三维Lotka-Volterra系统的双Hamilton结构
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摘要
考虑两个具有3个自由度的Lotka-Volterra系统,首先介绍三维系统中广义Poisson括号和广义Hamilton结构;然后通过构造局部同胚变换,观察得到系统的首次积分,建立代数方程,求解得到系统的Hamilton函数;最后给出Lotka-Volterra系统存在双Hamilton结构的充分性条件.
We considered two Lotka-Volterra system with three degrees of freedom.Firstly,we introduced the generalized Poisson-bracket and generalized Hamiltonian structures in threedimensional systems.Secondly,we observed the first integral of the system by constructing local homeomorphism transformation and established the algebraic equation.The Hamiltonian function of the system was obtained by solving the algebraic equation.Finally,we gave the sufficient conditions for the existence of Bi-Hamiltonian structure of Lotka-Volterra system.
We considered two Lotka-Volterra system with three degrees of freedom.Firstly,we introduced the generalized Poisson-bracket and generalized Hamiltonian structures in threedimensional systems.Secondly,we observed the first integral of the system by constructing local homeomorphism transformation and established the algebraic equation.The Hamiltonian function of the system was obtained by solving the algebraic equation.Finally,we gave the sufficient conditions for the existence of Bi-Hamiltonian structure of Lotka-Volterra system.
引文
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[7]WANG Qinlong,HUANG Wentao,LI Bailian.Limit Cycles and Singular Point Quantities for a 3D LotkaVolterra System[J].Appl Math Comput,2011,217(21):8856-8859.
[8]WANG Ruiping.Competition in a Patchy Environment with Cross-Diffusion in a Three-Dimensional LotkaVolterra System[J].Nonlinear Anal:Real World Appl,2010,11(4):2726-2730.
[9]TUDORAN R M,GRBAN A.On a Hamiltonian Version of a Three-Dimensional Lotka-Volterra System[J].Nonlinear Anal:Real World Appl,2012,13(5):2304-2312.
[10]BALLESTEROS A,BLASCO A,MUSSO F.Integrable Deformations of Lotka-Volterra Systems[J].Phys Lett A,2011,375(38):3370-3374.
[11]李继彬,赵晓华,刘正荣.广义哈密顿系统理论及其应用[M].2版.北京:科学出版社,2007.(LI Jibin,ZHAO Xiaohua,LIU Zhengrong.Generalized Hamilton System Theory and Its Application[M].2nd ed.Beijing:Science Press,2007.)
[12]AY A,GRSES M,ZHELTUKHIN K.Hamiltonian Equations in瓗3[J].J Math Phys,2003,44(12):5688-5705.
[13]CARINENA J F,GUHA P,RA珦NADA M F.Hamiltonian and Quasi-Hamiltonian Systems,Nambu-Poisson Structures and Symmetries[J].J Phys A,2008,41(33):335209-1-335209-11.
[14]ESEN O,GHOSE CHOUDHURY A,GUHA P.Bi-Hamiltonian Structures of 3DChaotic Dynamical Systems[J/OL].Internat J Bifu Chaos,2016-07-29.https://doi.org/10.1142/s0218127416502151.
[15]GAO Puyun.Hamiltonian Structure and First Integrals for the Lotka-Volterra Systems[J].Phys Lett A,2000,273(1/2):85-96.
[2]BRAUER F,CASTILLO-CHVEZ C.Mathematical Models in Population Biology and Epidemiology[M].New York:Springer,2001.
[3]MALCAI O,BIHAM O,RICHMOND P,et al.Theoretical Analysis and Simulations of the Generalized LotkaVolterra Model[J].Phys Rev E,2002,66(3):031102-1-031102-6.
[4]MOREAU Y,LOUIS S,VANDEWALLE J,et al.Embedding Recurrent Neural Networks into Predator-Prey Models[J].Neural Networks,1999,12(2):237-245.
[5]CHAUVET E,PAULLET J E,PREVITE J P,et al.A Lotka-Vokterra Three-Species Food Chain[J].Math Magazine,2002,75(4):243-255.
[6]LLIBRE J,VALLS C.Polynomial,Rational and Analytic First Integrals for a Family of 3-Dimensional LotkaVolterra Systems[J].Z Angew Math Phys,2011,62(5):761-777.
[7]WANG Qinlong,HUANG Wentao,LI Bailian.Limit Cycles and Singular Point Quantities for a 3D LotkaVolterra System[J].Appl Math Comput,2011,217(21):8856-8859.
[8]WANG Ruiping.Competition in a Patchy Environment with Cross-Diffusion in a Three-Dimensional LotkaVolterra System[J].Nonlinear Anal:Real World Appl,2010,11(4):2726-2730.
[9]TUDORAN R M,GRBAN A.On a Hamiltonian Version of a Three-Dimensional Lotka-Volterra System[J].Nonlinear Anal:Real World Appl,2012,13(5):2304-2312.
[10]BALLESTEROS A,BLASCO A,MUSSO F.Integrable Deformations of Lotka-Volterra Systems[J].Phys Lett A,2011,375(38):3370-3374.
[11]李继彬,赵晓华,刘正荣.广义哈密顿系统理论及其应用[M].2版.北京:科学出版社,2007.(LI Jibin,ZHAO Xiaohua,LIU Zhengrong.Generalized Hamilton System Theory and Its Application[M].2nd ed.Beijing:Science Press,2007.)
[12]AY A,GRSES M,ZHELTUKHIN K.Hamiltonian Equations in瓗3[J].J Math Phys,2003,44(12):5688-5705.
[13]CARINENA J F,GUHA P,RA珦NADA M F.Hamiltonian and Quasi-Hamiltonian Systems,Nambu-Poisson Structures and Symmetries[J].J Phys A,2008,41(33):335209-1-335209-11.
[14]ESEN O,GHOSE CHOUDHURY A,GUHA P.Bi-Hamiltonian Structures of 3DChaotic Dynamical Systems[J/OL].Internat J Bifu Chaos,2016-07-29.https://doi.org/10.1142/s0218127416502151.
[15]GAO Puyun.Hamiltonian Structure and First Integrals for the Lotka-Volterra Systems[J].Phys Lett A,2000,273(1/2):85-96.