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.
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.
Abstract-We describe a new optoelectronic device with timedelayed feedback that uses a Mach-Zehnder interferometer as passive nonlinearity and a semiconductor laser as a current-to-opticalfrequency 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.
The properties of nonlinear dynamics and chaos are shown to be fundamental to optimal communication signals subject to two practical and realistic design requirements: (i) operation in a noisy environment and (ii) simple hardware implementation. Starting with a simple electronic circuit, a linear filter receiver is presumed, and the matched optimal communication waveform that maximizes the receiver signal-to-noise performance is derived. A return map using samples from this optimal waveform is conjugate to a shift, thereby implying the waveform is chaotic. The optimal communication waveform for a second simple receiver is similarly derived, and it is found to be an exact solution to a physically realizable chaotic oscillator. Thus, a practical consequence of chaos in these waveforms is the potential for simple and efficient signal generation using chaotic oscillators. A conjecture is made that the optimal communication waveform for any stable infinite impulse response filter is similarly chaotic.
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