Topological signature of deterministic chaos in short nonstationary signals from an optical parametric oscillator

摘要

Most quantitative measures of chaos (e.g., fractal dimensions or Lyapunov exponents) rely on constructing an approximation of the natural measure on a strange attractor, which requires observing the system for at least a few hundreds of cycles at fixed control parameters. Thus, it is extremely difficult to assess deterministic chaos in a real system that experiences parameter drifts on a time scale comparable to the mean dynamical period. A natural question then is: can we infer the existence of an underlying chaotic dynamics from a very short, nonstationary, time series?We present an experimental case in which this question can be answered positively. By applying topological tools to a burst of irregular behavior recorded in a triply resonant optical parametric oscillator subject to thermal effects, we have extracted a clearcut signature of deterministic chaos from an extremely short time series segment of only 9 cycles. Indeed, this segment shadows an unstable periodic orbit whose knot type can only occur in a chaotic system. Moreover, this topological approach provides us with quantitative estimates of chaos, as a lower bound on the topological entropy of the system can be determined from the knot structure. Two positive-entropy periodic orbits are detected in a time series of about 40 cycles, suggesting that the presence of such orbits in a time series is common. Thus, nonstationarity is not necessarily an obstacle to the characterization of chaos.