A Novel Integrable Fourth-Order Difference Equation Admitting Three Invariants
75
25. J. Hietarinta, C.M. Viallet, Searching for integrable lattice maps using factorization. J. Phys. A
Math. Theor. 40, 12,629–12,643 (2007)
26. N. Joshi, C.M. Viallet, Rational maps with invariant surfaces. J. Int. Syst. 3, xyy017 (14pp)
(2018)
27. S.K. Lando, Lectures on Generating Functions (American Mathematical Society, 2003)
28. S. Maeda, Completely integrable symplectic mapping. Proc. Jpn. Acad. A Math. Sci. 63,
198–200 (1987)
29. W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes, 3rd edn.
(Cambridge University Press, Cambridge, 2007). The Art of Scientific Computing
30. G.R.W. Quispel, J.A.G. Roberts, C.J. Thompson, Integrable mappings and soliton equations.
Phys. Lett. A 126, 419 (1988)
31. G.R.W. Quispel, J.A.G. Roberts, C.J. Thompson, Integrable mappings and soliton equations II.
Physica D 34(1), 183–192 (1989)
32. G.W.R. Quispel, H.R. Capel, J.A.G. Roberts, Duality for discrete integrable systems. J. Phys.
A Math. Gen. 38(18), 3965 (2005)
33. J.A.G. Roberts, D. Jogia, Birational maps that send biquadratic curves to biquadratic curves. J.
Phys. A Math. Theor. 48, 08FT02 (2015)
34. T. Takenawa, Algebraic entropy and the space of initial values for discrete dynamical systems.
J. Phys. A 34, 10,533 (2001)
35. D.T. Tran, P.H. van der Kamp, G.R.W. Quispel, Closed-form expressions for integrals of
traveling wave reductions of integrable lattice equations. J. Phys. A 42, 225,201, (20pp) (2009)
36. D.T. Tran, P.H. van der Kamp, G.R.W. Quispel, Sufficient number of integrals for the pth-order
Lyness equation. J. Phys. A 43, 302,001, (11pp) (2010)
37. D.T. Tran, P.H. van der Kamp, G.R.W. Quispel, Involutivity of integrals of sine-Gordon,
modified KdV and potential KdV maps. J. Phys. A 44, 295,206, (13pp) (2011)
38. D.T. Tran, P.H. van der Kamp, G.R.W. Quispel, Poisson brackets of mappings obtained as
(q, −p) reductions of lattice equations. Regul. Chaotic Dyn. 21(6), 682–696 (2016)
39. S. Tsujimoto, R. Hirota, Pfaffian representation of solutions to the discrete BKP hierarchy in
bilinear form. J. Phys. Soc. Jpn. 65, 2797–2806 (1996)
40. P. van der Kamp, G.W.R. Quispel, The staircase method: integrals for periodic reductions of
integrable lattice equations. J. Phys. A Math. Theor. 43, 465,207 (2010)
41. A.P. Veselov, Integrable maps. Russ. Math. Surv. 46, 1–51 (1991)
42. A.P. Veselov, Growth and integrability in the dynamics of mappings. Commun. Math. Phys.
145, 181–193 (1992)
43. C.M. Viallet, On the algebraic structure of rational discrete dynamical systems. J. Phys. A
48(16), 16FT01 (2015)
44. C.M. Viallet, B. Grammaticos, A. Ramani, On the integrability of correspondences associated
to integral curves. Phys. Lett. A 322, 186–93 (2004)
75
25. J. Hietarinta, C.M. Viallet, Searching for integrable lattice maps using factorization. J. Phys. A
Math. Theor. 40, 12,629–12,643 (2007)
26. N. Joshi, C.M. Viallet, Rational maps with invariant surfaces. J. Int. Syst. 3, xyy017 (14pp)
(2018)
27. S.K. Lando, Lectures on Generating Functions (American Mathematical Society, 2003)
28. S. Maeda, Completely integrable symplectic mapping. Proc. Jpn. Acad. A Math. Sci. 63,
198–200 (1987)
29. W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes, 3rd edn.
(Cambridge University Press, Cambridge, 2007). The Art of Scientific Computing
30. G.R.W. Quispel, J.A.G. Roberts, C.J. Thompson, Integrable mappings and soliton equations.
Phys. Lett. A 126, 419 (1988)
31. G.R.W. Quispel, J.A.G. Roberts, C.J. Thompson, Integrable mappings and soliton equations II.
Physica D 34(1), 183–192 (1989)
32. G.W.R. Quispel, H.R. Capel, J.A.G. Roberts, Duality for discrete integrable systems. J. Phys.
A Math. Gen. 38(18), 3965 (2005)
33. J.A.G. Roberts, D. Jogia, Birational maps that send biquadratic curves to biquadratic curves. J.
Phys. A Math. Theor. 48, 08FT02 (2015)
34. T. Takenawa, Algebraic entropy and the space of initial values for discrete dynamical systems.
J. Phys. A 34, 10,533 (2001)
35. D.T. Tran, P.H. van der Kamp, G.R.W. Quispel, Closed-form expressions for integrals of
traveling wave reductions of integrable lattice equations. J. Phys. A 42, 225,201, (20pp) (2009)
36. D.T. Tran, P.H. van der Kamp, G.R.W. Quispel, Sufficient number of integrals for the pth-order
Lyness equation. J. Phys. A 43, 302,001, (11pp) (2010)
37. D.T. Tran, P.H. van der Kamp, G.R.W. Quispel, Involutivity of integrals of sine-Gordon,
modified KdV and potential KdV maps. J. Phys. A 44, 295,206, (13pp) (2011)
38. D.T. Tran, P.H. van der Kamp, G.R.W. Quispel, Poisson brackets of mappings obtained as
(q, −p) reductions of lattice equations. Regul. Chaotic Dyn. 21(6), 682–696 (2016)
39. S. Tsujimoto, R. Hirota, Pfaffian representation of solutions to the discrete BKP hierarchy in
bilinear form. J. Phys. Soc. Jpn. 65, 2797–2806 (1996)
40. P. van der Kamp, G.W.R. Quispel, The staircase method: integrals for periodic reductions of
integrable lattice equations. J. Phys. A Math. Theor. 43, 465,207 (2010)
41. A.P. Veselov, Integrable maps. Russ. Math. Surv. 46, 1–51 (1991)
42. A.P. Veselov, Growth and integrability in the dynamics of mappings. Commun. Math. Phys.
145, 181–193 (1992)
43. C.M. Viallet, On the algebraic structure of rational discrete dynamical systems. J. Phys. A
48(16), 16FT01 (2015)
44. C.M. Viallet, B. Grammaticos, A. Ramani, On the integrability of correspondences associated
to integral curves. Phys. Lett. A 322, 186–93 (2004)