  We explore transport properties in a disordered nonlinear chain of classical
harmonic oscillators and thereby identify a regime exhibiting behavior
analogous to that seen in quantum many-body-localized systems. Through
extensive numerical simulations of this system connected at its ends to heat
baths at different temperatures, we computed the heat current and the
temperature profile in the nonequilibrium steady state as a function of system
size $N$, disorder strength $\Delta$, and temperature $T$. The conductivity
$\kappa_N$, obtained for finite length ($N$) systems, saturates to a value
$\kappa_\infty >0$ in the large $N$ limit, for all values of disorder strength
$\Delta$ and temperature $T>0$. We show evidence that for any $\Delta>0$ the
conductivity goes to zero faster than any power of $T$ in the $(T/\Delta) \to
0$ limit, and find that the form $\kappa_\infty \sim e^{-B |\ln(C
\Delta/T)|^3}$ fits our data. This form has earlier been suggested by a theory
based on the dynamics of multi-oscillator chaotic islands. The finite-size
effect can be $\kappa_N < \kappa_{\infty}$ due to boundary resistance when the
bulk conductivity is high (the weak disorder case), or $\kappa_N >
\kappa_{\infty}$ due to direct bath-to-bath coupling through bulk localized
modes when the bulk is weakly conducting (the strong disorder case). We also
present results on equilibrium dynamical correlation functions and on the role
of chaos on transport properties. Finally, we explore the differences in the
growth and propagation of chaos in the weak and strong chaos regimes by
studying the classical version of the Out-of-Time-Ordered-Commutator.
