Stochastic Effects On A Hopf Adaptive Frequency Oscillator

This paper explores the stochastic dynamics of a Hopf adaptive frequency oscillator when driven by noise. Adaptive oscillators are nonlinear oscillators that store information via plastic states. As noise is ubiquitous in physical systems, it is important to gain an understanding of the stochastic effects on adaptive oscillators. Previously, it has been shown that a simplified analysis of the Fokker-Planck equation results in affecting the plastic frequency state of these oscillators. However, when the full Fokker-Planck equation is considered, new behaviors are observed due to changes in oscillation amplitudes in addition to frequencies. The plastic frequency state of these oscillators may benefit from enhanced learning due to small amplitudes of noise, converge to incorrect values for medium amplitudes of noise, and even collapse to zero in the limit of large amplitudes of noise. Interestingly, not all averaged states collapse equally, which leads a two dimensional limit cycle to collapse into single dimensional oscillations when considering the averaged dynamics. These behaviors are compared analytically through the Fokker-Planck equation, numerically using the Euler-Maruyama simulations, and finally validated experimentally using an analog, electronic circuit. These results show that noise can enhance, mislead, or even reduce the dimensionality of the averaged adaptive Hopf oscillator.