ⁢ π cos i ⁡ θ ⁢ sin j + 3 ⁡ θ ⁢ sin ⁡ ( ( 2 ⁢ h + 1 ) ⁢ θ ) ⁢ 𝑑 θ = { π ⁢ G i ⁢ j 2 ⁢ h + 1 if ⁢ i ⁢ even, ⁢ j ⁢ even , h = 0 , 1 , … , n 1 , π ⁢ G ~ i ⁢ j 2 ⁢ h + 1 if ⁢ i ⁢ even, ⁢ j ⁢ even , h = 0 , 1 , … , n 2 , π ⁢ G ^ i ⁢ j 2 ⁢ h + 1 if ⁢ i ⁢ even, ⁢ j ⁢ even , h = 0 , 1 , … , n 3 , π ⁢ G ¯ i ⁢ j 2 ⁢ h + 1 if ⁢ i ⁢ even, ⁢ j ⁢ even , h = 0 , 1 , … , n 4 , 0 otherwise ,

∫ 0 2 ⁢ π cos i ⁡ θ ⁢ sin j + 3 ⁡ θ ⁢ cos ⁡ ( ( 2 ⁢ h ) ⁢ θ ) ⁢ 𝑑 θ = { π ⁢ H i ⁢ j 2 ⁢ h if ⁢ i ⁢ even, ⁢ j ⁢ odd , h = 0 , 1 , … , n 1 , π ⁢ H ~ i ⁢ j 2 ⁢ h if ⁢ i ⁢ even, ⁢ j ⁢ odd , h = 0 , 1 , … , n 2 , π ⁢ H ^ i ⁢ j 2 ⁢ h if ⁢ i ⁢ even, ⁢ j ⁢ odd , h = 0 , 1 , … , n 3 , π ⁢ H ¯ i ⁢ j 2 ⁢ h if ⁢ i ⁢ even, ⁢ j ⁢ odd , h = 0 , 1 , … , n 4 , 0 otherwise ,

∫ 0 2 ⁢ π cos i ⁡ θ ⁢ sin j + 3 ⁡ θ ⁢ cos ⁡ ( ( 2 ⁢ h + 1 ) ⁢ θ ) ⁢ 𝑑 θ = { π ⁢ I i ⁢ j 2 ⁢ h + 1 if ⁢ i ⁢ odd, ⁢ j ⁢ odd , h = 0 , 1 , … , n 1 , π ⁢ I ~ i ⁢ j 2 ⁢ h + 1 if ⁢ i ⁢ odd, ⁢ j ⁢ odd , h = 0 , 1 , … , n 2 , π ⁢ I ^ i ⁢ j 2 ⁢ h + 1 if ⁢ i ⁢ odd, ⁢ j ⁢ odd , h = 0 , 1 , … , n 3 , π ⁢ I ¯ i ⁢ j 2 ⁢ h + 1 if ⁢ i ⁢ odd, ⁢ j ⁢ odd , h = 0 , 1 , … , n 4 , 0 otherwise ,

∫ 0 2 ⁢ π cos i ⁡ θ ⁢ sin j + 4 ⁡ θ ⁢ sin ⁡ ( ( 2 ⁢ h + 1 ) ⁢ θ ) ⁢ 𝑑 θ = { π ⁢ J i ⁢ j 2 ⁢ h + 1 if ⁢ i ⁢ even, ⁢ j ⁢ odd , h = 0 , 1 , … , n 1 , π ⁢ J ~ i ⁢ j 2 ⁢ h + 1 if ⁢ i ⁢ even, ⁢ j ⁢ odd , h = 0 , 1 , … , n 2 , π ⁢ J ^ i ⁢ j 2 ⁢ h + 1 if ⁢ i ⁢ even, ⁢ j ⁢ odd , h = 0 , 1 , … , n 3 , π ⁢ J ¯ i ⁢ j 2 ⁢ h + 1 if ⁢ i ⁢ even, ⁢ j ⁢ odd , h = 0 , 1 , … , n 4 , 0 otherwise ,

∫ 0 2 ⁢ π cos i ⁡ θ ⁢ sin j + 4 ⁡ θ ⁢ cos ⁡ ( ( 2 ⁢ h + 1 ) ⁢ θ ) ⁢ 𝑑 θ = { π ⁢ L i ⁢ j 2 ⁢ h + 1 if ⁢ i ⁢ odd, ⁢ j ⁢ even , h = 0 , 1 , … , n 1 , π ⁢ L ~ i ⁢ j 2 ⁢ h + 1 if ⁢ i ⁢ odd, ⁢ j ⁢ even , h = 0 , 1 , … , n 2 , π ⁢ L ^ i ⁢ j 2 ⁢ h + 1 if ⁢ i ⁢ odd, ⁢ j ⁢ even , h = 0 , 1 , … , n 3 , π ⁢ L ¯ i ⁢ j 2 ⁢ h + 1 if ⁢ i ⁢ odd, ⁢ j ⁢ even , h = 0 , 1 , … , n 4 , 0 otherwise ,

∫ 0 2 ⁢ π cos i + h + 1 ⁡ θ ⁢ sin j + k + 1 ⁡ θ ⁢ d ⁢ θ = { π ⁢ M i ⁢ j ⁢ h ⁢ k if ⁢ i ⁢ even, ⁢ j ⁢ even , h ⁢ odd, ⁢ k ⁢ odd , or ⁢ i ⁢ odd, ⁢ j ⁢ odd , h ⁢ even, ⁢ k ⁢ even , or ⁢ i ⁢ even, ⁢ j ⁢ odd , h ⁢ odd, ⁢ k ⁢ even , or ⁢ i ⁢ odd, ⁢ j ⁢ even , h ⁢ even, ⁢ k ⁢ odd , 0 otherwise ,

∫ 0 2 ⁢ π cos i + h + 1 ⁡ θ ⁢ sin j + k + 2 ⁡ θ ⁢ d ⁢ θ = { π ⁢ N i ⁢ j ⁢ h ⁢ k if ⁢ i ⁢ odd, ⁢ j ⁢ even , h ⁢ even, ⁢ k ⁢ even , or ⁢ i ⁢ even, ⁢ j ⁢ even , h ⁢ odd, ⁢ k ⁢ even , or ⁢ i ⁢ odd, ⁢ j ⁢ odd , h ⁢ even, ⁢ k ⁢ odd , or ⁢ i ⁢ even, ⁢ j ⁢ odd , h ⁢ odd, ⁢ k ⁢ odd , 0 otherwise ,

∫ 0 2 ⁢ π cos i + h + 1 ⁡ θ ⁢ sin j + k + 3 ⁡ θ ⁢ d ⁢ θ = { π ⁢ P i ⁢ j ⁢ h ⁢ k if ⁢ i ⁢ even, ⁢ j ⁢ even , h ⁢ odd, ⁢ k ⁢ odd , or ⁢ i ⁢ odd, ⁢ j ⁢ odd , h ⁢ even, ⁢ k ⁢ even , or ⁢ i ⁢ even, ⁢ j ⁢ odd , h ⁢ odd, ⁢ k ⁢ even , or ⁢ i ⁢ odd, ⁢ j ⁢ even , h ⁢ even, ⁢ k ⁢ odd , 0 otherwise ,

∫ 0 2 ⁢ π cos i + h + 1 ⁡ θ ⁢ sin j + k + 4 ⁡ θ ⁢ d ⁢ θ = { π ⁢ Q i ⁢ j ⁢ h ⁢ k if ⁢ i ⁢ odd, ⁢ j ⁢ even , h ⁢ even, ⁢ k ⁢ even , or ⁢ i ⁢ even, ⁢ j ⁢ even , h ⁢ odd, ⁢ k ⁢ even , or ⁢ i ⁢ odd, ⁢ j ⁢ odd , h ⁢ even, ⁢ k ⁢ odd , or ⁢ i ⁢ even, ⁢ j ⁢ odd , h ⁢ odd, ⁢ k ⁢ odd , 0 otherwise ,

∫ 0 2 ⁢ π cos i + h + 1 ⁡ θ ⁢ sin j + k + 5 ⁡ θ ⁢ d ⁢ θ = { π ⁢ R i ⁢ j ⁢ h ⁢ k if ⁢ i ⁢ even, ⁢ j ⁢ even , h ⁢ odd, ⁢ k ⁢ odd , or ⁢ i ⁢ odd, ⁢ j ⁢ odd , h ⁢ even, ⁢ k ⁢ even , or ⁢ i ⁢ even, ⁢ j ⁢ odd , h ⁢ odd, ⁢ k ⁢ even , or ⁢ i ⁢ odd, ⁢ j ⁢ even , h ⁢ even, ⁢ k ⁢ odd , 0 otherwise ,

∫ 0 2 ⁢ π cos i + h + 1 ⁡ θ ⁢ sin j + k + 6 ⁡ θ ⁢ d ⁢ θ = { π ⁢ S i ⁢ j ⁢ h ⁢ k if ⁢ i ⁢ odd, ⁢ j ⁢ even , h ⁢ even, ⁢ k ⁢ even , or ⁢ i ⁢ even, ⁢ j ⁢ even , h ⁢ odd, ⁢ k ⁢ even , or ⁢ i ⁢ odd, ⁢ j ⁢ odd , h ⁢ even, ⁢ k ⁢ odd , or ⁢ i ⁢ even, ⁢ j ⁢ odd , h ⁢ odd, ⁢ k ⁢ odd , 0 otherwise ,

∫ 0 2 ⁢ π cos i + h + 1 ⁡ θ ⁢ sin j + k + 7 ⁡ θ ⁢ d ⁢ θ = { π ⁢ T i ⁢ j ⁢ h ⁢ k if ⁢ i ⁢ even, ⁢ j ⁢ even , h ⁢ odd, ⁢ k ⁢ odd , or ⁢ i ⁢ odd, ⁢ j ⁢ odd , h ⁢ even, ⁢ k ⁢ even , or ⁢ i ⁢ even, ⁢ j ⁢ odd , h ⁢ odd, ⁢ k ⁢ even , or ⁢ i ⁢ odd, ⁢ j ⁢ even , h ⁢ even, ⁢ k ⁢ odd , 0 otherwise ,

where

α i ⁢ j , β i ⁢ j , δ i ⁢ j , γ i ⁢ j ,

A i ⁢ j 2 ⁢ h , A ~ i ⁢ j 2 ⁢ h , A ^ i ⁢ j 2 ⁢ h , A ¯ i ⁢ j 2 ⁢ h ,

B i ⁢ j 2 ⁢ h + 1 , B ~ i ⁢ j 2 ⁢ h + 1 , B ^ i ⁢ j 2 ⁢ h + 1 , B ¯ i ⁢ j 2 ⁢ h + 1 ,

C i ⁢ j 2 ⁢ h + 1 , C ~ i ⁢ j 2 ⁢ h + 1 , C ^ i ⁢ j 2 ⁢ h + 1 , C ¯ i ⁢ j 2 ⁢ h + 1 ,

D i ⁢ j 2 ⁢ h + 1 , D ~ i ⁢ j 2 ⁢ h + 1 , D ^ i ⁢ j 2 ⁢ h + 1 , D ¯ i ⁢ j 2 ⁢ h + 1 ,

E i ⁢ j 2 ⁢ h , E ~ i ⁢ j 2 ⁢ h , E ^ i ⁢ j 2 ⁢ h , E ¯ i ⁢ j 2 ⁢ h ,

F i ⁢ j 2 ⁢ h + 1 , F ~ i ⁢ j 2 ⁢ h + 1 , F ^ i ⁢ j 2 ⁢ h + 1 , F ¯ i ⁢ j 2 ⁢ h + 1 ,

G i ⁢ j 2 ⁢ h + 1 , G ~ i ⁢ j 2 ⁢ h + 1 , G ^ i ⁢ j 2 ⁢ h + 1 , G ¯ i ⁢ j 2 ⁢ h + 1 ,

H i ⁢ j 2 ⁢ h , H ~ i ⁢ j 2 ⁢ h , H ^ i ⁢ j 2 ⁢ h , H ¯ i ⁢ j 2 ⁢ h ,

I i ⁢ j 2 ⁢ h + 1 , I ~ i ⁢ j 2 ⁢ h + 1 , I ^ i ⁢ j 2 ⁢ h + 1 , I ¯ i ⁢ j 2 ⁢ h + 1 ,

J i ⁢ j 2 ⁢ h + 1 , J ~ i ⁢ j 2 ⁢ h + 1 , J ^ i ⁢ j 2 ⁢ h + 1 , J ¯ i ⁢ j 2 ⁢ h + 1 ,

L i ⁢ j 2 ⁢ h + 1 , L ~ i ⁢ j 2 ⁢ h + 1 , L ^ i ⁢ j 2 ⁢ h + 1 , L ¯ i ⁢ j 2 ⁢ h + 1 ,

M i ⁢ j ⁢ h ⁢ k , N i ⁢ j ⁢ h ⁢ k , P i ⁢ j ⁢ h ⁢ k , Q i ⁢ j ⁢ h ⁢ k , R i ⁢ j ⁢ h ⁢ k , S i ⁢ j ⁢ h ⁢ k , T i ⁢ j ⁢ h ⁢ k

are non-zero constants.

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Received: 2020-02-15

Revised: 2020-09-16

Accepted: 2020-12-05

Published Online: 2022-04-08

Published in Print: 2023-06-01

Maximum number of limit cycles for generalized Kukles differential system

Abstract

A Appendix

References

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