Abstract
In this paper, we study the distribution and number of limit cycles for quadratic systems with two foci. It is proved that a quadratic system with two foci has at most one limit cycles around one of the two foci, and hence the limit cycles of the quadratic system with two foci must be (0, 1)-distribution or (1,i)-distribution (i=0,1,2,…)
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References
Shi Songling,A concrete example of the existence of four limit cycles for plane quadratic systems, Sci. China Ser.11 (1979), 1051–1056.
Ye Yangqian et, al.,Theory of Limit Cycle, Translation of Math. Monographs 66, American Math. Society, Providence, R.I., 1986.
Zhang Zhifen, et, al.,Qualitative Theory of Differential Equation (in Chinese); Modem Math. Basic Series, Science Press, Beijing, 1985.
Zhang Pingguang,On the concentrative distribution and uniqueness of limit cycles for a quadratic system, Dynamical systems, Proc. Prog. Nanki Inst. Math. Tianjin China, Sept, 1990–1991, World Sci. Pallco. Ptc. Ltd. Singapore4 (1993), 297–310.
Zhang Pingguang,On the concentrative distribution and number of limit cycles for a quadratic system (in Chinese), Appl. Math. J. Chinese Univ13A (1998), 1–12.
Zhang Pingguang,Quadratic systems with two foci (in Chinese), Appl. Math. J. Chinese Univ14A (1999), 247–253.
Zhang Pingguang,On the distribution and number of limit cycles for quadratic systems with two foci (in Chinese), Acta Math. Sinica44 (2001), no. 1, 37–44.
Zhang Pingguang andLi Wenhua,A quadratic system with weak focus and strong focus, Ann. Differential Equations8 (1992), no. 1, 122–128.
Zhang Pingguang andCai Suilin,Quadratic systems with a weak focus, Bull. Austral. Math. Soc.44 (1991), no. 3, 511–526.
Zhang Pingguang,Quadratic systems with a weak focus and a strong focus, Appl. Math. J. Chinese Univ.14B (1999), no. 1, 7–14.
Zhang Pingguang andZhao Shenqi,On number of limit cycle surrounding strong focus for a quadratic systems with a weak focus and a strong focus, Appl. Math. J. Chinese Univ.16B (2001), no. 2, 127–132.
Li Chengzhi,Non-existence LCs around a fine focus of order 3 of quadratic systems, Chin. Ann. of Math.7B (1986), no. 2, 174–190.
Zhang Pingguang,Uniqueness of the limit cycle around a second-order weak focus for quadratic systems, (in Chinese) Acta Math. Sinica42 (1999), no. 2, 289–304.
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Pingguang, Z. On the distribution and number of limit cycles for quadratic systems with two foci. Qual. Th. Dyn. Syst. 3, 437–463 (2002). https://doi.org/10.1007/BF02969414
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DOI: https://doi.org/10.1007/BF02969414