Recurrence and symmetry of time series: Application to transition detection

Abstract

International audienceThe 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

Centered and Averaged Fuzzy Entropy to Improve Fuzzy Entropy Precision

“…However, from our point of view, only isometries (translation T , reflection R , inversion I and glide reflection G ) are suitable. This statement is supported by the recent work reported in [ 8 ] where the concept of symmetry was placed back on stage in the study of time series. Indeed, in [ 8 ], it was shown that the concept of recurrences could be generalized by taking into account the symmetry properties of m -patterns.…”

“…This statement is supported by the recent work reported in [ 8 ] where the concept of symmetry was placed back on stage in the study of time series. Indeed, in [ 8 ], it was shown that the concept of recurrences could be generalized by taking into account the symmetry properties of m -patterns. As entropy can be derived from the recurrence concept (the recurrence plot [ 9 ] is defined as with ), from [ 8 ], four new kinds of entropy ( , , , or , , , or , , , ) can be proposed.…”

“…The total number of similar centered 2-patterns is much larger than no-centered 2-patterns. The second approach is inspired by [ 8 ], where transformed patterns are compared to reference patterns. Thus, in the second approach, a transformed m -pattern (see below) is compared to a reference m -pattern .…”

“…Indeed, in [ 8 ], it was shown that the concept of recurrences could be generalized by taking into account the symmetry properties of m -patterns. As entropy can be derived from the recurrence concept (the recurrence plot [ 9 ] is defined as with ), from [ 8 ], four new kinds of entropy ( , , , or , , , or , , , ) can be proposed. Finally, as our ultimate goal is to increase the precision of FuzzyEn, it is more appropriate here to calculate the mean value of the four new fuzzy entropies.…”

Perspective Chapter: Predicting Vehicle-Track Interaction with Recurrence Plots

“…Pham and Yan [30] proposed the sample entropy to measure Recurrence Plots irregularities. Girault [31] measured the symmetry in Recurrence Plots to identify the system's dynamic behavior. Sipers et al [32] defined a procedure for retrieving a signal from a Recurrence Plot, but their method is valid only for embedded signals.…”