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.

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.

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.

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.

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.

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.

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.

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.

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.

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 four experiments per subject were therefore performed. Simulations and bifurcation analysis of the NMM were carried out and compared with the experimental findings.Main results.High intensity 10 Hz stimulation led to an increment in power at 5 Hz across all the four 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.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. NMMs could thus be valuable for understanding properties and pathologies of EEG dynamics, and may contribute to the engineering of brain-computer interface technologies.

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.

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.

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.

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).

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 τPC. 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 τPC 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 τPC 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.