Distinguishing anticipation from causality: anticipatory bias in the estimation of information flow.

We report that transfer entropy estimates obtained from low-resolution and/or small data sets show net information flow away from a purely anticipatory element whereas transfer entropy calculated using exact distributions show the flow towards it. This means that for real-world data sets anticipatory elements can appear to be strongly driving the network dynamics even when there is no possibility of such an influence. Furthermore, we show that in the low-resolution limit there is no statistic that can distinguish anticipatory elements from causal ones.

Paucity of attractors in nonlinear systems driven with complex signals.

We study the probability of multistability in a quadratic map driven repeatedly by a random signal of length N, where N is taken as a measure of the signal complexity. We first establish analytically that the number of coexisting attractors is bounded above by N. We then numerically estimate the probability p of a randomly chosen signal resulting in a multistable response as a function of N. Interestingly, with increasing drive signal complexity the system exhibits a paucity of attractors. That is, almost any drive signal beyond a certain complexity level will result in a single attractor response (p=0). This mechanism may play a role in allowing sensitive multistable systems to respond consistently to external influences.

Determinism in synthesized chaotic waveforms.

The output of a linear filter driven by a randomly polarized square wave, when viewed backward in time, is shown to exhibit determinism at all times when embedded in a three-dimensional state space. Combined with previous results establishing exponential divergence equivalent to a positive Lyapunov exponent, this result rigorously shows that such reverse-time synthesized waveforms appear equally to have been produced by a deterministic chaotic system.

Synthesizing folded band chaos.

A randomly driven linear filter that synthesizes Lorenz-like, reverse-time chaos is shown also to produce Rössler-like folded band wave forms when driven using a different encoding of the random source. The relationship between the topological entropy of the random source, dissipation in the linear filter, and the positive Lyapunov exponent for the reverse-time wave form is exposed. The two drive encodings are viewed as grammar restrictions on a more general encoding that produces a chaotic superset encompassing both the Lorenz butterfly and Rössler folded band paradigms of nonlinear dynamics.

Deconstructing spatiotemporal chaos using local symbolic dynamics.

We find that the global symbolic dynamics of a diffusively coupled map lattice is well approximated by a very small local model for weak to moderate coupling strengths. A local symbolic model is a truncation of the full symbolic model to one that considers only a single element and a few neighbors. Using interval analysis, we give rigorous results for a range of coupling strengths and different local model widths. Examples are presented of extracting a local symbolic model from data and of controlling spatiotemporal chaos.

Chaos without nonlinear dynamics.

A linear, second-order filter driven by randomly polarized pulses is shown to generate a waveform that is chaotic under time reversal. That is, the filter output exhibits determinism and a positive Lyapunov exponent when viewed backward in time. The filter is demonstrated experimentally using a passive electronic circuit, and the resulting waveform exhibits a Lorenz-like butterfly structure. This phenomenon suggests that chaos may be connected to physical theories whose underlying framework is not that of a traditional deterministic nonlinear dynamical system.

Symbolic dynamics of coupled map lattices.

We present a method to reduce the [FORMULA: SEE TEXT] dynamics of coupled map lattices (CMLs) of N invertibly coupled unimodal maps to a sequence of N-bit symbols. We claim that the symbolic description is complete and provides for the identification of all fixed points, periodic orbits, and dense orbits as well as an efficient representation for studying pattern formation in CMLs. We give our results for CMLs in terms of symbolic dynamical concepts well known for one-dimensional chaotic maps, including generating partitions, Gray orderings, and kneading sequences. An example utilizing coupled quadratic maps is given.

Complete and flexible replacement of chaotic uncertainty with transmitted information.

Natural chaos can be described as an information source emitting symbolic sequences with positive entropy. We use two algorithmic techniques from data compression in a nonstandard way along with a control scheme to replace the natural uncertainty in chaotic systems with an arbitrary digital message. Unlike previous targeting-based control, the controlled, deterministic, transmission appears statistically identical to natural chaos, with a message modulated on it at the intrinsic Kolmogorov-Sinai information generation rate of the chaotic oscillator. Thus, chaotic communication by targeting need not consume any additional channel capacity beyond that required by the message itself and the message-bearing signal may appear identical to the uncontrolled oscillator. We also demonstrate control and data transmission at the channel capacity of the oscillator, the maximum possible data rate compatible with the grammar.

Lag and anticipating synchronization without time-delay coupling.

We describe a new method for achieving approximate lag and anticipating synchronization in unidirectionally coupled chaotic oscillators. The method uses a specific parameter mismatch between the drive and response that is a first-order approximation to true time-delay coupling. As a result, an adjustable lag or anticipation effect can be achieved without the need for a variable delay line, making the method simpler and more economical to implement in many physical systems. We present a stability analysis, demonstrate the method numerically, and report experimental observation of the effect in radio-frequency electronic oscillators. In the circuit experiments, both lag and anticipation are controlled by tuning a single capacitor in the response oscillator.

Information flow in chaos synchronization: fundamental tradeoffs in precision, delay, and anticipation.

We use symbolic dynamics to examine the flow of information in unidirectionally coupled chaotic oscillators exhibiting synchronization. The theory of symbolic dynamics reduces chaos to a shift map that acts on a discrete set of symbols, each of which contains information about the system state. Using this transformation we explore so-called achronal synchronization, in which the response lags or leads the drive by a fixed amount of time. We find fundamental tradeoffs between the precision to which the drive state is detected, the quality of synchronization attained, and the delay or anticipation exhibited by the response system. To illustrate these tradeoffs, we provide a physical example using electronic circuits.

Synchronizing the information content of a chaotic map and flow via symbolic dynamics.

In this paper we report an extension to the concept of generalized synchronization for coupling different types of chaotic systems, including maps and flows. This broader viewpoint takes disparate systems to be synchronized if their information content is equivalent. We use symbolic dynamics to quantize the information produced by each system and compare the symbol sequences to establish synchronization. A general architecture is presented for drive-response coupling that detects symbols produced by a chaotic drive oscillator and encodes them in a response system using the methods of chaos control. We include experimental results demonstrating synchronization of information content in an electronic oscillator circuit driven by a logistic map.

Control of long-period orbits and arbitrary trajectories in chaotic systems using dynamic limiting.

We demonstrate experimental control of long-period orbits and arbitrary chaotic trajectories using a new chaos control technique called dynamic limiting. Based on limiter control, dynamic limiting uses a predetermined sequence of limiter levels applied to the chaotic system to stabilize natural states of the system. The limiter sequence is clocked by the natural return time of the chaotic system such that the oscillator sees a new limiter level for each peak return. We demonstrate control of period-8 and period-34 unstable periodic orbits in a low-frequency circuit and provide evidence that the control perturbations are minimal. We also demonstrate control of an arbitrary waveform by replaying a sequence captured from the uncontrolled oscillator, achieving a form of delayed self-synchronization. Finally, we discuss the use of dynamic limiting for high-frequency chaos communications. (c) 2002 American Institute of Physics.