  Weakly nonlinear dynamics in anti-de Sitter (AdS) spacetimes is reviewed,
keeping an eye on the AdS instability conjecture and focusing on the resonant
approximation that accurately captures in a simplified form the long-term
evolution of small initial data. Topics covered include turbulent and regular
motion, dynamical recurrences analogous to the Fermi-Pasta-Ulam phenomena in
oscillator chains, and relations between AdS dynamics and nonrelativistic
nonlinear Schrodinger equations in harmonic potentials. Special mention is
given to the way the classical dynamics of weakly nonlinear strongly resonant
systems is illuminated by perturbative considerations within the corresponding
quantum theories, in particular, in relation to quantum chaos theory.
