We discuss stationary solutions of the discrete nonlinear Schr\"odinger
equation (DNSE) with a potential of the ϕ4 type which is generically
applicable to several quantum spin, electron and classical lattice systems. We
show that there may arise chaotic spatial structures in the form of
incommensurate or irregular quantum states. As a first (typical) example we
consider a single electron which is strongly coupled with phonons on a 1D
chain of atoms --- the (Rashba)--Holstein polaron model. In the adiabatic
approximation this system is conventionally described by the DNSE. Another
relevant example is that of superconducting states in layered superconductors
described by the same DNSE. Amongst many other applications the typical example
for a classical lattice is a system of coupled nonlinear oscillators. We
present the exact energy spectrum of this model in the strong coupling limit
and the corresponding wave function. Using this as a starting point we go on to
calculate the wave function for moderate coupling and find that the energy
eigenvalue of these structures of the wave function is in exquisite agreement
with the exact strong coupling result. This procedure allows us to obtain
(numerically) exact solutions of the DNSE directly. When applied to our typical
example we find that the wave function of an electron on a deformable lattice
(and other quantum or classical discrete systems) may exhibit incommensurate
and irregular structures. These states are analogous to the periodic,
quasiperiodic and chaotic structures found in classical chaotic dynamics