ONSET OF CHAOS IN A PENDULUM COUPLED TO A THERMAL ENVIRONMENT

We consider the onset of chaos in a pendulum driven by a Brownian nois e environment. The behavior of the Lyapunov exponents of the system as a function of the strength of the coupling to the thermal environment is investigated for two models of thermal coupling (given by the Lang evin equation and by random stop-start motion). For sufficiently stron g coupling to the environment, the motion is nonchaotic and almost all trajectories are stable but, for each initial velocity and realizatio n of the Brownian noise, an exceptional set of unstable trajectories e xists, in analogy to the existence of basin boundaries in deterministi cally driven systems. The initial points of the unstable trajectories form a fractal set, the dimension of which is zero for coupling to the environment greater than a (temperature-independent) critical value, grows as the environmental coupling is decreased, and equals one at th e transition to chaotic behavior.