Spatially localized, time-periodic excitations, known as discrete breathers, have been found to occur in a wide variety of 1-dimensional (1-D) lattices of nonlinear oscillators with nearest-neighbour coupling. Eliminating the time-dependence from the differential-difference equations of motion, and taking into account only the N largest Fourier modes, we view these solutions as orbits of a (non-integrable) 2N-D map. For breathers to occur, the trivial rest state of the lattice must be a hyperbolic fixed point of the map, with an N-D stable and an N-D unstable manifold. The breathers and multibreathers (with one and more spatial oscillations respectively) are then directly related to the intersections of these manifolds, and hence to homoclinic orbits of the 2N-D map. This is explicitly shown here on a discretized nonlinear Schrödinger equation with only one Fourier mode (N=1), represented by a 2-D map. We then construct the 2N-D map for an array of nonlinear oscillators, with nearest neighbour coupling and a quartic on-site potential, and demonstrate how a one-Fourier-mode representation (via a 2-D map) can be used to provide remarkably accurate initial conditions for the breather and multibreather solutions of the system.