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Sufficient conditions for dynamical systems to have pre-symplectic or pre-implectic structures

Author

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  • Byrnes, G.B.
  • Haggar, F.A.
  • Quispel, G.R.W.

Abstract

We present a number of alternative sufficient conditions for the existence of pre-symplectic or pre-implectic (Poisson) structures, for both ordinary differential (ODE) and ordinary difference (OΔE) equations. Four alternative sets of conditions are obtained for ODEs and OΔEs in n dependent variables: (1) A vector field in involution with the ODE and an integral (or two symmetries for an OΔE) imply a pre-implectic structure; (2) volume preservation and n−2 symmetries imply a pre-symplectic structure; (3) volume preservation and n−2 integrals imply a pre-implectic structure; (4) complex implectic structure implies infinitely many real implectic structures. In all but the first case the methods can give a set of distinct structures.

Suggested Citation

  • Byrnes, G.B. & Haggar, F.A. & Quispel, G.R.W., 1999. "Sufficient conditions for dynamical systems to have pre-symplectic or pre-implectic structures," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 272(1), pages 99-129.
    • Handle: RePEc:eee:phsmap:v:272:y:1999:i:1:p:99-129
    DOI: 10.1016/S0378-4371(99)00094-1
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    References listed on IDEAS

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    1. Quispel, G.R.W. & Capel, H.W. & Papageorgiou, V.G. & Nijhoff, F.W., 1991. "Integrable mappings derived from soliton equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 173(1), pages 243-266.
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