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.
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.
A novel electromechanical chaotic oscillator is described that admits an exact analytic solution. The oscillator is a hybrid dynamical system with governing equations that include a linear second order ordinary differential equation with negative damping and a discrete switching condition that controls the oscillatory fixed point. The system produces provably chaotic oscillations with a topological structure similar to either the Lorenz butterfly or Rössler's folded-band oscillator depending on the configuration. Exact solutions are written as a linear convolution of a fixed basis pulse and a sequence of discrete symbols. We find close agreement between the exact analytical solutions and the physical oscillations. Waveform return maps for both configurations show equivalence to either a shift map or tent map, proving the chaotic nature of the oscillations.
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