arXiv subjects

Sagar Chakraborty

Publications and source records attributed to Sagar Chakraborty.

At least 55 records · Page 3Linked to original sources

arXiv2021

Weight of fitness deviation governs strict physical chaos in replicator dynamics

Replicator equation -- a paradigm equation in evolutionary game dynamics -- mathematizes the frequency dependent selection of competing strategies vying to enhance their fitness (quantified by the average payoffs) with respect to the average fitnesses of the evolving population under consideration. In this paper, we deal with two discrete versions of the replicator equation employed to study evolution in a population where any two players, interaction is modeled by a two-strategy symmetric normal-form game. There are twelve distinct classes of such games, each typified by a particular ordinal relationship among the elements of the corresponding payoff matrix. Here we find the sufficient conditions for the existence of asymptotic solutions of the replicator equations such that the solutions -- fixed points, periodic orbits, and chaotic trajectories -- are all strictly physical, meaning that the frequency of any strategy lies inside the closed interval zero to one at all times. Thus, we elaborate which of the twelve types of games are capable of showing meaningful physical solutions and for which of the two types of replicator equation. Subsequently, we introduce the concept of the weight of fitness deviation that is the scaling factor in a positive affine transformation connecting two payoff matrices such that the corresponding one-shot games have exactly same Nash equilibria and evolutionary stable states. The weight also quantifies how much the excess of fitness of a strategy over the average fitness of the population affects the per capita change in the frequency of the strategy. Intriguingly, the weight's variation is capable of making the Nash equilibria and the evolutionary stable states useless by introducing strict physical chaos in the replicator dynamics based on the normal-form game.

nlin.CD↗

arXiv2021

Effect of chaotic agent dynamics on coevolution of cooperation and synchronization

The effect of the chaotic dynamical states of the agents on the coevolution of cooperation and synchronization in a structured population of the agents remains unexplored. With a view to gaining insights into this problem, we construct a coupled map lattice of the paradigmatic chaotic logistic map by adopting the Watts--Strogatz network algorithm. The map models the agent's chaotic state dynamics. In the model, an agent benefits by synchronizing with its neighbours and in the process of doing so, it pays a cost. The agents update their strategies (cooperation or defection) by using either a stochastic or a deterministic rule in an attempt to fetch themselves higher payoffs than what they already have. Among some other interesting results, we find that beyond a critical coupling strength, that increases with the rewiring probability parameter of the Watts--Strogatz model, the coupled map lattice is spatiotemporally synchronized regardless of the rewiring probability. Moreover, we observe that the population does not desynchronize completely -- and hence finite level of cooperation is sustained -- even when the average degree of the coupled map lattice is very high. These results are at odds with how a population of the non-chaotic Kuramoto oscillators as agents would behave. Our model also brings forth the possibility of the emergence of cooperation through synchronization onto a dynamical state that is a periodic orbit attractor.

nlin.AO↗

arXiv2021

Deciphering chaos in evolutionary games

Discrete-time replicator map is a prototype of evolutionary selection game dynamical models that have been very successful across disciplines in rendering insights into the attainment of the equilibrium outcomes, like the Nash equilibrium and the evolutionarily stable strategy. By construction, only the fixed point solutions of the dynamics can possibly be interpreted as the aforementioned game-theoretic solution concepts. Although more complex outcomes like chaos are omnipresent in the nature, it is not known to which game-theoretic solutions they correspond. Here we construct a game-theoretic solution that is realized as the chaotic outcomes in the selection monotone game dynamic. To this end, we invoke the idea that in a population game having two-player--two-strategy one-shot interactions, it is the product of the fitness and the heterogeneity (the probability of finding two individuals playing different strategies in the infinitely large population) that is optimized over the generations of the evolutionary process.

q-bio.PE↗

arXiv2021

Replicator equations induced by microscopic processes in nonoverlapping population playing bimatrix games

This paper is concerned with exploring the microscopic basis for the discrete versions of the standard replicator equation and the adjusted replicator equation. To this end, we introduce frequency-dependent selection -- as a result of competition fashioned by game-theoretic consideration -- into the Wright--Fisher process, a stochastic birth-death process. The process is further considered to be active in a generation-wise nonoverlapping finite population where individuals play a two-strategy bimatrix population game. Subsequently, connections among the corresponding master equation, the Fokker--Planck equation, and the Langevin equation are exploited to arrive at the deterministic discrete replicator maps in the limit of infinite population size.

q-bio.PE↗

arXiv2021

Evolutionary dynamics of the delayed replicator-mutator equation: Limit cycle and cooperation

Game theory deals with strategic interactions among players and evolutionary game dynamics tracks the fate of the players' populations under selection. In this paper, we consider the replicator equation for two-player-two-strategy games involving cooperators and defectors. We modify the equation to include the effect of mutation and also delay that corresponds either to the delayed information about the population state or in realizing the effect of interaction among players. By focusing on the four exhaustive classes of symmetrical games -- the Stag Hunt game, the Snowdrift game, the Prisoners' Dilemma game, and the Harmony game -- we analytically and numerically analyze delayed replicator-mutator equation to find the explicit condition for the Hopf bifurcation bringing forth stable limit cycle. The existence of the asymptotically stable limit cycle imply the coexistence of the cooperators and the defectors; and in the games, where defection is a stable Nash strategy, a stable limit cycle does provide a mechanism for evolution of cooperation. We find that while mutation alone can never lead to oscillatory cooperation state in two-player-two-strategy games, the delay can change the scenario. On the other hand, there are situations when delay alone cannot lead to the Hopf bifurcation in the absence of mutation in the selection dynamics.

nlin.AO↗

arXiv2021

Cooperators overcome migration dilemma through synchronization

Synchronization, cooperation, and chaos are ubiquitous phenomena in nature. In a population composed of many distinct groups of individuals playing the prisoner's dilemma game, there exists a migration dilemma: No cooperator would migrate to a group playing the prisoner's dilemma game lest it should be exploited by a defector; but unless the migration takes place, there is no chance of the entire population's cooperator-fraction to increase. Employing a randomly rewired coupled map lattice of chaotic replicator maps, modelling replication-selection evolutionary game dynamics, we demonstrate that the cooperators -- evolving in synchrony -- overcome the migration dilemma to proliferate across the population when altruism is mildly incentivized making few of the demes play the leader game.

q-bio.PE↗

arXiv2020

Suppressing birhythmicity by parametrically modulating nonlinearity in limit cycle oscillators

Multirhythmicity, a form of multistability, in an oscillator is an intriguing phenomenon found across many branches of science. From an application point of view, while the multirhythmicity is sometimes desirable as it presents us with many possible coexisting stable oscillatory states to tap into, it can also be a nuisance because a random perturbation may make the system settle onto an unwanted stable state. Consequently, it is not surprising that there are many natural and artificial mechanisms available that can control the multirhythmicity. We propose in this paper the idea of incorporating parametric (periodic) modulation of the nonlinear damping in the limit cycle oscillators with a view to exciting resonance and antiresonance responses at particular angular driving frequencies, and controlling the resulting birhythmicity by changing the amplitude of the modulation. To this end, we employ analytical (perturbative) and numerical techniques on the van der Pol oscillator---a paradigmatic limit cycle system---having additional position dependent time delay term and its modified autonomous birhythmic version. We also bring the fact to the fore that introduction of delay---a commonly adopted method of controlling multirhythmicity---in such a system can sometimes bring forth unwanted birhythmicity; and interestingly, our method of controlling birhythmicity through periodic modulation can suppress such a delay induced birhythmic response.

nlin.AO↗

arXiv2020

Bridging Inertial and Dissipation Range Statistics in Rotating Turbulence

We investigate the connection between the inertial range and the dissipation range statistics of rotating turbulence through detailed simulations of a helical shell model and a multifractal analysis. In particular, by using the latter, we find an explicit relation between the (anomalous) scaling exponents of equal-time structure functions in the inertial range in terms of the generalised dimensions associated with the energy dissipation rate. This theoretical prediction is validated by detailed simulations of a helical shell model for various strengths of rotation from where the statistics of dissipation rate, and thus the generalised dimensions, as well as the inertial range, in particular the anomalous scaling exponents, are extracted. Our work also underlines a surprisingly good agreement---such as in the spatial structure of the energy dissipation rates and the decrease in inertial range intermittency with increasing strengths of rotation---between solutions of the Navier--Stokes equation in a rotating frame with those obtained from low-dimensional, dynamical systems such as the shell model which are not explicitly anisotropic. Finally, we perform direct numerical simulations of the Navier--Stokes equation, with the Coriolis force incorporated, to confirm the robustness of the conclusions drawn from our multifractal and shell model studies.

physics.flu-dyn↗

arXiv2020

Giant Tunable Mechanical Nonlinearity in Graphene-Silicon Nitride Hybrid Resonator

High quality factor mechanical resonators have shown great promise in developing classical or quantum technologies. Simultaneously, progress has been made in developing controlled mechanical nonlinearity. Here we combine these two directions of progress in a single platform consisting of coupled Silicon Nitride (SiNx) and graphene mechanical resonators. We show that nonlinear response can be induced on a large area SiNx resonator mode and can be efficiently controlled by coupling it to a gate-tunable, freely suspended graphene mode. The induced nonlinear response of the hybrid modes, as measured on the SiNx resonator surface is giant, with one of the highest measured Duffing constants. We observe a novel phononic frequency comb which we use as an alternate validation of the measured values, along with numerical simulations which are in overall agreement with measurements.

cond-mat.mes-hall↗

arXiv2020

Comprehending deterministic and stochastic occasional uncoupling induced synchronizations through each other

In this paper, we numerically study the stochastic and the deterministic occasional uncoupling methods of effecting identical synchronized states in low dimensional, dissipative, diffusively coupled, chaotic flows that are otherwise not synchronized when continuously coupled at the same coupling strength parameter. In the process of our attempt to understand the mechanisms behind the success of the occasional uncoupling schemes, we devise a hybrid between the transient uncoupling and the stochastic on-off coupling, and aptly name it the transient stochastic uncoupling---yet another stochastic occasional uncoupling method. Our subsequent investigation on the transient stochastic uncoupling allows us to surpass the effectiveness of the stochastic on-off coupling with very fast on-off switching rate. Additionally, through the transient stochastic uncoupling, we establish that the indicators quantifying the local contracting dynamics in the corresponding transverse manifold are generally not useful in finding the optimal coupling region of the phase space in the case of the deterministic transient uncoupling. In fact, we highlight that the autocorrelation function---a non-local indicator of the dynamics---of the corresponding response system's chaotic time-series dictates when the deterministic uncoupling could be successful. We illustrate all our heuristic results using a few well-known examples of diffusively coupled chaotic oscillators.

nlin.AO↗

arXiv2019

Re-examining Einstein's $B$ coefficient and rate equations with the Rabi model

Starting from the Rabi Hamiltonian, which is useful in arriving at non-perturbative results within the rotating wave approximation, we have found Einstein's $B$ coefficient to be time-dependent: $B(t)\propto|J_0(ω_γt)|$ for a two-level system (atom or molecule) in thermal radiation field. Here $ω_γ$ is the corresponding Rabi flopping (angular) frequency and $J_0$ is the zeroth order Bessel function of the first kind. The resulting oscillations in the $B$ coefficient---even for very small $ω_γ$---drives the system away from thermodynamic equilibrium at any finite temperature contrary to Einstein's assumption. The time-dependent generalized $B$ coefficient facilitates a path to go beyond Pauli's formalism of non-equilibrium statistical mechanics involving the quantum statistical Boltzmann (master) equation. In this context, we have obtained entropy production of the two-level system by revising Einstein's rate equations, while considering the $A$ coefficient to be the original time-independent one and the $B$ coefficient to be time-dependent.

quant-ph↗

arXiv2018

Occasional uncoupling overcomes measure desynchronization

Owing to the absence of the phase space attractors in the Hamiltonian dynamical systems, the concept of the identical synchronization between the dissipative systems is inapplicable to the Hamiltonian systems for which, thus, one defines a related generalized phenomenon known as the measure synchronization. A coupled pair of Hamiltonian systems---the full coupled system also being Hamiltonian---can possibly be in two types of measure synchronized states: quasiperiodic and chaotic. In this paper, we take representative systems belonging to each such class of the coupled systems and highlight that, as the coupling strengths are varied, there may exist intervals in the ranges of the coupling parameters at which the systems are measure desynchronized. Subsequently, we illustrate that as a coupled system evolves in time, occasionally switching off the coupling when the system is in the measure desynchronized state can bring the system back in measure synchrony. Further, for the case of the occasional uncoupling being employed periodically and the corresponding time-period being small, we analytically find the values of the on-fraction of the time-period using which measure synchronization is effected on the corresponding desynchronized state.

nlin.CD↗

arXiv2018

On the energy spectrum of rapidly rotating forced turbulence

In this paper, we investigate the statistical features of the fully developed, forced, rapidly rotating, {turbulent} system using numerical simulations, and model {the} energy {spectrum} that {fits} well with the numerical data. Among the wavenumbers ($k$) larger than the Kolmogorov dissipation wavenumber, the energy is distributed such that the suitably non-dimensionized energy spectrum is ${\bar E}({\bar k})\approx \exp(-0.05{\bar k})$, where overbar denotes appropriate non-dimensionalization. {For the wavenumbers smaller than that of forcing, the energy in a horizontal plane is much more than that along the vertical rotation-axis.} {For} such wavenumbers, we find that the anisotropic energy spectrum, $E(k_\perp,k_\parallel)$ follows the power law scaling, $k_\perp^{-5/2}k_\parallel^{-1/2}$, where `$\perp$' and `$\parallel$' respectively refer to the directions perpendicular and parallel to the rotation axis; this result is in line with the Kuznetsov--Zakharov--Kolmgorov spectrum predicted by the weak inertial-wave turbulence theory for the rotating fluids.

physics.flu-dyn↗

arXiv2018

Finding Hannay angle in dissipative oscillatory systems via conservative perturbation theory

Usage of a Hamiltonian perturbation theory for a nonconservative system is counterintuitive and in general, a technical impossibility by definition. However, the time-independent dual Hamiltonian formalism for the nonconservative systems have opened the door for using various conservative perturbation theories for investigating the dynamics of such systems. Here we demonstrate that the Lie transform Hamiltonian perturbation theory can be adapted to find the perturbative solutions and the frequency corrections for the dissipative oscillatory systems. As a further application, we use the perturbation theory to analytically calculate the Hannay angle for the van der Pol oscillator's limit cycle trajectory when its parameters-the strength of the nonlinearity and the frequency of the linear part-evolve cyclically and adiabatically. For this van der Pol oscillator, we also numerically calculate the corresponding geometric phase and establish its equivalence with the Hannay angle.

math-ph↗

arXiv2018

Understanding transient uncoupling induced synchronization through modified dynamic coupling

An important aspect of the recently introduced transient uncoupling scheme is that it induces synchronization for large values of coupling strength at which the coupled chaotic systems resist synchronization when continu- ously coupled. However, why this is so is an open problem. To answer this question, we recall the conventional wisdom that the eigenvalues of the Jacobian of the transverse dynamics measure whether a trajectory at a phase point is locally contracting or diverging with respect to another nearby trajectory. Subsequently, we go on to highlight a lesser appreciated fact that even when, under the corresponding linearised flow, the nearby trajectory asymptotically diverges away, its distance from the reference trajectory may still be contracting for some intermediate period. We term this phenomenon transient decay in line with the phenomenon of the transient growth. Using these facts, we show that an optimal coupling region, i.e., a region of the phase space where coupling is on, should ideally be such that at any of the constituent phase point either the maximum of the real parts of the eigenvalues is negative or the magnitude of the positive maximum is lesser than that of the negative minimum. We also invent and employ modified dynamics coupling scheme---a significant improvement over the well-known dynamic coupling scheme---as a decisive tool to justify our results.

nlin.CD↗

arXiv2018

Statistical features of rapidly rotating decaying turbulence: enstrophy and energy spectra, and coherent structures

In this paper we investigate the properties of rapidly rotating decaying turbulence using numerical simulations and phenomenological modelling. We find that as the turbulent flow evolves in time, the Rossby number decreases to $\sim 10^{-3}$, and the flow becomes quasi-two-dimensional with strong coherent columnar structures arising due to the inverse cascade of energy. We establish that a major fraction of energy is confined in Fourier modes $(\pm1,0,0)$ and $(0,\pm1,0)$ that correspond to the largest columnar structure in the flow. For wavenumbers ($k$) greater than the enstrophy dissipation wavenumber ($k_d$), our phenomenological arguments and numerical study show that the enstrophy flux and spectrum of a horizontal cross-section perpendicular to the axis of rotation are given by $ε_ω\exp(-C(k/k_d)^2)$ and $Cε_ω^{2/3}k^{-1}\exp(-C(k/k_d)^2)$ respectively; for this 2D flow, $ε_ω$ is the enstrophy dissipation rate, and $C$ is a constant. Using these results, we propose a new form for the energy spectrum of rapidly rotating decaying turbulence: $E(k)=Cε_ω^{2/3}k^{-3}\exp(-C(k/k_d)^2)$. This model of the energy spectrum is based on wavenumber-dependent enstrophy flux, and it deviates significantly from power law energy spectrum reported earlier.

physics.flu-dyn↗

arXiv2017

On the Applicability of Low-Dimensional Models for Convective Flow Reversals at Extreme Prandtl Numbers

Constructing simpler models, either stochastic or deterministic, for exploring the phenomenon of flow reversals in fluid systems is in vogue across disciplines. Using direct numerical simulations and nonlinear time series analysis, we illustrate that the basic nature of flow reversals in convecting fluids can depend on the dimensionless parameters describing the system. Specifically, we find evidence of low-dimensional determinism in flow reversals occurring at zero Prandtl number, whereas we fail to find such signatures for reversals at infinite Prandtl number. Thus, even in a single system, as one varies the system parameters, one can encounter reversals that are fundamentally different in nature. Consequently, we conclude that a single general low-dimensional deterministic model cannot faithfully characterize flow reversals for every set of parameter values.

physics.flu-dyn↗

arXiv2017

Effect of Composition Gradient on Magnetothermal Instability Modified by Shear and Rotation

We model the intracluster medium as a weakly collisional plasma that is a binary mixture of the hydrogen and the helium ions, along with free electrons. When, owing to the helium sedimentation, the gradient of the mean molecular weight (or equivalently, composition or helium ions' concentration) of the plasma is not negligible, it can have appreciable influence on the stability criteria of the thermal convective instabilities, e.g., the heat-flux-buoyancy instability and the magnetothermal instability (MTI). These instabilities are consequences of the anisotropic heat conduction occurring preferentially along the magnetic field lines. In this paper, without ignoring the magnetic tension, we first present the mathematical criterion for the onset of composition gradient modified MTI. Subsequently, we relax the commonly adopted equilibrium state in which the plasma is at rest, and assume that the plasma is in a sheared state which may be due to differential rotation. We discuss how the concentration gradient affects the coupling between the Kelvin--Helmholtz instability and the MTI in rendering the plasma unstable or stable. We derive exact stability criterion by working with the sharp boundary case in which the physical variables---temperature, mean molecular weight, density, and magnetic field---change discontinuously from one constant value to another on crossing the boundary. Finally, we perform the linear stability analysis for the case of the differentially rotating plasma that is thermally and compositionally stratified as well. By assuming axisymmetric perturbations, we find the corresponding dispersion relation and the explicit mathematical expression determining the onset of the modified MTI.

astro-ph.GA↗