Limit cycles and global dynamic of planar cubic semi-quasi-homogeneous systems

Zecen He; Haihua Liang; Xiang Zhang

doi:10.3934/dcdsb.2021049

Article Contents

2022, Volume 27, Issue 1: 421-441. Doi: 10.3934/dcdsb.2021049

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Limit cycles and global dynamic of planar cubic semi-quasi-homogeneous systems

1.
School of Mathematics and Systems Science, Guangdong Polytechnic Normal University, Guangzhou 510665, China
2.
School of Mathematical Sciences and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, China

^* Corresponding author: Haihua Liang
^* Corresponding author: Haihua Liang

Received: September 2020

Revised: December 2020

Early access: February 2021

Published: January 2022

The second author is supported by the NNSF of China grant 11771101, by the major research program of colleges and universities in Guangdong Province grant 2017KZDXM054, and by the Science and Technology Program of Guangzhou of China grant 201805010001. The third author is partially supported by NNSF of China grant 11671254, 11871334 and 12071284.

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Abstract

Denote by CH, CSH, CQH, and CSQH the planar cubic homogeneous, cubic semi-homogeneous, cubic quasi-homogeneous and cubic semi-quasi-homogeneous differential systems, respectively. The problems on limit cycles and global dynamics of these systems have been solved for CH, and partially for CSH. This paper studies the same problems for CQH and CSQH. We prove that CQH have no limit cycles and CSQH can have at most one limit cycle with the limit cycle realizable. Moreover, we classify all the global phase portraits of CSQH.

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Mathematics Subject Classification: Primary: 34C07; Secondary: 34C05, 34C41.

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Figure 1. The global phase portraits of system (2) with $ a<-1 $

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Figure 2. The topological equivalence classes of the global phase portraits of planar cubic semi-quasi-homogeneous systems

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Figure 3.

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Figure 4.

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Figure 6. The global phase portraits of systems $ (A_{1}) $ and $ (B_{1, k})\ (k = 1, 2, 3) $

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Figure 5. Local phase portrait of system (10) at the origin

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Figure 7. The phase portrait of systems $ (M^{\pm}_{1, k, l}) $

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Figure 8. The global phase portraits of systems $ (A^{\pm}_{2, k}) $

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Figure 9. The global phase portraits of systems $ (C_{1, k}), $ $ (D_1), $ $ (E_{1, k}) $ and $ (F_{1}) $

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Figure 10. The global phase portraits of systems $ (G^{\pm}_{1, k, l}) $

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Figure 11. The global phase portraits of systems $ (H^{\pm}_{1, k}) $ and $ (I^{\pm}_{1, k}) $

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Figure 12. The phase portraits of systems $ (J^{\pm}_{1, k, l}) $ and $ L^{\pm}_{1, k, l}) $

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Figure 13. The global phase portraits of systems $ (B^{\pm}_{2, k}) $

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Figure 14. The global phase portraits of systems $ (C_{2, 1}) $ and $ (C^{\pm}_{2, 2}) $

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Figure 15. The global phase portraits of systems $ (D^{\pm}_{2, k}) $

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Figure 16. The global phase portraits of systems $ (E_{2, k}) $

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Figure 17. The global phase portraits of systems $ (F_{2, k}) $

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Table 1. The known results about the CH, CSH, CQH, and CSQH before this paper

Type of cubic systems	The maximum number of limit cycles	The global dynamics
Homogeneous	0	completed
Semi-Homogeneous	≥ 1	for some subclasses
Quasi-Homogeneous	?	for some subclasses
Semi-Quasi-Homogeneous	?	?

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Table 2. The known results about the CH, CSH, CQH, and CSQH after this paper

Type of cubic systems	The maximum number of limit cycles	The global dynamics
Homogeneous	0	completed
Semi-Homogeneous	≥	1 for some subclasses
Quasi-Homogeneous	0	for some subclasses
Semi-Quasi-Homogeneous	1	completed

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