Continuous and Discrete Linearizable Systems: The Riccati Saga

  • B. Grammaticos
  • A. Ramani
Part of the CRM Series in Mathematical Physics book series (CRM)


We investigate the extensions of the discrete Riccati equation, as a linearizable system, to higher dimensions. We first study the continuous and discrete (second-order) Gambier equation, which is a coupling of two Riccati equations in cascade. In the N-dimensional case, three new integrable mappings are obtained: they are the linearizable discretizations of the well-known projective, matrix, and conformai Riccati systems.


Riccati Equation Singular Behavior Discrete Linearisable System Matrix Riccati Equation Confinement Condition 
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  1. 1.
    M.J. Ablowitz, A. Ramani, and H. Segur, Nonlinear evolution equations and ordinary differential equations of Painlevé-type, Nuovo Cimento 23 (1978), 333–338.MathSciNetGoogle Scholar
  2. 2.
    R.L. Anderson, J. Harnad, and P. Winternitz, Systems of ordinary differential equations with nonlinear superposition principles, Phys. D 4 (1982), 164–182.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    J. Chazy, Sur les équations différentielles du troisième ordre et d’ordre supérieur dont l’intégrale générale a ses points critiques fixes, Acta Math. 34 (1910), 317–385.MathSciNetCrossRefGoogle Scholar
  4. 4.
    B. Gambier, Sur les équations différentielles du second ordre et du premier degré dont l’intégrale générale est à points critiques fixes, Acta Math. 33 (1910), 1–55.MathSciNetCrossRefGoogle Scholar
  5. 5.
    B. Grammaticos and A. Ramani, The Gambier mapping, Phys. A 223 (1995), 125–136.Google Scholar
  6. 6.
    B. Grammaticos and A. Ramani, Retracing the Painlevé-Gambier classification for discrete systems, Methods Appl. Anal. 4 (1997), no. 2, 196–211.MathSciNetMATHGoogle Scholar
  7. 7.
    B. Grammaticos, A. Ramani and K.M. Tamizhmani, Nonproliferation of preimages in integrable mappings, J. Phys. A 7 (1994), 559–566.MathSciNetADSGoogle Scholar
  8. 8.
    B. Grammaticos, A. Ramani, and P. Winternitz, Dicretizing families of linearizable equations, Phys. Lett. A 245 (1998), no. 5, 382–388.MathSciNetADSGoogle Scholar
  9. 9.
    A. Ramani, B. Grammaticos, and G. Karra, Linearizable mappings, Phys. A 181 (1992), 115–127.MathSciNetGoogle Scholar

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© Springer Science+Business Media New York 2001

Authors and Affiliations

  • B. Grammaticos
  • A. Ramani

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