  We generalise the concept of duality to systems of ordinary difference
equations (or maps). We propose a procedure to construct a chain of systems of
equations which are dual, with respect to an integral $H$, to the given system,
by exploiting the integral relation, defined by the upshifted version and the
original version of $H$. When the numerator of the integral relation is
biquadratic or multi-linear, we point out conditions where a dual fails to
exists. The procedure is applied to several two-component systems obtained as
periodic reductions of 2D lattice equations, including the nonlinear
Schr\"{o}dinger system, the two-component potential Korteweg-De Vries equation,
the scalar modified Korteweg-De Vries equation, and a modified Boussinesq
system.
