John F. Lindner

Techniques for the control of chaos

Abstract The concepts of chaos and its control are reviewed. Both are discussed from an experimental as well as a theoretical viewpoint. Examples are then given of the control of chaos in adiverse set of experimental systems. Current and future applications are discussed.

Removal, Suppression, and Control of Chaos by Nonlinear Design

Techniques to remove, suppress, and control the chaotic behavior of nonlinear systems are reviewed. Analysis of a forced damped nonlinear oscillator provides a brief overview of the relevant nonlinear dynamics of dissipative systems. Various techniques for suppression and control of chaos are then outlined, compared and contrasted. A unified mathematical notation facilitates the comparison. The successes of each strategy in numerical simulations and physical experiments are carefully noted. Their strengths and weaknesses are analyzed, and they are evaluated according to whether they employ feedback, are goal-oriented, are model-based, merely remove chaos–or truly exploit it. An elementary derivation of the important OGY control equation is supplied. Critical references provide an entry into the literature. It is argued that nonlinearity can be a real-world advantage, and it is hoped that this review will serve as summary of, and invitation to, the nascent field of nonlinear design.

Chaotic resonance : a simulation

Stochastic resonance is a statistical phenomenon that has been observed in periodically modulated, noise-driven, bistable systems. The characteristic signatures of the effect include an increase in the signal-to-noise of the output as noise is added to the system, and exponentially decreasing peaks in the probability density as a function of residence times in one state. Presented are the results of a numerical simulation where these same signatures were observed by adding achaotic driving term instead of a white noise term. Although the probability distributions of the noise and chaos inputs were significantly different, the stochastic and chaotic resonances were equal within the experimental error.

Nonlinearity and computation: implementing logic as a nonlinear dynamical system

Abstract Recently, Sinha and Ditto [Phys. Rev. Lett. 81 (1998) 2156] demonstrated the computational possibilities of an array of coupled maps. We generalize this nonlinear dynamical system to improve its computational usefulness. We then consider a second nonlinear system, a parameterized map, and use it to illustrate why logic requires nonlinearity.

Optimal disorders for taming spatiotemporal chaos

Abstract We study a coupled array of torqued damped nonlinear pendulums. Disordering this system can eliminate chaos. Here, we numerically investigate this phenomenon. For each of several types of disorder, we find an optimal magnitude of disorder which minimizes the systems largest Lyapunov exponent.

Self‐organized criticality: An experiment with sandpiles

In 1987, Bak, Tang, and Wiesenfeld introduced the notion of self‐organized criticality (SOC) in the guise of a computer simulation: a ‘‘sandpile cellular automaton machine.’’ They supposed that a real, many‐bodied, physical system in an external field assembles itself into a critical state. The system then relaxes about the critical state creating spatial and temporal self similarities which give rise to fractal objects and 1/f noise. Their computer modeling was of a system like a sandpile at its critical angle of repose. This provided a new paradigm for many‐body dynamics. Understanding SOC may well allow substantial strides to occur in the theory of flow and transport. The simplest model system, one for which computer simulations and corresponding real experiments are feasible, is a ‘‘sandpile’’ near its critical angle of repose. The size and duration of avalanches occurring as subsequent ‘‘sand’’ grains are added can provide detailed information about the ‘‘sandpile’’ as a model of SOC, and for SOC as ...

Pulse-enhanced stochastic resonance

Abstract By adding constant-amplitude pulses to a noisy bistable system, we enhance its response to monochromatic signals, significantly magnifying its unpulsed stochastic resonance. We observe the enhancement in both numerical simulations and in analog electronic experiments. This simple noninvasive control technique should be especially useful in noisy bistable systems that are difficult or impossible to modify internally.

Noise tolerant spatiotemporal chaos computing

We introduce and design a noise tolerant chaos computing system based on a coupled map lattice (CML) and the noise reduction capabilities inherent in coupled dynamical systems. The resulting spatiotemporal chaos computing system is more robust to noise than a single map chaos computing system. In this CML based approach to computing, under the coupled dynamics, the local noise from different nodes of the lattice diffuses across the lattice, and it attenuates each others effects, resulting in a system with less noise content and a more robust chaos computing architecture.

PULSATION PERIOD VARIATIONS IN THE RRc LYRAE STAR KIC 5520878

Learned et al. proposed that a sufficiently advanced extra-terrestrial civilization may tickle Cepheid and RR Lyrae variable stars with a neutrino beam at the right time, thus causing them to trigger early and jogging the otherwise very regular phase of their expansion and contraction. This would turn these stars into beacons to transmit information throughout the galaxy and beyond. The idea is to search for signs of phase modulation (in the regime of short pulse duration) and patterns, which could be indicative of intentional, omnidirectional signaling. We have performed such a search among variable stars using photometric data from the Kepler space telescope. In the RRc Lyrae star KIC 5520878, we have found two such regimes of long and short pulse durations. The sequence of period lengths, expressed as time series data, is strongly autocorrelated, with correlation coefficients of prime numbers being significantly higher (p = 99.8%). Our analysis of this candidate star shows that the prime number oddity originates from two simultaneous pulsation periods and is likely of natural origin. Simple physical models elucidate the frequency content and asymmetries of the KIC 5520878 light curve. Despite this SETI null result, we encourage testing of other archival and future time-series photometry for signs of modulated stars. This can be done as a by-product to the standard analysis, and can even be partly automated.

Can Neurons Distinguish Chaos from Noise

Recent studies suggesting evidence for determinism in the stochastic activity of the heart and brain have sparked an important scientific debate: Do biological systems exploit chaos or are they merely noisy? Here, we analyze the spike interval statistics of a simple integrate-and-fire model neuron to investigate how a real neuron might process noise and chaos, and possibly differentiate between the two. In some cases, our model neuron readily distinguishes noise from chaos, even discriminating among chaos characterized by different Lyapunov exponents. However, in other cases, the model neuron does not decisively differentiate noise from chaos. In these cases, the spectral content of the input signal may be more significant than its phase space structure, and higher-order spectral characterizations may be necessary to distinguish its response to chaotic or noisy inputs.