Jonathan N. Blakely

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.

A matched filter for chaos.

A novel chaotic oscillator is shown to admit an exact analytic solution and a simple matched filter. The oscillator is a hybrid dynamical system including both a differential equation and a discrete switching condition. The analytic solution is written as a linear convolution of a symbol sequence and a fixed basis function, similar to that of conventional communication waveforms. Waveform returns at switching times are shown to be conjugate to a chaotic shift map, effectively proving the existence of chaos in the system. A matched filter in the form of a delay differential equation is derived for the basis function. Applying the matched filter to a received waveform, the bit error rate for detecting symbols is derived, and explicit closed-form expressions are presented for special cases. The oscillator and matched filter are realized in a low-frequency electronic circuit. Remarkable agreement between the analytic solution and the measured chaotic waveform is observed.

Experimental observation of delay-induced radio frequency chaos in a transmission line oscillator

We report an experimental study of fast chaotic dynamics in a delay dynamical system. The system is an electronic device consisting of a length of coaxial cable terminated on one end with a diode and on the other with a negative resistor. When the negative resistance is large, the system evolves to a steady state. As the negative resistance is decreased, a Hopf bifurcation occurs. By varying the length of the transmission line we observe Hopf frequencies from 7-53 MHz. With the transmission line length fixed, we observe a period doubling route to chaos as the negative resistance is further reduced providing the first experimental confirmation of an existing theoretical model for nonlinear dynamics in transmission line oscillators [Corti et al., IEEE Trans. Circ. Syst., I: Fundam. Theory Appl. 41, 730 (1994)]. However, other experimental results indicate limitations to this model including an inability to predict the Hopf frequency or to produce realistic continuous wave forms. We extend the model to include finite bandwidth effects present in a real negative resistor. The resulting model is a neutral delay differential equation that provides better agreement with experimental results.

High-speed chaos in an optical feedback system with flexible timescales

We describe a new optoelectronic device with time-delayed feedback that uses a Mach-Zehnder interferometer as passive nonlinearity and a semiconductor laser as a current-to-optical-frequency converter. Band-limited feedback allows tuning of the characteristic time scales of both the periodic and high dimensional chaotic oscillations that can be generated with the device. Our implementation of the device produces oscillations in the frequency range of tens to hundreds of megahertz. We develop a model and use it to explore the experimentally observed Andronov-Hopf bifurcation of the steady state and to estimate the dimension of the chaotic attractor.

A simple Lorenz circuit and its radio frequency implementation.

A remarkably simple electronic circuit design based on the chaotic Lorenz system is described. The circuit consists of just two active nonlinear elements (high-speed analog multipliers) and a few passive linear elements. Experimental implementations of the circuit exhibit the classic butterfly attractor and the hysteretic transition from steady state to chaos observed in the Lorenz equations. The simplicity of the circuit makes it suitable for radio frequency applications. The power spectrum of the observed oscillations displays a peak frequency as high as 930 kHz and significant power beyond 1 MHz.

Exact folded-band chaotic oscillator

An exactly solvable chaotic oscillator with folded-band dynamics is shown. The oscillator is a hybrid dynamical system containing a linear ordinary differential equation and a nonlinear switching condition. Bounded oscillations are provably chaotic, and successive waveform maxima yield a one-dimensional piecewise-linear return map with segments of both positive and negative slopes. Continuous-time dynamics exhibit a folded-band topology similar to Rösslers oscillator. An exact solution is written as a linear convolution of a fixed basis pulse and a discrete binary sequence, from which an equivalent symbolic dynamics is obtained. The folded-band topology is shown to be dependent on the symbol grammar.

Time shifts and correlations in synchronized chaos.

We introduce a new method for predicting characteristics of the synchronized state achieved by a wide class of unidirectional coupling schemes. Specifically, we derive a transfer function from the coupling model that provides estimates of the correlation between the drive and response waveforms, and of the time shift (i.e., lag or anticipation) of the synchronized state. To demonstrate the method, we apply it to a simulated system of coupled Rossler oscillators as well as to an experimental system of coupled chaotic electronic circuits. Finally, we show that the transfer function can be exploited to design novel coupling schemes that significantly improve the correlation and increase the maximum achievable time shift.

Concept for low-cost chaos radar using coherent reception

We describe a new approach to random-signal radar based on the recent discovery of analytically solvable chaotic oscillators. These surprising nonlinear systems generate random, aperiodic waveforms that offer an exact analytic representation, allowing the implementation of simple matched filters and coherent reception. Notably, this approach enables nearly optimal detection of noise-like waveforms without need for expensive variable delay lines to store wideband waveforms for correlation. Mathematically, the waveform is expressed as a linear convolution of a bit sequence with a fixed basis function. We realize a simple matched filter for the waveform using a linear filter whose impulse response function is the time reverse of the basis function. Importantly, linear filters matched to finite bit sequences can be defined, enabling pulse compression and spread spectrum radar. We present an example oscillator, its matched filter, and simulation results demonstrating the pulse compression radar concept.

Acoustic Detection and Ranging Using Solvable Chaos

Acoustic experiments demonstrate a novel approach to ranging and detection that exploits the properties of a solvable chaotic oscillator. This nonlinear oscillator includes an ordinary differential equation and a discrete switching condition. The chaotic waveform generated by this hybrid system is used as the transmitted waveform. The oscillator admits an exact analytic solution that can be written as the linear convolution of binary symbols and a single basis function. This linear representation enables coherent reception using a simple analog matched filter and without need for digital sampling or signal processing. An audio frequency implementation of the transmitter and receiver is described. Successful acoustic ranging measurements in the presence of noise and interference from a second chaotic emitter are presented to demonstrate the viability of the approach.

Experimental investigation of high-quality synchronization of coupled oscillators

We describe two experiments in which we investigate the synchronization of coupled periodic oscillators. Each experimental system consists of two identical coupled electronic periodic oscillators that display bursts of desynchronization events similar to those observed previously in coupled chaotic systems. We measure the degree of synchronization as a function of coupling strength. In the first experiment, high-quality synchronization is achieved for all coupling strengths above a critical value. In the second experiment, no high-quality synchronization is observed. We compare our results to the predictions of the several proposed criteria for synchronization. We find that none of the criteria accurately predict the range of coupling strengths over which high-quality synchronization is observed. (c) 2000 American Institute of Physics.