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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. 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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3. 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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4. Synchronization-based model for turbulent thermoacoustic systems

   https://doi.org/10.1007/s11071-023-08368-z  

   Weng, Yue ; Unni, Vishnu R. ; Sujith, R. I. ; Saha, Abhishek ( March 2023 , Nonlinear Dynamics)

   We present a phenomenological reduced-order model to capture the transition to thermoacoustic instability in turbulent combustors. Based on the synchronization framework, the model considers the acoustic field and the unsteady heat release rate from turbulent reactive flow as two nonlinearly coupled sub-systems. To model combustion noise, we use a pair of nonlinearly coupled second-order ODEs to represent the unsteady heat release rate. This simple configuration, while nonlinearly coupled to another oscillator that represents the independent sub-system of acoustics (pressure oscillations) in the combustor, is able to produce chaos. Previous experimental studies have reported a route from low amplitude chaotic oscillation (i.e., combustion noise) to periodic oscillation through intermittency in turbulent combustors. By varying the coupling strength, the model can replicate the route of transition observed and reflect the coupled dynamics arising from the interplay of unsteady heat release rate and pressure oscillations.

    

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5. Stability of oscillator Ising machines: Not all solutions are created equal

   https://doi.org/10.1063/5.0157107  

   Bashar, Mohammad Khairul ; Lin, Zongli ; Shukla, Nikhil ( October 2023 , Journal of Applied Physics)

   Nonlinear dynamical systems such as coupled oscillators are being actively investigated as Ising machines for solving computationally hard problems in combinatorial optimization. Prior works have established the equivalence between the global minima of the cost function describing the coupled oscillator system and the ground state of the Ising Hamiltonian. However, the properties of the oscillator Ising machine (OIM) from a nonlinear control viewpoint, such as the stability of the OIM solutions, remain unexplored. Therefore, in this work, using nonlinear control-theoretic analysis, we (i) identify the conditions required to ensure the functionality of the coupled oscillators as an Ising machine, (ii) show that all globally optimal phase configurations may not always be stable, resulting in some configurations being more favored over others and, thus, creating a biased OIM, and (iii) elucidate the impact of the stability of locally optimal phase configurations on the quality of the solution computed by the system. Our work, fostered through the unique convergence between nonlinear control theory and analog systems for computing, provides a new toolbox for the design and implementation of dynamical system-based computing platforms.

    

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