A memory-based approach to model glorious uncertainties of love

We propose a minimal yet intriguing model for a relationship between two individuals. The feeling of an individual is modeled by a complex variable and, hence, has two degrees of freedom [Jafari et al., Nonlinear Dyn. 83, 615-622 (2016)]. The effect of memory of the other individual's behavior in the past has now been incorporated via a conjugate coupling between each other's feelings. A region of parameter space exhibits multi-stable solutions wherein trajectories with different initial conditions end up in different aperiodic trajectories. This aligns with the natural observation that most relationships are aperiodic and unique not only to themselves but, more importantly, to the initial conditions too. Thus, the inclusion of memory makes the task of predicting the trajectory of a relationship hopelessly impossible.

Pathway selection by an active droplet

We report the movement of an active 1-pentanol drop within a closed Y-shaped channel subjected to geometrical and chemical asymmetry. A Y-shaped channel was configured with an angle of 120° between any two arms, which serves as the closed area of movement for the active drop. The arm where the 1-pentanol drop is introduced in the beginning is considered the source arm, and the center of the Y-shaped structure is the decision region. The drop always selects a specific route to move away from the decision region. The total probability of pathway selection excludes the possibility of the drop choosing the source channel. Remarkably, the active drop exhibits a strong sense of navigation for both geometrically and chemically asymmetric environments with accuracy rates of 80% and 100%, respectively. The pathway selection in a chemically asymmetric channel is a demonstration of the artificial negative chemotaxis, where the extra confined drop acts as a chemo-repellent. To develop a better understanding of our observations, a numerical model is constructed, wherein the particle is subjected to a net force which is a combined form of - (i) Yukawa-like repulsive interaction force (acting between the drop and the walls), (ii) a self-propulsion force, (iii) a drag, and (iv) a stochastic force. The numerics can capture all the experimental findings both qualitatively and quantitatively. Finally, a statistical analysis validates conclusions derived from both experiments and numerics.

In-phase and mixed-phase measure synchronization of camphor rotors

The numerical, analytical, and experimental analyses are presented for synchronizing two rotors under the Yukawa interaction. We report that the rotors exhibit in-phase and mixed-phase measure synchronizations for a pair of coupled rotors. Here, the analytical condition for synchronization is derived, tested numerically, and confirmed experimentally using coupled camphor infused rotors as a test bed. Moreover, the concept of measure synchronization is discussed. We report that, in conservative systems, not only the critical coupling parameter but initial conditions also play an essential role for estimating the measure synchronization region.

How to capture active Marangoni surfers

Surfers at the air-water interface form a large subset of the domain of active matter systems. They range from the water strider in the biological world to soluto-capillary effect driven artificial boats. In this work, we propose a general protocol to capture soluto-capillary effect driven interfacial surfers. By locally modifying the air-water interface using the perturbation from a micro-air-pump, these boats are reliably captured in the region of influence (ROI) of the perturbation. The surfers begin to explore the available space freely again once the perturbation is switched off. This method is successfully generalized to a couple of distinct surface-active chemicals used as fuel for the boats. Control experiments involving passive particles validate the results as being significantly better than purely mechanical "herding" of the particles. A possible mechanism behind the observed "trapping" is proposed.

Quenching of oscillations via attenuated coupling for dissimilar electrochemical systems

The coupled dynamics of two similar and disparate electrochemical cells oscillators are analyzed. For the similar case, the cells are intentionally operated at different system parameters such that they exhibit distinct oscillatory dynamics ranging from periodic to chaotic. It is observed that when such systems are subjected to an attenuated coupling, implemented bidirectionally, they undergo a mutual quenching of oscillations. The same holds true for the configuration wherein two entirely different electrochemical cells are coupled via bidirectional attenuated coupling. Therefore, the attenuated coupling protocol seems to be universally efficient in achieving oscillation suppression in coupled oscillators (similar or heterogeneous oscillators). The experimental observations were verified by numerical simulations using appropriate electrodissolution model systems. Our results indicate that quenching of oscillations via attenuated coupling is robust and therefore could be ubiquitous in coupled systems with a large spatial separation prone to transmission losses.

Phase-flip transition in volume-mismatched pairs of coupled 1-pentanol drops

We have explored a variety of synchronization domains and observed phase-flip transition in a pair of coupled 1-pentanol drops as a function of the volume mismatch. Both experimental observations and numerical studies are presented. The experiments were carried out in a rectangular channel in a ferroin deionized water solution premixed with some volume of pentanol. A single pentanol drop (≥ 3μL) performs back and forth oscillations along the length of the channel due to the well-known Marangoni forces. In the present work, for a pair of drops, the drop 1 volume was changed from 3 to 5 μL in steps of 1μL, whereas the drop 2 volume was varied from 1 to 3 μL in steps of 0.5μL. A systematic investigation of all the possible combinations of the drop volumes showed the presence of three different types of synchrony-in-phase, antiphase, and phase-switched. In-phase synchronization was robust for a volume mismatch of >3.0μL between the two drops. On the other hand, antiphase synchronization was robust when the volume mismatch was <2.0μL. The phase-switched state is a synchronized state involving a phase-flip transition in the time domain. This state was observed for the intermediate range of volume mismatch. Numerically, the system has been investigated using two Stuart-Landau oscillators interacting via a coupling function in the form of Lennard-Jones potential. The numerical results suitably capture both in-phase and antiphase oscillations for a pair of volume-mismatched pentanol drops.

Independent-noise provoked spiking, synchronized via coupling

We report experimental and numerical evidence of synchronized spiking phenomena provoked by the interaction of two bidirectionally coupled electrochemical systems subjected to independent stochastic input signals. To this end, the anodic potentials of two such systems were diffusively coupled. The corresponding anodic currents of these systems exhibited excitable fixed point behavior in the vicinity of a homoclinic bifurcation. Following this, the anodic potentials were perturbed by independent noise signals. The invoked oscillatory dynamics are analyzed using normalized variance and cross-correlation coefficient. By systematically varying the coupling strength between the systems and the level of external noise, regions exhibiting synchronized spiking behavior were identified.

Generation of aperiodic motion due to sporadic collisions of camphor ribbons

We present numerical and experimental results for the generation of aperiodic motion in coupled active rotators. The numerical analysis is presented for two point particles constrained to move on a unit circle under the Yukawa-like interaction. Simulations exhibit that the collision among the rotors results in chaotic motion of the rotating point particles. Furthermore, the numerical model predicts a route to chaotic motion. Subsequently, we explore the effect of separation between the rotors on their chaotic dynamics. The numerically calculated fraction of initial conditions which led to chaotic motion shed light on the observed effects. We reproduce a subset of the numerical observations with two self-propelled ribbons rotating at the air-water interface. A pinned camphor rotor moves at the interface due to the Marangoni forces generated by surface tension imbalance around it. The camphor layer present at the common water surface acts as chemical coupling between two ribbons. The separation distance of ribbons (L) determines the nature of coupled dynamics. Below a critical distance (L_{T}), rotors can potentially, by virtue of collisions, exhibit aperiodic oscillations characterized via a mixture of co- and counterrotating oscillations. These aperiodic dynamics qualitatively matched the chaotic motion observed in the numerical model.

Electrical coupling of individual electrocatalytic oscillators

The catalytic electro-oxidation of some small organic molecules is known to display kinetic instabilities, which reflect on potential and/or current oscillations. Under oscillatory conditions, those systems can be considered electrocatalytic oscillators and, therefore, can be described by their amplitude, frequency, and waveform. Just like mechanical oscillators, the electrocatalytic ones can be coupled and their dynamics can be changed by setting different coupling parameters. In the present work, we study the unidirectional coupling of electrocatalytic oscillators, namely, those comprehending the catalytic electro-oxidation of methanol and formic acid on polycrystalline platinum in acidic media under potentiostatic control. Herein, we explore two different scenarios (the coupling of compositionally identical and non-identical oscillators) and investigate the effects of the master's identity and of the coupling constant on the slave's dynamics. For the master (methanol)-slave (methanol) coupling, the oscillators exhibited phase lag synchronization and complete phase synchronization. On the other hand, for the master (formic acid)-slave (methanol) coupling, the oscillators exhibited complete phase synchronization with phase-locking with a 2:3 ratio, complete phase synchronization with phase-locking with a 1:2 ratio, phase lag synchronization, and complete phase synchronization. The obtained results suggest that both the master's identity and the coupling constant (sign and magnitude) are parameters that play an important role on the coupled systems, in such a way that even for completely different systems, synchronization could emerge by setting a suitable coupling constant. To the best of our knowledge, this is the first report concerning the electrical coupling of hidden N-shaped-negative differential resistance type systems.

Intensity dependence of sub-harmonics in cortical response to photic stimulation

Objective. Periodic photic stimulation of human volunteers at 10 Hz is known to entrain their Electroencephalography (EEG) signals. This entrainment manifests as an increment in power at 10, 20, 30 Hz. We observed that this entrainment is accompanied by the emergence of sub-harmonics, but only at specific frequencies and higher intensities of the stimulating signal. Thereafter, we describe our results and explain them using the physiologically inspired Jansen and Rit Neural Mass Model (NMM).Approach. Four human volunteers were separately exposed to both high and low intensity 10 Hz and 6 Hz stimulation. A total of 4 experiments per subject were therefore performed. Simulations and bifurcation analysis of the NMM were carried out and compared with the experimental findings. <i> Main results. High intensity 10 Hz stimulation led to an increment in power at 5 Hz across all the 4 subjects. No increment of power was observed with low intensity stimulation. However, when the same protocol was repeated with a 6 Hz photic stimulation, neither high nor low intensity stimulation were found to cause a discernible change in power at 3 Hz. We found that the NMM was able to recapitulate these results. A further numerical analysis indicated that this arises from the underlying bifurcation structure of the NMM. <i> Significance. The excellent match between theory and experiment suggest that the bifurcation properties of the NMM are mirroring similar features possessed by the actual neural masses producing the EEG dynamics. Neural Mass Models could thus be valuable for understanding properties and pathologies of EEG dynamics, and may contribute to the engineering of brain-computer interface technologies.

Regulating dynamics through intermittent interactions

In this article we experimentally demonstrate an efficient scheme to regulate the behavior of coupled nonlinear oscillators through dynamic control of their interaction. It is observed that introducing intermittency in the interaction term as a function of time or the system state predictably alters the dynamics of the constituent oscillators. Choosing the nature of the interaction, attractive or repulsive, allows for either suppression of oscillations or stimulation of activity. Two parameters Δ and τ, that reign the extent of interaction among subsystems, are introduced. They serve as a harness to access the entire range of possible behaviors from fixed points to chaos. For fixed values of system parameters and coupling strength, changing Δ and τ offers fine control over the dynamics of coupled subsystems. We show this experimentally using coupled Chua's circuits and elucidate their behavior for a range of coupling parameters through detailed numerical simulations.

Interventions and their efficacy in controlling the spread of an epidemic: A numerical study

The effect of interventions on the progression of an epidemic is studied by numerically modeling attributes, such as lockdowns and vaccinations within a stochastic, highly connected, mobile community using an agent-based model. Based on real life assumptions, we are able to gauge the effectiveness of various strategies to contain the spread of a disease through a population. The fine-tuning of control parameters makes the model coherent with real life scenarios and robust from a policy-maker's perspective.

Quenching of oscillations in a liquid metal via attenuated coupling

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.

Modes of synchrony in self-propelled pentanol drops

We report various modes of synchrony observed for a population of two, three and four pentanol drops in a rectangular channel at the air-water interface. Initially, the autonomous oscillations of a single 1-pentanol drop were studied in a ferroin DI water solution pre-mixed with some volume of pentanol. A pentanol drop performs continuous motion on the air-water interface due to Marangoni forces. A linear channel was prepared to study the uniaxial movement of the drop(s). Thereafter, a systematic study of the self-propelled motion of a 1-pentanol drop was reported as a function of the drop volume. Subsequently, the coupled dynamics were studied for two, three and four drops, respectively. We observed anti-phase oscillations in a pair of pentanol drops. In the case of three drops, relay synchronization was observed, wherein consecutive pairs of drops were exhibiting out-of-phase oscillations and alternate drops were performing in-phase oscillations. Four pentanol drops showed two different modes of synchrony: one was relay synchrony and the other was out-of-phase oscillations between two pairs of drops (within a pair, the drops exhibit in-phase oscillations).

An alternate approach to simulate the dynamics of perturbed liquid drops

Liquid drops when subjected to external periodic perturbations can execute polygonal oscillations. In this work, a simple model is presented that demonstrates these oscillations and their characteristic properties. The model consists of a spring-mass network such that masses are analogous to liquid molecules and the springs correspond to intermolecular links. Neo-Hookean springs are considered to represent these intermolecular links. The restoring force of a neo-Hookean spring depends nonlinearly on its length such that the force of a compressed spring is much higher than the force of the spring elongated by the same amount. This is analogous to the incompressibility of liquids, making these springs suitable to simulate the polygonal oscillations. It is shown that this spring-mass network can imitate most of the characteristic features of experimentally reported polygonal oscillations. Additionally, it is shown that the network can execute certain dynamics, which so far have not been observed in a perturbed liquid drop. The characteristics of dynamics that are observed in the perturbed network are polygonal oscillations, rotation of network, numerical relations (rational and irrational) between the frequencies of polygonal oscillations and the forcing signal, and that the shape of the polygons depends on the parameters of perturbation.

Emergent noise-aided logic through synchronization

In this article, we present a dynamical scheme to obtain a reconfigurable noise-aided logic gate that yields all six fundamental two-input logic operations, including the xor operation. The setup consists of two coupled bistable subsystems that are each driven by one subthreshold logic input signal, in the presence of a noise floor. The synchronization state of their outputs robustly maps to two-input logic operations of the driving signals, in an optimal window of noise and coupling strengths. Thus the interplay of noise, nonlinearity, and coupling leads to the emergence of logic operations embedded within the collective state of the coupled system. This idea is manifested using both numerical simulations and proof-of-principle circuit experiments. The regions in parameter space that yield reliable logic operations were characterized through a stringent measure of reliability, using both numerical and experimental data.

Aperiodic bursting dynamics of active rotors

We report experiments on an active camphor rotor. A camphor rotor is prepared by infusing camphor on a regular rectangular paper strip. It performs self-propelled motion at the air-water interface due to Marangoni driven forces. After some transient (periodic) dynamics, the rotor enters into the aperiodic bursting regime, which is characterized as an irregularly repeated rest (halt) and run (motion) of the rotor. Subsequently, this aperiodic (irregular) rotor is entrained to a periodic (regular) regime with the help of a suitable external periodic forcing. Furthermore, we conducted experiments on two such coupled aperiodic camphor rotors. In this set of experiments, synchronized bursting was observed. During this bursting motion, one rotor follows the movement of the other rotor. A numerical point particle model, incorporating excitable underlying equations, successfully replicated experimentally observed aperiodic bursting.

Ill-matched timescales in coupled systems can induce oscillation suppression

We explore the behavior of two coupled oscillators, considering combinations of similar and dissimilar oscillators, with their intrinsic dynamics ranging from periodic to chaotic. We first investigate the coupling of two different real-world systems, namely, the chemical mercury beating heart oscillator and the electronic Chua oscillator, with the disparity in the timescales of the constituent oscillators. Here, we are considering a physical situation that is not commonly addressed: the coupling of sub-systems whose characteristic timescales are very different. Our findings indicate that the oscillations in coupled systems are quenched to oscillation death (OD) state, at sufficiently high coupling strength, when there is a large timescale mismatch. In contrast, phase synchronization occurs when their timescales are comparable. In order to further strengthen the concept, we demonstrate this timescale-induced oscillation suppression and phase synchrony through numerical simulations, with the disparity in the timescales serving as a tuning or control parameter. Importantly, oscillation suppression (OD) occurs for a significantly smaller timescale mismatch when the coupled oscillators are chaotic. This suggests that the inherent broad spectrum of timescales underlying chaos aids oscillation suppression, as the temporal complexity of chaotic dynamics lends a natural heterogeneity to the timescales. The diversity of the experimental systems and numerical models we have chosen as a test-bed for the proposed concept lends support to the broad generality of our findings. Last, these results indicate the potential prevention of system failure by small changes in the timescales of the constituent dynamics, suggesting a potent control strategy to stabilize coupled systems to steady states.

Competitive interplay of repulsive coupling and cross-correlated noises in bistable systems

The influence of noise on synchronization has potential impact on physical, chemical, biological, and engineered systems. Research on systems subject to common noise has demonstrated that noise can aid synchronization, as common noise imparts correlations on the sub-systems. In our work, we revisit this idea for a system of bistable dynamical systems, under repulsive coupling, driven by noises with varying degrees of cross correlation. This class of coupling has not been fully explored, and we show that it offers new counter-intuitive emergent behavior. Specifically, we demonstrate that the competitive interplay of noise and coupling gives rise to phenomena ranging from the usual synchronized state to the uncommon anti-synchronized state where the coupled bistable systems are pushed to different wells. Interestingly, this progression from anti-synchronization to synchronization goes through a domain where the system randomly hops between the synchronized and anti-synchronized states. The underlying basis for this striking behavior is that correlated noise preferentially enhances coherence, while the interactions provide an opposing drive to push the states apart. Our results also shed light on the robustness of synchronization obtained in the idealized scenario of perfectly correlated noise, as well as the influence of noise correlation on anti-synchronization. Last, the experimental implementation of our model using bistable electronic circuits, where we were able to sweep a large range of noise strengths and noise correlations in the laboratory realization of this noise-driven coupled system, firmly indicates the robustness and generality of our observations.

Phase coalescence in a population of heterogeneous Kuramoto oscillators

Phase coalescence (PC) is an emerging phenomenon in an ensemble of oscillators that manifests itself as a spontaneous rise in the order parameter. This increment in the order parameter is due to the overlaying of oscillator phases to a pre-existing system state. In the current work, we present a comprehensive analysis of the phenomenon of phase coalescence observed in a population of Kuramoto phase oscillators. The given population is divided into responsive and non-responsive oscillators depending on the position of the phases of the oscillators. The responsive set of oscillators is then reset by a pulse perturbation. This resetting leads to a temporary rise in a macroscopic observable, namely, order parameter. The provoked rise thus induced in the order parameter is followed by unprovoked increments separated by a constant time τ_PC. These unprovoked increments in the order parameter are caused due to a temporary gathering of the oscillator phases in a configuration similar to the initial system state, i.e., the state of the network immediately following the perturbation. A theoretical framework corroborating this phenomenon as well as the corresponding simulation results are presented. Dependence of τ_PC and the magnitude of spontaneous order parameter augmentation on various network parameters such as coupling strength, network size, degree of the network, and frequency distribution are then explored. The size of the phase resetting region would also affect the magnitude of the order parameter at τ_PC since it directly affects the number of oscillators reset by the perturbation. Therefore, the dependence of order parameter on the size of the phase resetting region is also analyzed.

Revealing the deterministic components in active avalanche-like dynamics

Avalanche dynamics in an ensemble of self-propelled camphor boats are studied. The self-propelled agents are camphor infused circular paper disks moving on the surface of water. The ensemble exhibits bursts of activity in the autonomous state triggered by stochastic fluctuations. This type of dynamics has been previously reported in a slightly different system (J. Phys. Soc. Jpn., 2015, 84, 034802). Fourier analysis of the autonomous ensemble's average speed reveals a unimodal spectrum, indicating the presence of a preferred time scale in the dynamics. We therefor, entrain such an ensemble by external forcing by using periodic air perturbations on the surface of the water. This forcing is able to replace the stochastic fluctuations which trigger a burst in the autonomous ensemble, thus entraining the system. Upon varying the periodic forcing frequency, an optimal frequency is revealed at which the quality of entrainment of the ensemble by the forcing is augmented. This optimal frequency is found to be in the vicinity of the Fourier spectrum peak of the autonomous ensemble's average speed. This indicates the existence of an underlying deterministic component in the apparent aperiodic bursts of motion of the autonomous ensemble of active particles. A qualitative reasoning for the observed phenomenon is presented.

Chimeralike states in a minimal network of active camphor ribbons

A chimeralike state is the spatiotemporal pattern in an ensemble of homogeneous coupled oscillators, described as the emergence of coexisting coherent (synchronized) and incoherent (unsynchronized) groups. We demonstrate the existence of these states in three active camphor ribbons, which are camphor infused rectangular pieces of paper. These pinned ribbons rotate on the surface of the water due to Marangoni effect driven forces generated by the surface tension gradients. The ribbons are coupled via a camphor layer on the surface of the water. In the minimal network of globally coupled camphor ribbons, chimeralike states are characterized by the coexistence of two synchronized and one unsynchronized ribbons. We present a numerical model, simulating the coupling between ribbons as repulsive Yukawa interactions, which was able to reproduce these experimentally observed states.

Periodic oscillations in a string of camphor infused disks

The rhythmic beating motion of autonomously motile filaments has many practical applications. Here, we present an experimental study on a filament made of camphor infused paper disks, stitched together adjacent to each other using nylon thread. The filament displays spontaneous translatory motion when it is placed on the surface of water due to the surface tension gradients created by camphor molecules on the water surface. When this filament is clamped on one end, we obtain regular oscillatory motion instead of translation. The filament shows qualitatively different dynamics at different activity levels, which is controlled by the amount of camphor infused into the paper disks. For a better physical understanding of the filament dynamics, we develop a minimal numerical model involving a semi-flexible filament made of active polar disks, where the polarity is coupled to the instantaneous velocity of the particle. This model qualitatively reproduces different oscillatory modes of the filament. Moreover, our model reveals a rich dynamical state diagram of the system, as a function of filament activity and the coupling strength.

Asymmetry induced suppression of chaos

We explore the dynamics of a group of unconnected chaotic relaxation oscillators realized by mercury beating heart systems, coupled to a markedly different common external chaotic system realized by an electronic circuit. Counter-intuitively, we find that this single dissimilar chaotic oscillator manages to effectively steer the group of oscillators on to steady states, when the coupling is sufficiently strong. We further verify this unusual observation in numerical simulations of model relaxation oscillator systems mimicking this interaction through coupled differential equations. Interestingly, the ensemble of oscillators is suppressed most efficiently when coupled to a completely dissimilar chaotic external system, rather than to a regular external system or an external system identical to those of the group. So this experimentally demonstrable controllability of groups of oscillators via a distinct external system indicates a potent control strategy. It also illustrates the general principle that symmetry in the emergent dynamics may arise from asymmetry in the constituent systems, suggesting that diversity or heterogeneity may have a crucial role in aiding regularity in interactive systems.

Predictive modeling of disease propagation in a mobile, connected community using cellular automata

We present numerical results obtained from the modeling of a stochastic, highly connected, and mobile community. The spread of attributes like health and disease among the community members is simulated using cellular automata on a planar two-dimensional surface. With remarkably few assumptions, we are able to predict the future course of propagation of such a disease as a function of time and the fine-tuning of parameters related to interactions among the automata.

First passage of an active particle in the presence of passive crowders

We experimentally study the stochastic transport of a self-propelled camphor boat, driven by Marangoni forces, through a crowd of passive paper discs floating on water. We analyze the statistics of the first passage times of the active particle to travel from the center of a circular container to its boundary. While the mean times rise monotonically as a function of the covered area fraction φ of the passive paper discs, their fluctuations show a non-monotonic behavior - being higher at low and high value of φ compared to intermediate values. The reason is traced to an interplay of two distinct sources of fluctuations - one intrinsic to the dynamics, while the other due to the crowding.

Rotational synchronization of camphor ribbons in different geometries

We present experiments on multiple pinned self-propelled camphor ribbons, which is a rectangular piece of paper with camphor infused in its matrix. Experiments were performed on three, four, and five ribbons placed in linear and polygonal geometries. The pinned ribbons rotate on the surface of water, due to the surface tension gradient introduced by the camphor layer in the neighborhood of the ribbon. This camphor layer leads to a chemical coupling between the ribbons. In different geometries, the ribbons have been observed to rotationally synchronize in all the possible configurations. A numerical model, emulating the interactions between the ribbons as Yukawa interaction was studied, which was qualitatively able to reproduce the experimental findings.

Echo in complex networks

Large populations of globally coupled or uncoupled oscillators have been recently shown to exhibit an intriguing echo behavior [Ott, Platig, Antonsen, and Girvan, Chaos: An Interdiscip. J. Nonlinear Sci. 18, 037115 (2008)CHAOEH1054-150010.1063/1.2973816; Chen, Tinsley, Ott, and Showalter, Phys. Rev. X 6, 041054 (2016)2160-330810.1103/PhysRevX.6.041054], wherein a system is perturbed by two successive pulses at times T and T+τ inducing a spontaneous increase in the order parameter at the given times. These two provoked increments in the order parameter are followed by an unprovoked spontaneous increment in the order parameter at time T+2τ termed as an echo. In this paper, the effects of network topology on the emergence of an echo are explored. Two principal network parameters, namely, average degree and network randomness, are varied for this purpose. The networks are rewired to increase randomness in the network connections using the Watts-Strogatz algorithm to generate small world networks [Watts and Strogatz, Nature (London) 393, 440 (1998)10.1038/30918]. Thus, the whole span of networks ranging from a regular ring to a completely random network is explored. The average degree of the underlying connectivity, starting from nearest neighbor connections, is also monotonically increased and its effects on the echo behavior are analyzed. We find that for rings with low average degrees and high coupling strengths a discernible echo is not observed. Remarkably, an echo reemerges in the presence of sufficient randomness in the connections for such networks. For a regular ring network, increasing the average degree after a critical value also yields a transition to echo behavior. However, for completely random networks echoes are present in networks of all average degrees. This suggests that randomizing connections can induce echoes in systems even when the average degree of connections is very low. Another subtle feature arises for intermediate randomness, where the system exhibits a nonmonotonic dependence of the echo size on average degree. The echo size was found to be minimum at an intermediate value of the average degree. Lastly we consider the influence of dynamically changing links on the echo size and demonstrate that time-varying connections destroy the echo in low average degree networks, while the echo persists under dynamic links in high average degree networks. So our results clearly demarcate the class of networks that are robust candidates for exhibiting echoes, as well as provide caveats for the observation of echoes in networks.

Stochastic resonance via parametric adaptation: Experiments and numerics

In contrast with the conventionally observed mechanism of stochastic resonance (SR) wherein the level of additive noise is systematically varied with a fixed set-point parameter, in this work we report the emergence of the SR phenomena in an electrochemical system maintaining the same level of noise and varying the parametric distance from a homoclinic bifurcation inherent to the system. The experimental system involves the corrosion of a metal disk in an acidic medium under potentiostatic conditions. The applied potential is used as a control parameter and the anodic current generated during the electrodissolution of the metal is the accessible system variable. In the presence of noise, it was observed that the system was able to enhance its output's fidelity with a weak subthreshold input signal when the set point was kept at an optimal parametric distance from the bifurcation. Numerical simulations were performed on a model for this system to corroborate the experimental observations. This type of SR may be critical in scenarios where a biological entity has control only on its sensory parameters and not on the environmental noise amplitude.

Velocity controlled pattern writing: An application of stochastic resonance

In the present work, the concept of stochastic resonance is employed for pattern fabrication. In particular, the interplay of noise amplitudes and intrinsic system time scales is investigated. This interplay enabled us to obtain preordained patterns. Experiments were performed galvanostatically in a two electrode electrochemical cell onto a n-type Si substrate using a coherent wavelength laser source of 5 mW intensity. A focused laser beam was swept along the silicon substrate unidirectionally by moving the electrochemical cell at different velocities. By systematic tuning of the velocity, we have observed a unimodal variation in the contrast of the pattern. This indicates the occurrence of the stochastic resonance phenomena. Corresponding numerical simulations, performed on a spatial array of diffusively coupled FitzHugh-Nagumo oscillators in the presence of external noise, reveal good agreement with the experimental observations.

Entrainment of aperiodic and periodic oscillations in the Mercury Beating Heart system using external periodic forcing

We report experimental results indicating entrainment of aperiodic and periodic oscillatory dynamics in the Mercury Beating Heart (MBH) system under the influence of superimposed periodic forcing. Aperiodic oscillations in MBH were controlled to generate stable topological modes, namely, circle, ellipse, and triangle, evolving in a periodic fashion at different parameters of the forcing signal. These periodic dynamics show 1:1 entrainment for circular and elliptical modes, and additionally the controlled system exhibits 1:2 entrainment for elliptical and triangular modes at a different set of parameters. The external periodic forcing of the periodic MBH system reveals the existence of domains of entrainment (1:1, 1:2, 1:3, and 1:4) represented in the Arnold tongue structures. Moreover, Devil's staircase is obtained when the amplitude-frequency space of parameters of the applied signal is scanned.

Pattern selection and regulation using noise in a liquid metal

Electric forcing can be used to select and to regulate the shape of liquid metals. In this work, we present a transition among different patterns in a liquid mercury drop regulated by noise. A stochastic resonancelike phenomenon was observed for two different structural transitions of the liquid metal. In the first set of experiments, the transition from irregular (I) → triangular (T) → irregular (I) patterns was obtained by increasing the amplitude of biased white noise. In the second part, we observed the transition from irregular (I) → elliptical (E) → irregular (I) patterns using the same kind of noise. Periodic stochastic resonance was corroborated in our experiments by employing the cross-correlation coefficient technique.

Oscillatory activity regulation in an ensemble of autonomous mercury beating heart oscillators

Collective behavior of an ensemble of directly or indirectly coupled oscillators can be a function of population density. Experiments using autonomous mercury beating heart (MBH) oscillators coupled through their surroundings are employed, to study the existence of quorum-like (population dependent) phenomena. Two coupling mechanisms are used, namely, static and dynamic coupling. For the static coupling scheme, the transitions of a subset of the coupled oscillators occur from active (oscillatory) to inactive (quiescent) state and vice versa. A continuous variation of collective dynamics was observed as the population of the oscillators increased. For the dynamic coupling scheme, the time for which the coupled oscillators are active changes sharply as the population increases beyond a certain threshold.

Rotational synchronization of camphor ribbons

Experiments on interacting pinned self-propelled rotators are presented. The rotators are made from paper with camphor infused in its matrix. The ribbons rotate due to Marangoni effect driven forces arising by virtue of surface tension gradients. Two such self-rotating camphor ribbons are observed to experience a repulsive coupling via the camphor layer in the common water medium. Lag synchronization in both corotating (same sense of rotation) and counterrotating (opposite sense of rotation) ribbons is reported for the experiments. This synchronization is found to be dependent on the pivot to pivot distance l. For distances less than the span of both the ribbons, l_{c}, the rotators successfully synchronize. Furthermore, it is experimentally perceived that synchronization in the counterrotating ribbons is more robust than in the corotating ribbons. We rationalize the mechanism of this synchronization via a theoretical model involving a Yukawa type interaction which is analyzed numerically.

Provoking predetermined aperiodic patterns in human brainwaves

In the present work, electroencephalographic recordings of healthy human participants were performed to study the entrainment of brainwaves using a variety of stimuli. First, the periodic entrainment of the brainwaves was studied using two different stimuli in the form of periodic auditory and visual signals. The entrainment with the periodic visual stimulation was consistently observed, whereas the auditory entrainment was inconclusive. Hence, a photic (visual) stimulus, where two frequencies were presented to the subject simultaneously, was used to further explore the bifrequency entrainment of human brainwaves. Subsequently, the evolution of brainwaves as a result of an aperiodic stimulation was explored, wherein an entrainment to the predetermined aperiodic pattern was observed. These results suggest that the aperiodic entrainment could be used as a tool for guided modification of brainwaves. This could find possible applications in processes such as epilepsy suppression and biofeedback.

Intrinsic stochastic resonance via set-point variation

In the present paper, the possibility of invoking stochastic resonance (SR, periodic and aperiodic) by regulating the operating value of an appropriate parameter is explored. The operating values of these parameters are defined as the set point of the system throughout the present paper. Brusselator, a mathematical model [I. Prigogine and R. Lefever, J. Chem. Phys. 48, 1695 (1968)JCPSA60021-960610.1063/1.1668896] of nonlinear chemical reactions, is used for this purpose. We consider the effect of intrinsic noise in the Brusselator due to the Markovian nature of the chemical reactions. The stochastic time evolution is studied using the Gillespie algorithm [D. T. Gillespie, J. Comput. Phys. 22, 403 (1976)JCTPAH0021-999110.1016/0021-9991(76)90041-3], which is an exact stochastic simulation algorithm. We analyze the dependence of the resonance point on both the strength of the intrinsic noise as well as the distance from the bifurcation point. Subsequently, the phenomena of SR is explored using both periodic and aperiodic stimulus. It was found that, for a given system size, in both cases, SR is achieved by variation of the set point. Set-point variation can be achieved by regulating either the source concentration or the rate constants. Resonance is observed in both cases. However, this resonance occurs at different values of the set point, even with a fixed system size. This is clearly seen in the set-point versus system-size plane, where the resonance line has different slopes for the two scenarios. Our semianalytic treatment points to the fact that for a given system size intrinsic noise is affected differently for different methods involving the variation of the set point. This is explained by writing the corresponding chemical Langevin equation and comparing the various intrinsic noise sources.

Control, synchronization, and enhanced reliability of aperiodic oscillations in the Mercury Beating Heart system

Experiments involving the Mercury Beating Heart (MBH) oscillator, exhibiting irregular (aperiodic) dynamics, are performed. In the first set of experiments, control over irregular dynamics of the MBH oscillator was obtained via a superimposed periodic voltage signal. These irregular (aperiodic) dynamics were recovered once the control was switched off. Subsequently, two MBH oscillators were coupled to attain synchronization of their aperiodic oscillations. Finally, two uncoupled MBH oscillators were subjected, repeatedly, to a common stochastic forcing, resulting in an enhancement of their mutual phase correlation.

An alternate protocol to achieve stochastic and deterministic resonances

Periodic and Aperiodic Stochastic Resonance (SR) and Deterministic Resonance (DR) are studied in this paper. To check for the ubiquitousness of the phenomena, two unrelated systems, namely, FitzHugh-Nagumo and a particle in a bistable potential well, are studied. Instead of the conventional scenario of noise amplitude (in the case of SR) or chaotic signal amplitude (in the case of DR) variation, a tunable system parameter ("a" in the case of FitzHugh-Nagumo model and the damping coefficient "j" in the bistable model) is regulated. The operating values of these parameters are defined as the "setpoint" of the system throughout the present work. Our results indicate that there exists an optimal value of the setpoint for which maximum information transfer between the input and the output signals takes place. This information transfer from the input sub-threshold signal to the output dynamics is quantified by the normalised cross-correlation coefficient ( |CCC|). |CCC| as a function of the setpoint exhibits a unimodal variation which is characteristic of SR (or DR). Furthermore, |CCC| is computed for a grid of noise (or chaotic signal) amplitude and setpoint values. The heat map of |CCC| over this grid yields the presence of a resonance region in the noise-setpoint plane for which the maximum enhancement of the input sub-threshold signal is observed. This resonance region could be possibly used to explain how organisms maintain their signal detection efficacy with fluctuating amounts of noise present in their environment. Interestingly, the method of regulating the setpoint without changing the noise amplitude was not able to induce Coherence Resonance (CR). A possible, qualitative reasoning for this is provided.

Electrical resonance of Amphotericin B channel activity in lipidic membranes

In our previous work [J. Membrane Biol. 237, 31 (2010)], we showed the dependence of the time average conductance of Nystatin channels as a function of the applied potential. Specifically, it was observed that greater potential induced enhanced channel activity. This indicates that the supramolecular structure could be stabilized by a large field, possibly by giving a preferential orientation to the monomers. In the present work, we entertain the notion that the process of pore formation in the lipidic membranes has an underlying deterministic component. To verify this hypothesis, experiments were performed under potentio-dynamic conditions, i.e., a square train of pulses of different frequencies (0.05-2 Hz) were applied to a 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine membrane having 30 mol. % cholesterol and the presence of 35 μM Amphotericin B. An emergence of a resonant frequency, in the present experiments, is tantamount to observing fingerprints of determinism in the activity of these channels in lipidic membranes.

Scaling dependence and synchronization of forced mercury beating heart systems

We perform experiments on a nonautonomous Mercury beating heart system, which is forced to pulsate using an external square wave potential. At suitable frequencies and volumes, the drop exhibits pulsation with polygonal shapes having n corners. We find the scaling dependence of the forcing frequency ν_{n} on the volume V of the drop and establish the relationship ν_{n}∝n/sqrt[V]. It is shown that the geometrical shape of substrate is important for obtaining closer match to these scaling relationships. Furthermore, we study synchronization of two nonidentical drops driven by the same frequency and establish that synchrony happens when the relationship n_{2}/n_{1}=sqrt[V_{2}/V_{1}] is satisfied.

Intrinsic periodic and aperiodic stochastic resonance in an electrochemical cell

In this paper we show the interaction of a composite of a periodic or aperiodic signal and intrinsic electrochemical noise with the nonlinear dynamics of an electrochemical cell configured to study the corrosion of iron in an acidic media. The anodic voltage setpoint (V_{0}) in the cell is chosen such that the anodic current (I) exhibits excitable fixed point behavior in the absence of noise. The subthreshold periodic (aperiodic) signal consists of a train of rectangular pulses with a fixed amplitude and width, separated by regular (irregular) time intervals. The irregular time intervals chosen are of deterministic and stochastic origins. The amplitude of the intrinsic internal noise, regulated by the concentration of chloride ions, is then monotonically increased, and the provoked dynamics are analyzed. The signal to noise ratio and the cross-correlation coefficient versus the chloride ions' concentration curves have a unimodal shape indicating the emergence of an intrinsic periodic or aperiodic stochastic resonance. The abscissa for the maxima of these unimodal curves correspond to the optimum value of intrinsic noise where maximum regularity of the invoked dynamics is observed. In the particular case of the intrinsic periodic stochastic resonance, the scanning electron microscope images for the electrode metal surfaces are shown for certain values of chloride ions' concentrations. These images, qualitatively, corroborate the emergence of order as a result of the interaction between the nonlinear dynamics and the composite signal.

Synchronization using environmental coupling in mercury beating heart oscillators

We report synchronization of Mercury Beating Heart (MBH) oscillators using the environmental coupling mechanism. This mechanism involves interaction of the oscillators with a common medium/environment such that the oscillators do not interact among themselves. In the present work, we chose a modified MBH system as the common environment. In the absence of coupling, this modified system does not exhibit self sustained oscillations. It was observed that, as a result of the coupling of the MBH oscillators with this common environment, the electrical and the mechanical activities of both the oscillators synchronized simultaneously. Experimental results indicate the emergence of both lag and the complete synchronization in the MBH oscillators. Simulations of the phase oscillators were carried out in order to better understand the experimental observations.

Experimental evidence of explosive synchronization in mercury beating-heart oscillators

We report experimental evidence of explosive synchronization in coupled chemo-mechanical systems, namely in mercury beating-heart (MBH) oscillators. Connecting four MBH oscillators in a star network configuration and setting natural frequencies of each oscillator in proportion to the number of its links, a gradual increase of the coupling strength results in an abrupt and irreversible (first-order-like) transition from the system's unordered to ordered phase. On its turn, such a transition indicates the emergence of a bistable regime wherein coexisting states can be experimentally revealed. Finally, we prove how such a regime allows an experimental implementation of magneticlike states of synchronization, by the use of an external signal.

Tracking stochastic resonance curves using an assisted reference model

The optimal noise amplitude for Stochastic Resonance (SR) is located employing an Artificial Neural Network (ANN) reference model with a nonlinear predictive capability. A modified Kalman Filter (KF) was coupled to this reference model in order to compensate for semi-quantitative forecast errors. Three manifestations of stochastic resonance, namely, Periodic Stochastic Resonance (PSR), Aperiodic Stochastic Resonance (ASR), and finally Coherence Resonance (CR) were considered. Using noise amplitude as the control parameter, for the case of PSR and ASR, the cross-correlation curve between the sub-threshold input signal and the system response is tracked. However, using the same parameter the Normalized Variance curve is tracked for the case of CR. The goal of the present work is to track these curves and converge to their respective extremal points. The ANN reference model strategy captures and subsequently predicts the nonlinear features of the model system while the KF compensates for the perturbations inherent to the superimposed noise. This technique, implemented in the FitzHugh-Nagumo model, enabled us to track the resonance curves and eventually locate their optimal (extremal) values. This would yield the optimal value of noise for the three manifestations of the SR phenomena.

Kuramoto transition in an ensemble of mercury beating heart systems

We have studied, experimentally, the collective behavior of the electrically coupled autonomous Mercury Beating Heart (MBH) systems exhibiting the breathing mode, by varying both the coupling strength and the population size (from N = 3 to N = 16). For a fixed N, the electrical and the mechanical activities of the MBH systems achieve complete synchronization at different coupling strengths. The electrical activity of each MBH system is measured by the corresponding electrode potential (Ei = Vi). Additionally, the mechanical activity of each MBH oscillator is visually observed (snapshots and video clips). Subsequently, this activity is quantified by calculating the temporal variation in the area (Ai) of the Hg drop. As a result, the synchronization of the electrical (Ei = Vi) and the mechanical (Ai) activities can be measured. The extent of synchronization was quantified by employing the order parameter (r). Our experimental results are found to be in agreement with the Kuramoto theory.

Stochastic resonance with a mesoscopic reaction-diffusion system

In a mesoscopic reaction-diffusion system with an Oregonator reaction model, we show that intrinsic noise can drive a resonant stable pattern in the presence of the initial subthreshold perturbations. Both spatially periodic and aperiodic stochastic resonances are demonstrated by employing the Gillespies stochastic simulation algorithm. The mechanisms for these phenomena are discussed.

Quorum sensing via static coupling demonstrated by Chua's circuits

Dynamical quorum sensing, the population based phenomenon, is believed to occur when the elements of a system interact via dynamic coupling. In the present work, we demonstrate an alternate scenario, involving static coupling, that could also lead to quorum sensing behavior. These static and dynamic coupling terms have already been employed by Konishi [Int. J. Bifurcation Chaos Appl. Sci. Eng. 17, 2781 (2007)]. In our context, the coupling is defined as static or dynamic, on the basis of the relative time scales at which the surrounding dynamics and the elements' dynamics evolve. According to this, if the variation in the surrounding dynamics happens on a much larger (fast) time scale than that at which the elements' dynamics are varying (such as seconds and μs), then the coupling is considered to be static, otherwise it is considered to be dynamic. A series of experiments have been performed starting from a system of three Chua's circuits to a system of 20 Chua's circuits to study two types of quorum transitions: the emergence and the extinction of global oscillations (period-1). The numerics involving up to 100 Chua's circuits validate the experimental observations.

Conjugate feedback induced suppression and generation of oscillations in the Chua circuit: experiments and simulations

We study the suppression (amplitude death) and generation of oscillations (rhythmogenesis) in the Chua circuit using a feedback term consisting of conjugate variables (conjugate feedback). When the independent Chua circuit (without feedback) is placed in the oscillatory domain, this conjugate feedback induces amplitude death in the system. On the contrary, introducing the conjugate feedback in the system exhibiting fixed point behavior results in the generation of rhythms. Furthermore, it is observed that the dynamics of the Chua circuit could be tuned efficiently by varying the strength of this feedback term. Both experimental and numerical results are presented.