The odds of meeting a ghost

Abstract. We consider oscillators whose parameters randomly switch between two values at equal time intervals. If random switching is fast compared to the oscillator’s intrinsic time scale, one expects the switch-ing system to follow the averaged system, obtained by replacing the random variables with their mean. The averaged system is multistable and one of its attractors is not shared by the switching system and acts as a ghost attractor for the switching system. Starting from the attraction basin of the averaged system’s ghost attractor, the trajec-tory of the switching system can converge near the ghost attractor with high probability or may escape to another attractor with low proba-bility. Applying our recent general results on convergent properties of randomly switching dynamical systems [1,2], we derive explicit bounds that connect these probabilities, the switching frequency, and the cho-sen initial conditions.