Folded torus in the forced Rayleigh oscillator with a diode pair

It is known that the periodically forced Rayleigh equation is the first differential equation for which an aperiodic solution was ever discovered. However, it has not yet been clarified whether or not observable chaos exists in this equation. Chaotic oscillations observed in the forced Rayleigh oscillator are investigated in detail by using the piecewise-linear and degeneration technique. The model is a negative resistance LC oscillator with a pair of diodes driven by a sinusoidal source. The piecewise-linear constrained equation is derived from this circuit by idealizing the diode pair as a switch. The Poincare map of this equation is derived strictly as a one-dimensional return mapping on a circle (so-called circle map). This mapping becomes noninvertible when the amplitude of the forcing term is tuned larger. The folded torus observed in this oscillator is well explained by this mapping.<
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