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Necessary conditions for the existence of quasi-polynomial invariants: the quasi-polynomial and Lotka–Volterra systems

Author

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  • Figueiredo, A.
  • Filho, T.M.Rocha
  • Brenig, L.

Abstract

We show that any quasi-polynomial invariant of a quasi-polynomial dynamical system can be transformed into a quasi-polynomial invariant of a homogeneous quadratic Lotka–Volterra dynamical system. We show how this quasi-polynomial invariant can be decomposed in a simple manner. This decomposition permits to conclude that the existence of polynomial semi-invariants in Lotka–Volterra systems is a necessary condition for the existence of quasi-polynomial invariants. We derive a method which allows to construct the necessary conditions for existence of semi-invariants on Lotka–Volterra dynamical systems. Applications are given.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:phsmap:v:262:y:1999:i:1:p:158-180
    DOI: 10.1016/S0378-4371(98)00396-3
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    References listed on IDEAS

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    1. 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.
    2. 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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    Cited by:

    1. Figueiredo, Annibal & Rocha Filho, Tarcisio M., 2009. "Basins of attraction of invariant regular manifolds," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 1877-1889.

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