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On the complete and partial integrability of non-Hamiltonian systems

Author

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  • Bountis, T.C.
  • Ramani, A.
  • Grammaticos, B.
  • Dorizzi, B.

Abstract

The methods of singularity analysis are applied to several third order non-Hamiltonian systems of physical significance including the Lotka-Volterra equations, the three-wave interaction and the Rikitake dynamo model. Complete integrability is defined and new completely integrable systems are discovered by means of the Painlevé property. In all these cases we obtain integrals, which reduce the equations either to a final quadrature or to an irreducible second order ordinary differential equation (ODE) solved by Painlevé transcendents. Relaxing the Painlevé property we find many partially integrable cases whose movable singularities are poles at leading order, with In(t-t0) terms entering at higher orders. In an Nth order, generalized Rössler model a precise relation is established between the partial fulfillment of the Painlevé conditions and the existence of N - 2 integrals of the motion.

Suggested Citation

  • Bountis, T.C. & Ramani, A. & Grammaticos, B. & Dorizzi, B., 1984. "On the complete and partial integrability of non-Hamiltonian systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 128(1), pages 268-288.
    • Handle: RePEc:eee:phsmap:v:128:y:1984:i:1:p:268-288
    DOI: 10.1016/0378-4371(84)90091-8
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    References listed on IDEAS

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    1. Hijmans, J. & Schram, H.M., 1983. "On the bifurcations occuring in the parameter space of the sixteen vertex model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 121(3), pages 479-512.
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    Cited by:

    1. Figueiredo, A. & Filho, T.M.Rocha & Brenig, L., 1999. "Necessary conditions for the existence of quasi-polynomial invariants: the quasi-polynomial and Lotka–Volterra systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 262(1), pages 158-180.
    2. Bountis, Tassos & Tsarouhas, George, 1988. "On the application of normal forms near attracting fixed points of dynamical systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 153(1), pages 160-178.
    3. Grammaticos, B. & Moulin-Ollagnier, J. & Ramani, A. & Strelcyn, J.-M. & Wojciechowski, S., 1990. "Integrals of quadratic ordinary differential equations in R3: The Lotka-Volterra system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 163(2), pages 683-722.

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