Peak-to-peak dynamics: A critical survey

This paper is devoted to the study of a particular form of deterministic chaos, here called peak-to-peak dynamics (PPD). When a continuous-time system of order n has PPD, the amplitude and the time of occurrence of the next peak of its output variable can be predicted from information concerning at most two previous peaks. In other words, n differential equations can be substituted by a reduced order model, if attention is restricted to the peaks of the variable of concern. The observation of the output peaks is equivalent to the observation of the system on a Poincare section. This is why the existence of PPD is simply related to the dimension of the attractor. The usefulness of peak-to-peak analysis for the retrieval of one-dimensional dynamics within the attractor and for the estimate of the first Liapunov exponent is demonstrated through examples. Particular attention is devoted to the possibility of exploiting the PPD reduced order models for forecasting the next peak and for the regularization of the dynamics of chaotic systems by means of piecewise constant controls.