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In a joint work with Piotr Bizon, we study the nonlinear evolution of a weakly perturbed anti-de Sitter (AdS) space by solving numerically the four-dimensional spherically symmetric Einstein-massless-scalar field equations with a negative cosmological constant. Our results suggest that AdS spacetime is unstable under arbitrarily small generic perturbations. We conjecture that this instability is triggered by a resonant mode mixing which gives rise to diffusion of energy from low to high frequencies. A brief summary of our results can be found in arXiv:1104.3702.