Discrete mathematical modelling seminar

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Organizer(s) Tetsuji Tokihiro, Ralph Willox

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17:15-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Dinh T. Tran (School of Mathematics and Statistics, The University of Sydney)
Integrability for four-dimensional recurrence relations
[ Abstract ]
In this talk, we study some fourth-order recurrence relations (or mappings). These recurrence relations were obtained by assuming that they possess two polynomial integrals with certain degree patterns.
For mappings with cubic growth, we will reduce them to three-dimensional ones using a procedure called deflation. These three-dimensional maps have two integrals; therefore, they are integrable in the sense of Liouville-Arnold. Furthermore, we can reduce the obtained three-dimensional maps to two-dimensional maps of Quispel-Roberts-Thompsons (QRT) type. On the other hand, for recurrences with quadratic growth and two integrals, we will show that they are integrable in the sense of Liouville-Arnold by providing their Poisson brackets. Non-degenerate Poisson brackets can be found by using the existence of discrete Lagrangians and a discrete analogue of the Ostrogradsky transformation.
This is joint work with G. Gubbiotti, N. Joshi, and C-M. Viallet.