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Applicable Analysis
An International Journal
Volume 23, 1986 - Issue 1-2
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Original Articles

Proof of the uniqueness theorem of limit cycles of generalized liénard equationsFootnote*

Zhang Zhifen Peking University
Pages 63-76 | Received 23 Feb 1986, Published online: 02 May 2007

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Yang Kuang. (1988) Nonuniqueness of limit cycles of gause-type predator-prey systems. Applicable Analysis 29:3-4, pages 269-287.
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Yuan Liu, André Zegeling & Wentao Huang. (2024) The Application of Liénard Transformations to Predator–Prey Systems. Qualitative Theory of Dynamical Systems 23:2.
Crossref
Hui Zhou & Yueding Yuan. (2023) On the uniqueness of limit cycles for generalized Liénard systems. Open Mathematics 21:1.
Crossref
Maíra Duran Baldissera, Jaume Llibre & Regilene Oliveira. (2022) Dynamics of a Generalized Rayleigh System. Differential Equations and Dynamical Systems.
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Min Lu, Jicai Huang, Shigui Ruan & Pei Yu. (2020) Global Dynamics of a Susceptible-Infectious-Recovered Epidemic Model with a Generalized Nonmonotone Incidence Rate. Journal of Dynamics and Differential Equations 33:4, pages 1625-1661.
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Hebai Chen, Jaume Llibre & Yilei Tang. (2021) The Limit Cycles of the Higgins–Selkov Systems. Journal of Nonlinear Science 31:5.
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Robert E. Kooij & André Zegeling. (2021) Co-existence of a Period Annulus and a Limit Cycle in a Class of Predator–Prey Models with Group Defense. International Journal of Bifurcation and Chaos 31:10, pages 2150154.
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Ragupathy Venkatachalam & Ying-Fang Kao. 2021. Keynesian, Sraffian, Computable and Dynamic Economics. Keynesian, Sraffian, Computable and Dynamic Economics 273 310 .
Sze-Bi Hsu, Zhihua Liu & Pierre Magal. (2020) A Holling Predator-Prey Model with Handling and Searching Predators. SIAM Journal on Applied Mathematics 80:4, pages 1778-1795.
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Robert E. Kooij & André Zegeling. (2019) Predator-prey models with non-analytical functional response. Chaos, Solitons & Fractals 123, pages 163-172.
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H. Qolizadeh Amirabad, O. RabieiMotlagh & H.M. MohammadiNejad. (2019) Permanency in predator–prey models of Leslie type with ratio-dependent simplified Holling type-IV functional response. Mathematics and Computers in Simulation 157, pages 63-76.
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Pia Brechmann & Alan D. Rendall. (2018) Dynamics of the Selkov oscillator. Mathematical Biosciences 306, pages 152-159.
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Tohid Kasbi & Vahid Roomi. (2018) Existence and uniqueness of limit cycles in planar system of Liénard type. Rendiconti del Circolo Matematico di Palermo Series 2 67:3, pages 547-556.
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Alexandr V. Osipov & Gunnar Söderbacka. (2017) Poincaré Map Construction for Some Classic Two Predators–One Prey Systems. International Journal of Bifurcation and Chaos 27:08, pages 1750116.
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Zhinan Xia, Meng Fan & Ravi P Agarwal. (2017) ALMOST AUTOMORPHIC DYNAMICS OF GENERALIZED LIÉNARD EQUATION. Journal of Applied Analysis & Computation 7:1, pages 20-38.
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Xianbo Sun & Liqin Zhao. (2016) Perturbations of a class of hyper-elliptic Hamiltonian systems of degree seven with nilpotent singular points. Applied Mathematics and Computation 289, pages 194-203.
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A-ying Wan, Zhi-qiang Song & Li-fei Zheng. (2016) Patterned solutions of a homogenous diffusive predator-prey system of Holling type-III. Acta Mathematicae Applicatae Sinica, English Series 32:4, pages 1073-1086.
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Jitao Sun, Qing-guo Wang, Junping Shi & Fangfang Jiang. (2016) On the existence and uniqueness of a limit cycle for a Liénard system with a discontinuity line. Communications on Pure and Applied Analysis 15:6, pages 2509-2526.
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Jitsuro Sugie & Yasuhisa Saito. (2012) Uniqueness of Limit Cycles in a Rosenzweig–MacArthur Model with Prey Immigration. SIAM Journal on Applied Mathematics 72:1, pages 299-316.
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Alberto d’Onofrio. (2010) Globally attractive oscillations in open monosubstrate allosteric enzyme reactions. Journal of Mathematical Chemistry 49:2, pages 531-545.
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Shigui Ruan, Yilei Tang & Weinian Zhang. (2010) Versal unfoldings of predator–prey systems with ratio-dependent functional response. Journal of Differential Equations 249:6, pages 1410-1435.
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Nurul Huda Gazi & Kalyan Das. (2010) Structural stability analysis of an algal bloom mathematical model in tropic interaction. Nonlinear Analysis: Real World Applications 11:4, pages 2191-2206.
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Alberto d’Onofrio. (2010) Uniqueness and global attractivity of glycolytic oscillations suggested by Selkov’s model. Journal of Mathematical Chemistry 48:2, pages 339-346.
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Angelo Antoci, Roberto Dei & Marcello Galeotti. (2009) Financing the adoption of environment preserving technologies via innovative financial instruments: An evolutionary game approach. Nonlinear Analysis: Theory, Methods & Applications 71:12, pages e952-e959.
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Michele V Bartuccelli, Jonathan H.B Deane & Guido Gentile. (2008) Frequency locking in an injection-locked frequency divider equation. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465:2101, pages 283-306.
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Dongmei Xiao & Zhifen Zhang. (2008) On the existence and uniqueness of limit cycles for generalized Liénard systems. Journal of Mathematical Analysis and Applications 343:1, pages 299-309.
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Yilei Tang & Weigu Li. (2007) Global analysis of an epidemic model with a constant removal rate. Mathematical and Computer Modelling 45:7-8, pages 834-843.
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Wei Wang & Jian Hua Sun. (2006) On the Predator–Prey System with Holling-(n + 1) Functional Response. Acta Mathematica Sinica, English Series 23:1, pages 1-6.
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Alberto d’Onofrio. (2006) On a family of models of cell division cycle. Chaos, Solitons & Fractals 27:5, pages 1205-1212.
Crossref
Tzy-Wei Hwang & Hsin-Jung Tsai. (2005) Uniqueness of limit cycles in theoretical models of certain oscillating chemical reactions. Journal of Physics A: Mathematical and General 38:38, pages 8211-8223.
Crossref
Alberto d’Onofrio. (2005) Mathematical analysis of the Tyson model of the regulation of the cell division cycle. Nonlinear Analysis: Theory, Methods & Applications 62:5, pages 817-831.
Crossref
Maoan Han, Ping Bi & Dongmei Xiao. (2004) Bifurcation of limit cycles and separatrix loops in singular Lienard systems. Chaos, Solitons & Fractals 20:3, pages 529-546.
Crossref
Jifa Jiang & Jinjie Liu. (2003) On uniqueness of limit cycle for the equation ẍ+f(x)φ(ẋ)ẋ+g(x)=0. Journal of Mathematical Analysis and Applications 284:1, pages 1-8.
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Dongmei Xiao & Zhifeng Zhang. (2003) On the uniqueness and nonexistence of limit cycles for predator prey systems. Nonlinearity 16:3, pages 1185-1201.
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Tianshou Zhou & Suochun Zhang. (2002) Echo waves and coexisting phenomena in coupled brusselators. Chaos, Solitons & Fractals 13:4, pages 621-632.
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Yu-quan Wang & Zhu-jun Jing. (2002) Cubic Lienard Equations with Quadratic Damping (II). Acta Mathematicae Applicatae Sinica, English Series 18:1, pages 103-116.
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Freddy Dumortier, Robert E. Kooij & Chengzhi Li. (2000) Cubic Liénard equations with quadratic damping having two antisaddles. Qualitative Theory of Dynamical Systems 1:2, pages 163-209.
Crossref
Jitsuro Sugie. (2000) Uniqueness of limit cycles in a predator-prey system with Holling-type functional response. Quarterly of Applied Mathematics 58:3, pages 577-590.
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Robert E. Kooij & Fotios Giannakopoulos. (2000) Periodic orbits in planar systems modelling neural activity. Quarterly of Applied Mathematics 58:3, pages 437-457.
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André Zegeling & Robert E Kooij. (1999) The Distribution of Limit Cycles in Quadratic Systems with Four Finite Singularities. Journal of Differential Equations 151:2, pages 373-385.
Crossref
R.E. Kooij & A. Zegeling. (1997) Qualitative properties of two-dimensional predator-prey systems. Nonlinear Analysis: Theory, Methods & Applications 29:6, pages 693-715.
Crossref
Freddy Dumortier & Chengzhi Li. (1997) Quadratic Liénard Equations with Quadratic Damping. Journal of Differential Equations 139:1, pages 41-59.
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Daniele Ritelli, Giancarlo Barbiroli & Paolo Fabbri. (1997) Predation among technologies on the market: A modellistic analysis. Journal of Mathematical Economics 27:3, pages 347-374.
Crossref
K. Jiang & M. Han. (1996) Boundedness of solutions and existence of limit cycles for a nonlinear system. Nonlinear Analysis: Theory, Methods & Applications 26:12, pages 1995-2006.
Crossref
Robert E. Kooij. (2008) Cubic systems with four real line invariants. Mathematical Proceedings of the Cambridge Philosophical Society 118:1, pages 7-19.
Crossref
Xie Xiangdong & Cai Suilin. (1995) Bifurcation of limit cycle for quadratic system with invariant parabola. Applied Mathematics 10:2, pages 195-206.
Crossref
Helmar Nunes Moreira & Wang Yuquan. (1994) Sufficient conditions for the Sal'nikov equation to have at least two limit cycles. Journal of Mathematical Chemistry 15:1, pages 63-72.
Crossref
André Zegeling & Robert E. Kooij. (2009) Uniqueness of limit cycles in polynomial systems with algebraic invariants. Bulletin of the Australian Mathematical Society 49:1, pages 7-20.
Crossref
S. A. Gils. 1994. Dynamics, Bifurcation and Symmetry. Dynamics, Bifurcation and Symmetry 333 341 .
Torsten Lindstr�m. (1993) Qualitative analysis of a predator-prey system with limit cycles. Journal of Mathematical Biology 31:6, pages 541-561.
Crossref
Helmar Nunes Moreira. (1992) Liénard-type equations and the epidemiology of malaria. Ecological Modelling 60:2, pages 139-150.
Crossref
F Dumortier & C Rousseau. (1990) Cubic Lienard equations with linear damping. Nonlinearity 3:4, pages 1015-1039.
Crossref
H.I. Freedman & Y. Kuang. (1990) Uniqueness of limit cycles in liénard-type equations. Nonlinear Analysis: Theory, Methods & Applications 15:4, pages 333-338.
Crossref
S. Lynch. (1990) Small-amplitude limit cycles of Liénard systems. Calcolo 27:1-2, pages 1-32.
Crossref
Yang Kuang. (1989) Limit Cycles in a Chemostat-Related Model. SIAM Journal on Applied Mathematics 49:6, pages 1759-1767.
Crossref
Yang Kuang. (1989) On the location and period of limit cycles in Gause-type predator-prey systems. Journal of Mathematical Analysis and Applications 142:1, pages 130-143.
Crossref
M. Galeotti & F. Gori. (2006) UNIQUENESS OF PERIODIC ORBITIS IN LIENARD‐TYPE BUSINESS‐CYCLE MODELS. Metroeconomica 40:2, pages 135-146.
Crossref
W. A. Coppel. (2009) Quadratic systems with a degenerate critical point. Bulletin of the Australian Mathematical Society 38:1, pages 1-10.
Crossref
Yang Kuang & H.I. Freedman. (1988) Uniqueness of limit cycles in Gause-type models of predator-prey systems. Mathematical Biosciences 88:1, pages 67-84.
Crossref
F.M.A. Salam, S.A. Van Gils & Zhang Zhi-Fen. (1988) Global bifurcation analysis of an adaptive control system. Global bifurcation analysis of an adaptive control system.

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