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Integrable Non-QRT Mappings of the Plane

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Abstract

We construct 9-parameter and 13-parameter dynamical systems of the plane which map bi-quadratic curves to other bi-quadratic curves and return to the original curve after two iterations. These generalize the QRT maps which map each such curve to itself. The new families of maps include those that were found as reductions of integrable lattices by Joshi et al. (Lett. Math. Phys. 78:27–37, 2006).

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References

  1. Atkinson J., Nijhoff F.W.: Solutions of Adler’s lattice equation associated with 2-cycles of the Bäcklund transformation. J. Nonlinear Math. Phys. 15, 34–42 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  2. Atkinson, J., Private communication

  3. Baxter R.J.: Exactly Solved Models in Statistical Mechanics, pp. 471. Associated Press, London (1982)

    MATH  Google Scholar 

  4. Fordy A.P., Kassotakis P.G: Multidimensional maps of QRT Type. J. Phys. A: Math. Gen. 39, 10773–10786 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  5. Grammaticos B., Ramani A.: Integrable mappings with transcendental invariants. Com. Non. Sci. Num. Simu. 12, 350–356 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Haggar F.A., Byrnes G.B., Quispel G.R.W., Capel H.W.: k-Integrals and k-Lie symmetries in discrete dynamical systems. Phys. A 233, 379–394 (1996)

    Article  MATH  Google Scholar 

  7. Hirota R., Kimura K., Yahagi H.: How to find conserved quantities of nonlinear discrete equations. J. Phys. A: Math. Gen. 34, 10377–10386 (2001)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  8. Hone, A.N.W.: Laurent Polynomials and Superintegrable Maps. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 3, 022, 18p. (2007)

  9. Iatrou A., Roberts J.A.: Integrable mappings of the plane preserving biquadratic invariant curves II. Nonlinearity 15, 459–89 (2002)

    Google Scholar 

  10. Joshi N., Grammaticos B., Tamizhmani T., Ramani A.: From integrable lattices to non-QRT mappings. Lett. Math. Phys. 78, 27–37 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. Kimura K., Yahagi H., Hirota R., Ramani A., Grammaticos B., Ohta Y.: A new class of integrable discrete systems. J. Phys. A: Math. Gen. 35, 9205–9212 (2002)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  12. Ramani A., Carstea A.S., Grammaticos B., Ohta Y.: On the autonomous limit of discrete Painlevé equations. Phys. A 305, 437–444 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Roberts J.A.G., Quispel G.R.W., Thompson C.J.: Integrable mappings and solitons equations. Phys. Lett. A 126, 419–421 (1988)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  14. Roberts J.A.G., Quispel G.R.W.: Chaos and time-reversal symmetry. Order and chaos in reversible dynamical systems. Phys. Rep. 216, 63–177 (1992)

    Article  MathSciNet  ADS  Google Scholar 

  15. Tsuda T., Grammaticos B., Ramani A., Takenawa T.: A class of integrable and nonintegrable mappings and their dynamics. Lett. Math. Phys. 82, 39–49 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  16. Veselov A.P.: What is an integrable mapping?. In: Zakharov, V.E.(eds) What Is Integrability?, pp. 251–272. Springer, Berlin (1990)

    Google Scholar 

  17. Viallet C., Ramani A., Grammaticos B.: On integrability of correspondences associated to integral curves. Phys. Lett. A 322, 186 (2004)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  18. Willox R., Grammaticos B., Ramani A.: A study of the antisymmetric QRT mappings. J. Phys. A: Math. Gen. 38, 5227–5236 (2005)

    Article  MATH  MathSciNet  ADS  Google Scholar 

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Correspondence to Pavlos Kassotakis.

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Kassotakis, P., Joshi, N. Integrable Non-QRT Mappings of the Plane. Lett Math Phys 91, 71 (2010). https://doi.org/10.1007/s11005-009-0360-1

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  • DOI: https://doi.org/10.1007/s11005-009-0360-1

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