About: Chaotic is a(n) research topic. Over the lifetime, 28560 publication(s) have been published within this topic receiving 483285 citation(s).
TL;DR: This chapter describes the linking of two chaotic systems with a common signal or signals and highlights that when the signs of the Lyapunov exponents for the subsystems are all negative the systems are synchronized.
Abstract: Certain subsystems of nonlinear, chaotic systems can be made to synchronize by linking them with common signals. The criterion for this is the sign of the sub-Lyapunov exponents. We apply these ideas to a real set of synchronizing chaotic circuits.
Abstract: The mutual information I is examined for a model dynamical system and for chaotic data from an experiment on the Belousov-Zhabotinskii reaction. An N logN algorithm for calculating I is presented. As proposed by Shaw, a minimum in I is found to be a good criterion for the choice of time delay in phase-portrait reconstruction from time-series data. This criterion is shown to be far superior to choosing a zero of the autocorrelation function.
TL;DR: A method for learning nonlinear systems, echo state networks (ESNs), which employ artificial recurrent neural networks in a way that has recently been proposed independently as a learning mechanism in biological brains is presented.
Abstract: We present a method for learning nonlinear systems, echo state networks (ESNs). ESNs employ artificial recurrent neural networks in a way that has recently been proposed independently as a learning mechanism in biological brains. The learning method is computationally efficient and easy to use. On a benchmark task of predicting a chaotic time series, accuracy is improved by a factor of 2400 over previous techniques. The potential for engineering applications is illustrated by equalizing a communication channel, where the signal error rate is improved by two orders of magnitude.
TL;DR: The new effect of phase synchronization of weakly coupled self-sustained chaotic oscillators is presented, and a relation between the phase synchronization and the properties of the Lyapunov spectrum is studied.
Abstract: We present the new effect of phase synchronization of weakly coupled self-sustained chaotic oscillators. To characterize this phenomenon, we use the analytic signal approach based on the Hilbert transform and partial Poincar\'e maps. For coupled R\"ossler attractors, in the synchronous regime the phases are locked, while the amplitudes vary chaotically and are practically uncorrelated. Coupling a chaotic oscillator with a hyperchaotic one, we observe another new type of synchronization, where the frequencies are entrained, while the phase difference is unbounded. A relation between the phase synchronization and the properties of the Lyapunov spectrum is studied.
Abstract: This Letter reports the finding of a new chaotic at tractor in a simple three-dimensional autonomous system, which resembles some familiar features from both the Lorenz and Rossler at tractors.