On invariants for difference equations and systems of difference equations of rational form (Q1290953)

The author generalizes results of \textit{C. J. Schinas} [J. Math. Anal. Appl. 216, No. 1, 164-179 (1997; Zbl 0889.39006)] on invariants of difference equations of rational form to second- and third-order autonomous and nonautonomous difference equations. A nontrivial function \(I_n=I(n,x_n,x_{n+1},\ldots, x_{n+N-1})\), \(x_n=x(n)\), is said to be an invariant or integral of motion for an \(N\)th-order ordinary difference equation \(x_{n+N}=f(n,x_n,x_{n+1},\ldots,x_{n+N-1})\) if \(I_{n+1}=I_n\). An autonomous second-order difference equation considered in the paper is \(x(n+1)=y(n)\), \(y(n+1)=P(x(n),y(n))/Q(x(n),y(n))\), where \(P(x,y)\) and \(Q(x,y)\) are biquadratic polynomials in \(x\) and \(y\). Two coupled difference equations are also considered.