Integrals of a Lotka-Volterra System of Odd Number of Variables

Itoh, Yoshiaki

doi:10.1143/ptp.78.507

Cited by 121 publications

(98 citation statements)

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“…It may be regarded as a special case of a parameter-dependent zero curvature condition. In this work we derive generalized Lotka-Volterra (LV) lattices (see [3,4,5,6,7], for example) from a "master equation" with a bicomplex formulation. Some consequences of the latter (conservation laws [1], Bäcklund transformations [2]) are briefly discussed.…”

Section: Introductionmentioning

confidence: 99%

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Dimakis

Müller‐Hoissen

2002

Czechoslovak Journal of Physics

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Section: Introductionmentioning

confidence: 99%

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Dimakis

Müller‐Hoissen

2002

Czechoslovak Journal of Physics

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“…For Hamiltonian systems, there is the Ziglin theory for the n degrees of freedom systems. This theory has been proved to be useful for the following systems: the motion of rigid body around a fixed point, 11,12 homogeneous potentials, 13,14 the Toda lattices, 15,16 a perturbed Kepler potential, 17 nonhomogeneous potentials, 18 and a reduced Yang-Mills potential. 19 Ziglin's theory is based on the monodromy properties around particular solutions ͑straight line notions͒.…”

Section: Introductionmentioning

confidence: 99%

Integrability, partial integrability, and nonintegrability for systems of ordinary differential equations

Goriely¹

1996

Journal of Mathematical Physics

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On the integrability aspects of the onedimensional classical continuum isotropic biquadratic Heisenberg spin chainThe integrability of systems of ordinary differential equations with polynomial vector fields is investigated by using the singularity analysis methods. Three types of results are obtained. First, a general relationship between the degrees of first integrals and the so-called Kowalevskaya exponents is derived. Second, it is shown that all solutions of algebraically integrable systems can be expanded in Puiseux series. Third, a new method to study partially integrable systems is studied. These different aspects allow us to study algorithmically the integrability, partial integrability, and nonintegrability of differential systems.

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“…Some special case of the lattice 1 was found also independently by Itoh [14]. The lattice 3 after the change of variables a k → a −1 k and t → −t turns intoȧ…”

Section: Continuous-time Bogoyavlensky Latticesmentioning

confidence: 63%

Integrable discretizations of the Bogoyavlensky lattices

Suris

1996

Journal of Mathematical Physics

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Abstract. Discretizations of the Bogoyavlensky lattices are introduced, belonging to the same hierarchies as the continuous-time systems. The construction exemplifies the general scheme for integrable discretization of systems on Lie algebras with r-matrix Poisson brackets. An initial value problem for the difference equations is solved in terms of a factorization problem in a group. Interpolating Hamiltonian flow is found.

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Integrals of a Lotka-Volterra System of Odd Number of Variables

Cited by 121 publications

References 5 publications

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Integrability, partial integrability, and nonintegrability for systems of ordinary differential equations

Integrable discretizations of the Bogoyavlensky lattices

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