Dynamics of a parametrically excited simple pendulum.

The dynamics of a parametric simple pendulum submitted to an arbitrary angle of excitation ϕ was investigated experimentally by simulations and analytically. Analytical calculations for the loci of saddle-node bifurcations corresponding to the creation of resonant orbits were performed by applying Melnikov's method. However, this powerful perturbative method cannot be used to predict the existence of odd resonances for a vertical excitation within first order corrections. Yet, we showed that period-3 resonances indeed exist in such a configuration. Two degenerate attractors of different phases, associated with the same loci of saddle-node bifurcations in parameter space, are reported. For tilted excitation, the degeneracy is broken due to an extra torque, which was confirmed by the calculation of two distinct loci of saddle-node bifurcations for each attractor. This behavior persists up to ϕ≈7π/180, and for inclinations larger than this, only one attractor is observed. Bifurcation diagrams were constructed experimentally for ϕ=π/8 to demonstrate the existence of self-excited resonances (periods smaller than three) and hidden oscillations (for periods greater than three).

Entropy Analysis of High-Definition Transcranial Electric Stimulation Effects on EEG Dynamics.

A foundation of medical research is time series analysis-the behavior of variables of interest with respect to time. Time series data are often analyzed using the mean, with statistical tests applied to mean differences, and has the assumption that data are stationary. Although widely practiced, this method has limitations. Here we present an alternative statistical approach with sample analysis that provides a summary statistic accounting for the non-stationary nature of time series data. This work discusses the use of entropy as a measurement of the complexity of time series, in the context of Neuroscience, due to the non-stationary characteristic of the data. To elucidate our argument, we conducted entropy analysis on a sample of electroencephalographic (EEG) data from an interventional study using non-invasive electrical brain stimulation. We demonstrated that entropy analysis could identify intervention-related change in EEG data, supporting that entropy can be a useful "summary" statistic in non-linear dynamical systems.