Large language models are increasingly deployed not as single assistants but as committees whose members deliberate and then vote or synthesize a decision. Such systems are often expected to be more robust than individual models. We show that iterative multi-LLM deliberation can instead amplify tiny perturbations into divergent conversational trajectories and different final decisions. In a fully deterministic self-hosted benchmark, exact reruns are identical, yet small meaning-preserving changes to the scenario text still separate over time and often alter the final recommendation. In deployed black-box API systems, nominally identical committee runs likewise remain unstable even at temperature 0, where many users expect near-determinism. Across 12 policy scenarios, these findings indicate that instability in collective AI is not only a consequence of residual platform-side stochasticity, but can arise from sensitivity to nearby initial conditions under repeated interaction itself. Additional deployed experiments show that committee architecture modulates this instability: role structure, model composition, and feedback memory can each alter the degree of divergence. Collective AI therefore faces a stability problem, not only an accuracy problem: deterministic execution alone does not guarantee predictable or auditable deliberative outcomes.