PHYSICAL REVIEW E 105, L032201 (2022) Letter Quenching of oscillations in a liquid metal via attenuated coupling Ishant Tiwari , * Richa Phogat , * Animesh Biswas , * and P. Parmananda Department of Physics, Indian Institute of Technology, Bombay, Powai, Mumbai 400076, India Sudeshna Sinha Department of Physical Sciences, Indian Institute of Science Education and Research Mohali, Knowledge City, SAS Nagar, Sector 81, Manauli, P.O. Box 140306, Punjab, India (Received 23 January 2022; accepted 23 February 2022; published 14 March 2022) In this work, we report a quenching of oscillations observed upon coupling two chemomechanical oscillators. Each one of these oscillators consists of a drop of liquid metal submerged in an oxidizing solution. These pseudoidentical oscillators have been shown to exhibit both periodic and aperiodic oscillatory behavior. In the experiments performed on these oscillators, we find that coupling two such oscillators via an attenuated resistive coupling leads the coupled system towards an oscillation quenched state. To further comprehend these experimental observations, we numerically explore and verify the presence of similar oscillation quenching in a model of coupled Hindmarsh-Rose (HR) systems. A linear stability analysis of this HR system reveals that attenuated coupling induces a change in eigenvalues of the relevant Jacobian, leading to stable quenched oscillation states. Additionally, the analysis yields a threshold of attenuation for oscillation quenching that is consistent with the value observed in numerics. So this phenomenon, demonstrated through experiments, as well as simulations and analysis of a model system, suggests a powerful natural mechanism that can potentially suppress periodic and aperiodic oscillations in coupled nonlinear systems. DOI: 10.1103/PhysRevE.105.L032201 I. INTRODUCTION Scientific literature is replete with examples of oscillating entities in both temporal and spatially [1,2] extended natu- ral and laboratory systems [3–8]. The mercury beating heart (MBH) is a system, wherein both the mechanical movements of the mercury drop and the redox system potential are oscil- latory in nature [9,10]. Given appropriate system parameters, this drop of mercury kept in the presence of an oxidizing agent may exhibit both periodic [9] and aperiodic oscillations [11]. The excitatory nature of these chemomechanical oscillations makes this system an ideal tabletop system to demonstrate and verify a plethora of intriguing behaviors observed or predicted in such systems. A few examples are the entrain- ment [6,12–14], synchronization [15,16], Kuramoto transition [17,18], quorum sensing [19–21], and cessation of oscillations [22–26]. Quenching of oscillation may prove to be both detrimental and advantageous to a system depending on a wide variety of circumstantial factors [27,28]. A sustained rhythmic activity would be a prerequisite for the proper functioning of cardiac cells [29], whereas it would be detrimental to a stable laser output [30]. In the current work, we explore the quenching as well as the revival of oscillations observed in a system of cou- pled periodic and aperiodic MBH oscillators. Two oscillating drops of mercury are coupled bidirectionally in such a way that each oscillator receives an attenuated copy of the other * These authors contributed equally to this work. oscillator’s redox time series. This was done to emulate the signal attenuation over long distances. A robust quenching of oscillations is observed when the attenuation of the signals is above a critical threshold. The experimental observations are numerically corroborated in a system of coupled Hindmarsh- Rose (HR) oscillators [31,32]. The HR oscillators are kept in both periodic and chaotic regimes and coupled to each other in varying degrees of attenuation, to mimic the experimental results involving periodic and aperiodic MBH systems. In ad- dition, linear stability analysis of the HR neurons is performed as a function of the attenuation factor α. This linear stability analysis reveals a stabilization of the system’s fixed points when α crosses a critical threshold. This work is organized in the following manner. In Sec. II, the experimental setup and their corresponding results are pre- sented. These results are followed by numerical simulations corroborating the experiments, using a model system of cou- pled Hindmarsh-Rose (HR) oscillators in Sec. III. Finally, the results of our work are summarized and discussed in Sec. IV. II. EXPERIMENTS A. Experimental setup A schematic diagram showing the experimental setup is presented in Fig. 1. It consists of two periodic (aperiodic) oscillators coupled bidirectionally with the help of an op- erational amplifier. The redox voltage from one MBH (O1) is initially scaled by an attenuation factor α by the first operational amplifier working as an inverting amplifier. The 2470-0045/2022/105(3)/L032201(5) L032201-1 ©2022 American Physical Society