Two Peculiar Classes of Solvable Systems Featuring 2 Dependent Variables Evolving in Discrete-Time via 2 Nonlinearly-Coupled First-Order Recursion Relations. Issue 4 (2nd October 2019) - Calogero, Francesco; Payandeh, Farrin
Abstract : In this paper we identify certain peculiar systems of 2 discrete-time evolution equations, which are algebraically solvable . Here l is the "discrete-time" independent variable taking integer values ( l = 0, 1, 2, . . .), xn ≡ xn ( l ) are 2 dependent variables, and are the corresponding 2 updated variables. In a previous paper the 2 functions F ( n ) ( x 1, x 2 ), n = 1, 2, were defined as follows: F ( n ) ( x 1, x 2 ) = P 2 ( xn, xn +1 ), n = 1, 2 mod[2], with P 2 ( x 1, x 2 ) a specific second-degree homogeneous polynomial in the 2 ( indistinguishable !) dependent variables x 1 ( l ) and x 2 ( l ). In the present paper we further clarify some aspects of that model and we present its extension to the case when a specific homogeneous function of arbitrary ( integer ) degree k (hence a polynomial of degree k when k > 0) in the 2 dependent variables x 1 ( l ) and x 2 ( l ).