Independent symbol and oscillation time scales in solvable chaos

We present and demonstrate in electronics a chaotic oscillator with an analytic solution that features independent and easily tunable oscillation, symbol, and information time scales. The oscillator is a generalization of a known chaotic oscillator based on an unstable second-order filter that admits an analytic solution. Like most chaotic oscillators, the known oscillator exhibits independently tunable information and oscillation times scales, but directly coupled symbol and oscillation time scales. The generalization presented here separates the symbol time scale from the oscillation time scale in a manner akin to a carrier in a traditional communications system. Importantly, the generalized oscillator also admits an analytic solution. We realize the oscillator at four different pairs of symbol and oscillation time scales in a mixed-signal, electronic circuit that operates at an oscillation frequency of ≈4.2 kHz. We use the theoretical analytic solution to validate the electronic implementation and find good agreement between theoretical system and measured results.