  We investigate the dynamics of a macroscopic system which consists of an
anharmonic subsystem embedded in an arbitrary harmonic lattice, including
quenched disorder. Elimination of the harmonic degrees of freedom leads to a
nonlinear Langevin equation for the anharmonic coordinates. For zero
temperature, we prove that the support of the Fourier transform of the memory
kernel and of the time averaged velocity-velocity correlations functions of the
anharmonic system can not overlap. As a consequence, the asymptotic solutions
can be constant, periodic,quasiperiodic or almost periodic, and possibly weakly
chaotic. For a sinusoidal trajectory with frequency $\Omega$ we find that the
energy $E_T$ transferred to the harmonic system up to time $T$ is proportional
to $T^{\alpha}$. If $\Omega$ equals one of the phonon frequencies $\omega_\nu$,
it is $\alpha=2$. We prove that there is a full measure set such that for
$\Omega$ in this set it is $\alpha=0$, i.e. there is no energy dissipation.
Under certain conditions there exists a zero measure set such that for $\Omega
\in this set the dissipation rate is nonzero and may be subdissipative $(0 \leq
\alpha < 1)$ or superdissipative $(1 <\alpha \leq 2)$. Consequently, the
harmonic bath does act as an anomalous thermostat. Intraband discrete breathers
are such solutions which do not relax. We prove for arbitrary anharmonicity and
small but finite coupling that intraband discrete breathers with frequency
$\Omega$ exist for all $\Omega$ in a Cantor set $\mathcal{C}(k)$ of finite
Lebesgue measure. This is achieved by estimating the contribution of small
denominators appearing in the memory kernel. For $\Omega\in\mathcal{C}(k)$ the
small denominators do not lead to divergencies such that this kernel is a
smooth and bounded function in $t$.
