Lyapunov exponents from experimental time series: Application to cymbal vibrations

Lyapunov exponents are among the most relevant and most informative invariants for detecting and quantifying chaos in a dynamical system. This method is applied here to the analysis of cymbal vibrations. The advantage of using a quadratic fit for determining the Jacobian of the dynamics is presented. In addition, the interest of using a time step for the evolution of the neighbourhood not equal to the timelag used for the reconstruction of the phase space is underlined. The robustness of the algorithm used yields a high degree of confidence in the characterization and in the quantification of the chaotic state. To illustrate these features in the case of cymbal vibrations, transitions from quasiperiodicity to chaos are exhibited. The quasiperiodic state of the system is characterized together by the power spectrum of the experimental signal and by calculation of the Lyapunov spectrum.