Recurrence and symmetry of time series: Application to transition detection

The study of transitions in low dimensional, nonlinear dynamical systems is a complex problem for which there is not yet a simple, global numerical method able to detect chaos-chaos, chaos-periodic bifurcations and symmetry-breaking, symmetry-increasing bifurcations. We present here for the first time a general framework focusing on the symmetry concept of time series that at the same time reveals new kinds of recurrence. We propose several numerical tools based on the symmetry concept allowing both the qualification and quantification of different kinds of possible symmetry. By using several examples based on periodic symmetrical time series and on logistic and cubic maps, we show that it is possible with simple numerical tools to detect a large number of bifurcations of chaos-chaos, chaos-periodic, broken symmetry and increased symmetry types.