Dynamics of nonlinear oscillators with random interactions

We develop a mean field theory fo a system of coupled oscillators with random interactions with variable symmetry. Numerical simulations of the resulting one-dimensional dynamics are in accordance with simulations of the N-oscillator dynamics. We find a transition in dependence on interaction strength J and symmetry parameter µ from a dynamically disordered phase to a phase with static disorder, where all oscillators are frozen in random positions. This transition between the"paramagnetic" phase and the spin glass phase appears to be of first order and is dynamically characterized by chaos (positive Lyapunov exponents) in the former case and regular motion (vanishing Lyapunov exponents) in the latter case. The Lyapunov spectrum shows an interesting symmetry for antisymmetric interaction (µ=-1).