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1. Dynamics of circular oscillator arrays subjected to noise

   https://doi.org/10.1007/s11071-021-07165-w  

   Balachandran, B.; Breunung, T.; Acar, G.; Alofi, A.; and Yorke, J. (January 2022, Nonlinear dynamics)

   Energy localization, which are spatially confined response patterns, have been observed in turbomachinery applications, micro-electromechanical systems, and atomic crystals. While confined energy can reduce a device’s life-span, in sensing and energy harvesting applications, it can be beneficial to steer a system’s response into a localized mode. Building on earlier studies, in this article, the authors extend the research on localization by considering an array of coupled Duffing oscillators arranged in a circle. The system is composed of multiple nonlinear oscillators each connected to two neighboring oscillators via springs. Due to the periodic boundary conditions waves can propagate through the boundaries. These oscillators are hardening in most of the considered cases, and softening in the others. In the studied parameter range, the system is characterized by multi-stable behavior and a localized mode as well as a unison-low-amplitude motion coexist. The possibility that white noise can drive the system response from the localized mode to the low amplitude mode and thus suppresses energy localization is investigated. For different noise levels, the duration needed to stop energy localization as well as the probability to suppress localization within a certain time is numerically studied. In addition, the effects of linear coupling and nonlinear coupling between the oscillators on the strength of localization and the minimum noise addition needed to suppress energy localization are examined in depth. Moreover, modeling of large array dynamics with smaller subsystems is explored and dynamics with non-Gaussian noise is also considered. 

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2. Noise-induced chaos and signal detection by the nonisochronous Hopf oscillator

   https://doi.org/10.1063/1.5091938  

   Faber, Justin; Bozovic, Dolores (April 2019, Chaos: An Interdisciplinary Journal of Nonlinear Science)

   The Hopf oscillator has been shown to capture many phenomena of the auditory and vestibular systems. These systems exhibit remarkable temporal resolution and sensitivity to weak signals, as they are able to detect sounds that induce motion in the angstrom regime. In the present work, we find the analytic response function of a nonisochronous Hopf oscillator to a step stimulus and show that the system is most sensitive in the regime where noise induces chaotic dynamics. We show that this regime also provides a faster response and enhanced temporal resolution. Thus, the system can detect a very brief, low-amplitude pulse. Finally, we subject the oscillator to periodic delta-function forcing, mimicking a spike train, and find the exact analytic expressions for the stroboscopic maps. Using these maps, we find a period-doubling cascade to chaos with increasing force strength. 

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3. Using noise to augment synchronization among oscillators

   https://doi.org/10.1038/s41598-021-83806-9  

   Vaidya, Jaykumar; Bashar, Mohammad Khairul; Shukla, Nikhil (December 2021, Scientific Reports)

   null (Ed.)

   Abstract Noise is expected to play an important role in the dynamics of analog systems such as coupled oscillators which have recently been explored as a hardware platform for application in computing. In this work, we experimentally investigate the effect of noise on the synchronization of relaxation oscillators and their computational properties. Specifically, in contrast to its typically expected adverse effect, we first demonstrate that a common white noise input induces frequency locking among uncoupled oscillators. Experiments show that the minimum noise voltage required to induce frequency locking increases linearly with the amplitude of the oscillator output whereas it decreases with increasing number of oscillators. Further, our work reveals that in a coupled system of oscillators—relevant to solving computational problems such as graph coloring, the injection of white noise helps reduce the minimum required capacitive coupling strength. With the injection of noise, the coupled system demonstrates frequency locking along with the desired phase-based computational properties at 5 × lower coupling strength than that required when no external noise is introduced. Consequently, this can reduce the footprint of the coupling element and the corresponding area-intensive coupling architecture. Our work shows that noise can be utilized as an effective knob to optimize the implementation of coupled oscillator-based computing platforms. 

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4. Noise-induced switching from a symmetry-protected shallow metastable state

   https://doi.org/10.1038/s41598-020-66243-y  

   Tadokoro, Yukihiro; Tanaka, Hiroya; Dykman, M. I. (December 2020, Scientific Reports)

   null (Ed.)

   Abstract We consider escape from a metastable state of a nonlinear oscillator driven close to triple its eigenfrequency. The oscillator can have three stable states of period-3 vibrations and a zero-amplitude state. Because of the symmetry of period-tripling, the zero-amplitude state remains stable as the driving increases. However, it becomes shallow in the sense that the rate of escape from this state exponentially increases, while the system still lacks detailed balance. We find the escape rate and show how it scales with the parameters of the oscillator and the driving. The results facilitate using nanomechanical, Josephson-junction based, and other mesoscopic vibrational systems for studying, in a well-controlled setting, the rates of rare events in systems lacking detailed balance. They also describe how fluctuations spontaneously break the time-translation symmetry of a driven oscillator. 

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5. Noise-enhanced learning in Duffing adaptive oscillator

   https://doi.org/10.1098/rspa.2025.0002  

   Perkins, Edmon (June 2025, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   Noise typically degrades the performance of physical systems. In some cases, however, noise can provide an enhanced effect. For instance, stochastic resonance may be observed in a nonlinear oscillator when a weak signal is boosted by noise. On the other hand, adaptive oscillators are a type of nonlinear oscillator that can learn and store information in dynamic states. Here, noise is shown to increase the learning rate of an adaptive oscillator, using a Duffing adaptive oscillator. To highlight this effect, stochastic simulations are employed, and the Fokker–Planck equation is semi-analytically solved using the cumulant neglect method. As adaptive oscillators can also be used as a powerful physical reservoir computer architecture, this work shows that the Duffing adaptive oscillator can be optimized for fast learning in noisy environments. 

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