Path Memory and Multistability in Coupled Oscillator Networks: A Numerical Study

We report numerical evidence that a coupled oscillator network with a history-dependent internal variable exhibits path memory and multistability. Using a ring of N oscillators with local and global coupling, we find: 1. A critical coupling threshold for global synchronization. Below the threshold, oscillators are desynchronized; above it, they lock into a single synchronized state. 2. Path memory. With a memory mechanism, systems with identical current coupling strength but different histories produce different future probability distributions. This violates the Markov property. 3. Multistability. For fixed coupling parameters, repeated trials with random initial conditions yield multiple distinct stable states. The number of attractors increases with system size (N=20: 2 attractors; N=50: 4; N=200: 5+). These results suggest a physical mechanism for path-dependent memory in dynamical systems: memory is encoded in the attractor structure of the network, not in static weights or external storage. Keywords: coupled oscillators, synchronization, path memory, multistability, attractors This preprint reports a numerical study. It does not validate any broader theoretical framework. The equations of the memory mechanism are not included; they will be presented in a forthcoming publication.