Chaos in a Pendulum Adaptive Frequency Oscillator Circuit Experiment

Abstract

Adaptive oscillators can learn and encode information in dynamic, plastic states. The pendulum has recently been proposed as the base oscillator of an adaptive system. In a mechanical setup, the horizontally forced pendulum adaptive frequency oscillator seeks a resonance condition by modifying the length of the pendulum's rod. This system stores the external forcing frequency when the external amplitude is small, while it can store the resonance frequency, which is affected by the nonlinearity of the pendulum, when the external amplitude is large. Furthermore, for some frequency ranges, the pendulum adaptive frequency oscillator can exhibit chaotic motion when the amplitudes are large. This adaptive oscillator could be used as a smart vibratory energy harvester device, but this chaotic region could degrade its performance by using supplementary energy to modify the rod length. The pendulum adaptive frequency oscillator’s equations of motions are discussed, and a field-programmable analog array is used as an experimental realization of this system as an electronic circuit. Bifurcation diagrams are shown for both the numerical simulations and experiments, while period-3 motion is shown for the numerical simulations. As little work has been done on the stability of adaptive oscillators, the authors believe that this work is the first demonstration of chaos in an adaptive oscillator.

The van der Pol physical reservoir computer

“…FPAAs have been popularly used to implement many nonlinear oscillators. These include a pendulum adaptive oscillator [56], a four-state adaptive oscillator [57], a fractional order chaotic system [58], the Lorenz oscillator [59], a chaotic system with infinite attractors [60], the Sprott N chaotic system [61], the Nahrain chaotic system [62], and a hyperjerk oscillator [63]. This FPAA implementation of the van der Pol oscillator was chosen to provide a rapid prototype, which could be easily reproduced by other researchers.…”

Comparative analysis and FPGA realization of different control synchronization approaches for chaos-based secured communication systems

“…In this paper we decided to use digital implementation using FPGA, even though we appreciate the value in using analog implementation using Field-Programmable Analog Array FPAA [ 47 ]. Digital tools allow us to repeat experiments more accurately, which is important when we want to see how our control method interacts with nonlinear systems.…”