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The concept of evolutionarily stability and its relation with the fixed points of the replicator equation are important aspects of evolutionary game dynamics. In the light of the fact that oscillating state of a population and individuals (or players) of different roles are quite natural occurrences, we ask the question how the concept of evolutionarily stability can be generalized so as to associate game-theoretic meaning to oscillatory behaviours of players asymmetrically interacting, i.e., if there are both intraspecific and interspecific interactions between two subpopulations in the population. We guide our scheme of generalization such that the evolutionary stability is related to the dynamic stability of the corresponding periodic orbits of a time-discrete replicator dynamics. We name the generalization of evolutionarily stable state as two-species heterogeneity stable orbit. Furthermore, we invoke the principle of decrease of relative entropy in order to associate the generalization of evolutionary stability with an information-theoretic meaning. This particular generalization is aptly termed as two-species information stable orbit.

Evolutionarily stable strategy (ESS) is the defining concept of evolutionary game theory. It has a fairly unanimously accepted definition for the case of symmetric games which are played in a homogeneous population where all individuals are in same role. However, in asymmetric games, which are played in a population with multiple subpopulations (each of which has individuals in one particular role), situation is not as clear. Various generalizations of ESS defined for such cases differ in how they correspond to fixed points of replicator equation which models evolutionary dynamics of frequencies of strategies in the population. Moreover, some of the definitions may even be equivalent, and hence, redundant in the scheme of things. Along with reporting some new results, this paper is partly indented as a contextual mini-review of some of the most important definitions of ESS in asymmetric games. We present the definitions coherently and scrutinize them closely while establishing equivalences -- some of them hitherto unreported -- between them wherever possible. Since it is desirable that a definition of ESS should correspond to asymptotically stable fixed points of replicator dynamics, we bring forward the connections between various definitions and their dynamical stabilities. Furthermore, we find the use of principle of relative entropy to gain information-theoretic insights into the concept of ESS in asymmetric games, thereby establishing a three-fold connection between game theory, dynamical system theory, and information theory in this context. We discuss our conclusions also in the backdrop of asymmetric hypermatrix games where more than two individuals interact simultaneously in the course of getting payoffs., Comment: The earlier version (arXiv:2409.19320v1) had an inadvertent error that led to incorrect results. This revised version rectifies those results

Synergy between evolutionary dynamics of cooperation and fluctuating state of shared resource being consumed by the cooperators is essential for averting the tragedy of the commons. Not only in humans, but also in the cognitively-limited organisms, this interplay between the resource and the cooperation is ubiquitously witnessed. The strategically interacting players engaged in such game-environment feedback scenarios naturally pick strategies based on their perception of the environmental state. Such perception invariably happens through some sensory information channels that the players are endowed with. The unfortunate reality is that any sensory channel must be noisy due to various factors; consequently, the perception of the environmental state becomes faulty rendering the players incapable of adopting the strategy that they otherwise would. Intriguingly, situation is not as bad as it sounds. Here we introduce the hitherto neglected information channel between players and the environment into the paradigm of stochastic evolutionary games with a view to bringing forward the counterintuitive possibility of emergence and sustenance of cooperation on account of the noise in the channel. Our primary study is in the simplest non-trivial setting of two-state stochastically fluctuating resource harnessed by a large unstructured population of cooperators and defectors adopting either memory-1 strategies or reactive strategies while engaged in repeated two-player interactions. The effect of noisy information channel in enhancing the cooperation in reactive-strategied population is unprecedented. We find that the propensity of cooperation in the population is inversely related to the mutual information (normalized by the channel capacity) of the corresponding information channel.

Information is everywhere in nature which is very uncertain and unpredictable. But information, in itself, is a very ambiguous term. In this cursory write-up, we attempt to understand the formal meaning of information by quantifying uncertainty and discuss how it naturally appears in two core topics of classical physics -- classical mechanics and statistical mechanics. In the process, we witness how the concepts of the information theory render a unique viewpoint in physics. Except for the presentation of the material, the author claims no originality; however, the responsibility of every inadvertent error lies entirely with the author., Comment: Essentially the write-up is summarized lecture-notes of a course taken by UG and PG physics students of IIT Kanpur, India

While quite a few recent papers have explored game-resource feedback using the framework of evolutionary game theory, almost all the studies are confined to using time-continuous dynamical equations. Moreover, in such literature, the effect of ubiquitous chaos in the resulting eco-evolutionary dynamics is rather missing. Here, we present a deterministic eco-evolutionary discrete-time dynamics in generation-wise non-overlapping population of two types of harvesters, one harvesting at a faster rate than the other, consuming a self-renewing resource capable of showing chaotic dynamics. In the light of our finding that sometimes chaos is confined exclusively to either the dynamics of the resource or that of the consumer fractions, an interesting scenario is realized: The resource state can keep oscillating chaotically, and hence, it does not vanish to result in the tragedy of the commons, extinction of the resource due to selfish indiscriminate exploitation, and yet the consumer population, whose dynamics depends directly on the state of the resource, may end up being composed exclusively of defectors, i.e., high harvesters. This appears non-intuitive because it is well known that prevention of tragedy of the commons usually requires substantial cooperation to be present.

We present a systematic investigation of the quantum games, constructed using a novel repeated game protocol, when played repeatedly ad infinitum. We focus on establishing that such repeated games -- by virtue of inherent quantum-mechanical randomness -- can be mapped to the paradigm of stochastic games. Subsequently, using the setup of two-player--two-action games, we explore the pure reactive strategies belonging to the set of reactive strategies, whose support in the quantum games is no longer countably finite but rather non-denumerably infinite. We find that how two pure strategies fare against each other is crucially dependent on the discount factor (the probability of occurrence of every subsequent round) and how much entangled the quantum states of the players are. We contrast the results obtained with the corresponding results in the classical setup and find fundamental differences between them: e.g, when the underlying game is the prisoner's dilemma, in the quantum game setup, always-defect strategy can be beaten by the tit-for-tat strategy for high enough discount factor.

We generalize the Bush--Mosteller learning, the Roth--Erev learning, and the social learning to include mistakes such that the nonlinear replicator-mutator equation with either additive or multiplicative mutation is generated in an asymptotic limit. Subsequently, we exhaustively investigate the ubiquitous Rock-Paper-Scissors game for some analytically tractable motifs of mutation pattern. We consider both symmetric and asymmetric game interactions, and reveal that mistakes can some-times help the players learn. While the replicator-mutator flow exhibits rich dynamics that include limit cycles and chaotic orbits, it can control chaos as well to lead to rational Nash equilibrium outcome. Moreover, we also report an instance of hitherto unknown Hamiltonian structure of the replicator-mutator equation., Comment: 16 pages, 14 figures

Strategies for sustaining cooperation and preventing exploitation by selfish agents in repeated games have mostly been restricted to Markovian strategies where the response of an agent depends on the actions in the previous round. Such strategies are characterized by lack of learning. However, learning from accumulated evidence over time and using the evidence to dynamically update our response is a key feature of living organisms. Bayesian inference provides a framework for such evidence-based learning mechanisms. It is therefore imperative to understand how strategies based on Bayesian learning fare in repeated games with Markovian strategies. Here, we consider a scenario where the Bayesian player uses the accumulated evidence of the opponent's actions over several rounds to continuously update her belief about the reactive opponent's strategy. The Bayesian player can then act on her inferred belief in different ways. By studying repeated Prisoner's dilemma games with such Bayesian inferential strategies, both in infinite and finite populations, we identify the conditions under which such strategies can be evolutionarily stable. We find that a Bayesian strategy that is less altruistic than the inferred belief about the opponent's strategy can outperform a larger set of reactive strategies, whereas one that is more generous than the inferred belief is more successful when the benefit-to-cost ratio of mutual cooperation is high. Our analysis reveals how learning the opponent's strategy through Bayesian inference, as opposed to utility maximization, can be beneficial in the long run, in preventing exploitation and eventual invasion by reactive strategies., Comment: 13 pages, 9 figures

State-of-the-art offline Optical Character Recognition (OCR) frameworks perform poorly on semi-structured handwritten domain-specific documents due to their inability to localize and label form fields with domain-specific semantics. Existing techniques for semi-structured document analysis have primarily used datasets comprising invoices, purchase orders, receipts, and identity-card documents for benchmarking. In this work, we build the first semi-structured document analysis dataset in the legal domain by collecting a large number of First Information Report (FIR) documents from several police stations in India. This dataset, which we call the FIR dataset, is more challenging than most existing document analysis datasets, since it combines a wide variety of handwritten text with printed text. We also propose an end-to-end framework for offline processing of handwritten semi-structured documents, and benchmark it on our novel FIR dataset. Our framework used Encoder-Decoder architecture for localizing and labelling the form fields and for recognizing the handwritten content. The encoder consists of Faster-RCNN and Vision Transformers. Further the Transformer-based decoder architecture is trained with a domain-specific tokenizer. We also propose a post-correction method to handle recognition errors pertaining to the domain-specific terms. Our proposed framework achieves state-of-the-art results on the FIR dataset outperforming several existing models, Comment: This paper has been accepted in 17th International Conference on Document Analysis and Recognition(ICDAR) as an Oral presentation

In this paper, the replicator dynamics of the two-locus two-allele system under weak mutation and weak selection is investigated in a generation-wise non-overlapping unstructured population of individuals mating at random. Our main finding is that the dynamics is gradient-like when the point mutations at the two loci are independent. This is in stark contrast to the case of one-locus multi-allele where the existence gradient behaviour is contingent on a specific relationship between the mutation rates. When the mutations are not independent in the two-locus two-allele system, there is the possibility of non-convergent outcomes, like asymptotically stable oscillations, through either the Hopf bifurcation or the Neimark--Sacker bifurcation depending on the strength of the weak selection. The results can be straightforwardly extended for multi-locus two-allele systems., Comment: 10 pages, 1 figure

Discrete time crystals (DTCs) are emergent non-equilibrium phases of periodically driven many-body systems, with potential applications ranging from quantum computing to sensing and metrology. There has been significant recent interest in understanding mechanisms leading to DTC formation and a search for novel DTC phases beyond subharmonic entrainment. Here, we report observation of multiple DTC phases in a nanoelectromechanical system (NEMS) based on coupled graphene and silicon nitride membranes. We confirm the time-crystalline nature of these symmetry broken phases by establishing their many-body characters, long-range time and spatial order, and rigidity against parameter fluctuation or noise. Furthermore, we employ controlled mechanical strain to drive the transitions between phases with different symmetries, thereby mapping the emergent time-crystalline phase diagram. Overall, our work takes a step towards establishing time crystals as a system with complexity rivaling that of solid state crystals., Comment: 8 pages, 4 figures

Even though existence of non-convergent evolution of the states of populations in ecological and evolutionary contexts is an undeniable fact, insightful game-theoretic interpretations of such outcomes are scarce in the literature of evolutionary game theory. As a proof-of-concept, we tap into the information-theoretic concept of relative entropy in order to construct a game-theoretic interpretation for periodic orbits in a wide class of deterministic discrete-time evolutionary game dynamics, primarily investigating the two-player two-strategy case. Effectively, we present a consistent generalization of the evolutionarily stable strategy -- the cornerstone of the evolutionary game theory -- and aptly term the generalized concept: information stable orbit. The information stable orbit captures the essence of the evolutionarily stable strategy in that it compares the total payoff obtained against an evolving mutant with the total payoff that the mutant gets while playing against itself. Furthermore, we discuss the connection of the information stable orbit with the dynamical stability of the corresponding periodic orbit., Comment: 12 pages, 3 figures

The aim of this short note is that if $\{ a_{n}\}$ and $\{ b_{n}\}$ are two sequences of positive real numbers such that $a_{n}\to +\infty$ and $b_n$ satisfying the asymptotic formula $b_n\sim k\cdot a_{n}$, where $k>0$, then $\lim\limits_{n\to\infty}\left(1+\frac{1}{a_{n}}\right)^{b_{n}}= e^{k}$., Comment: 1 page

The tragedy of the commons (TOC) is a ubiquitous social dilemma witnessed in interactions between a population of living entities and shared resources available to them: The individuals in the population tend to selfishly overexploit a common resource as it is arguably the rational choice, or in case of non-human beings, it may be an evolutionarily uninvadable action. How to avert the TOC is a significant problem related to the conservation of resources. It is not hard to envisage situations where the resource could be self-renewing and the size of the population may be dependent on the state of the resource through the fractions of the population employing different exploitation rates. If the self-renewal rate of the resource lies between the maximum and the minimum exploitation rates, it is not a priori obvious under what conditions the TOC can be averted. In this paper, we address this question analytically and numerically using the setup of an evolutionary game theoretical replicator equation that models the Darwinian tenet of natural selection. Through the replicator equation, while we investigate how a population of replicators exploit the shared resource, the latter's dynamical feedback on the former is also not ignored. We also present a transparent bottom-up derivation of the game-resource feedback model to facilitate future studies on the stochastic effects on the findings presented herein., Comment: 10 pages, 3 figures

We investigate the scaling form of appropriate time-scales extracted from time-dependent correlation functions in rotating, turbulent flows. In particular, we obtain precise estimates of the dynamic exponents $z_p$, associated with the time-scales, and their relation with the more commonly measured equal-time exponents $\zeta_p$. These theoretical predictions, obtained by using the multifractal formalism, are validated through extensive numerical simulations of a shell model for such rotating flows., Comment: 7 pages, 2 figures

A completely non-generous and reciprocal population of players can create a robust cooperating state that cannot be invaded by always defecting free riders if the interactions among players are repeated for long enough. However, strict non-generosity and strict reciprocity are ideal concepts, and may not even be desirable sometimes. Therefore, to what extent generosity or non-reciprocity can be allowed while still not be swamped by the mutants, is a natural question. In this paper, we not only ask this question but furthermore ask how generosity comparatively fares against non-reciprocity in this context. For mathematical concreteness, we work within the framework of multiplayer repeated prisoner's dilemma game with reactive strategies in a finite and an infinite population; and explore the aforementioned questions through the effects of the benefit to cost ratio, the interaction group size, and the population size., Comment: 10 pages, 5 figures

The tragedy of the commons (TOC) is an unfortunate situation where a shared resource is exhausted due to uncontrolled exploitation by the selfish individuals of a population. Recently, the paradigmatic replicator equation has been used in conjunction with a phenomenological equation for the state of the shared resource to gain insight into the influence of the games on the TOC. The replicator equation, by construction, models a fixed infinite population undergoing microevolution. Thus, it is unable to capture any effect of the population growth and the carrying capacity of the population although the TOC is expected to be dependent on the size of the population. Therefore, in this paper, we present a mathematical framework that incorporates the density dependent payoffs and the logistic growth of the population in the eco-evolutionary dynamics modelling the game-resource feedback. We discover a bistability in the dynamics: a finite carrying capacity can either avert or cause the TOC depending on the initial states of the resource and the initial fraction of cooperators. In fact, depending on the type of strategic game-theoretic interaction, a finite carrying capacity can either avert or cause the TOC when it is exactly the opposite for the corresponding case with infinite carrying capacity., Comment: 12 pages, 5 figures, accepted in Physical Review E

Populations composed of a collection of subpopulations (demes) with random migration between them are quite common occurrences. The emergence and sustenance of cooperation in such a population is a highly researched topic in the evolutionary game theory. If the individuals in every deme are considered to be either cooperators or defectors, the migration dilemma can be envisaged: The cooperators would not want to migrate to a defector-rich deme as they fear of facing exploitation; but without migration, cooperation can not be established throughout the network of demes. With a view to studying the aforementioned scenario, in this paper, we set up a theoretical model consisting of a coupled map lattice of replicator maps based on two-player--two-strategy games. The replicator map considered is capable of showing a variety of evolutionary outcomes, like convergent (fixed point) outcomes and nonconvergent (periodic and chaotic) outcomes. Furthermore, this coupled network of the replicator maps undergoes the phenomenon of amplitude death leading to non-oscillatory stable synchronized states. We specifically explore the effect of (i) the nature of coupling that models migration between the maps, (ii) the heterogenous demes (in the sense that not all the demes have same game being played by the individuals), (iii) the degree of the network, and (iv) the cost associated with the migration. In the course of investigation, we are intrigued by the effectiveness of the random migration in sustaining a uniform cooperator fraction across a population irrespective of the details of the replicator dynamics and the interaction among the demes.

In evolutionary game theory, it is customary to be partial to the dynamical models possessing fixed points so that they may be understood as the attainment of evolutionary stability, and hence, Nash equilibrium. Any show of periodic or chaotic solution is many a time perceived as a shortcoming of the corresponding game dynamic because (Nash) equilibrium play is supposed to be robust and persistent behaviour, and any other behaviour in nature is deemed transient. Consequently, there is a lack of attempt to connect the non-fixed point solutions with the game theoretic concepts. Here we provide a way to render game theoretic meaning to periodic solutions. To this end, we consider a replicator map that models Darwinian selection mechanism in unstructured infinite-sized population whose individuals reproduce asexually forming non-overlapping generations. This is one of the simplest evolutionary game dynamic that exhibits periodic solutions giving way to chaotic solutions (as parameters related to reproductive fitness change) and also obeys the folk theorems connecting fixed point solutions with Nash equilibrium. Interestingly, we find that a modified Darwinian fitness -- termed heterogeneity payoff -- in the corresponding population game must be put forward as (conventional) fitness times the probability that two arbitrarily chosen individuals of the population adopt two different strategies. The evolutionary dynamics proceeds as if the individuals optimize the heterogeneity payoff to reach an evolutionarily stable orbit, should it exist. We rigorously prove that a locally asymptotically stable period orbit must be heterogeneity stable orbit -- a generalization of evolutionarily stable state.

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., Comment: 11 pages, 6 figures

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.

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.

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., Comment: 12 pages, 8 figures

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.

The complexities involved in modelling the transmission dynamics of COVID-19 has been a roadblock in achieving predictability in the spread and containment of the disease. In addition to understanding the modes of transmission, the effectiveness of the mitigation methods also needs to be built into any effective model for making such predictions. We show that such complexities can be circumvented by appealing to scaling principles which lead to the emergence of universality in the transmission dynamics of the disease. The ensuing data collapse renders the transmission dynamics largely independent of geopolitical variations, the effectiveness of various mitigation strategies, population demographics, etc. We propose a simple two-parameter model -- the Blue Sky model -- and show that one class of transmission dynamics can be explained by a solution that lives at the edge of a blue sky bifurcation. In addition, the data collapse leads to an enhanced degree of predictability in the disease spread for several geographical scales which can also be realized in a model-independent manner as we show using a deep neural network. The methodology adopted in this work can potentially be applied to the transmission of other infectious diseases and new universality classes may be found. The predictability in transmission dynamics and the simplicity of our methodology can help in building policies for exit strategies and mitigation methods during a pandemic., Comment: 16 pages and 6 figures (version corresponding to the published version)

In connection to the two fascinating constants $e$ and $\pi$, there are many beautiful visual proofs to the inequality $\pi^{e}

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., Comment: Accepted (Physica D: Nonlinear Phenomena)

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., Comment: 18 pages with 7 figures. We add a figure on scale-by-scale intermittency and supporting evidence for our results, rearrange the text for better readability, and add new references

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., Comment: Accepted in Eur. Phys. J. B

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.

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., Comment: 11 pages, 7 figures, accepted in Physics of FLuids

Two meromorphic functions $f$ and $g$ are said to share the set $S\subset \mathbb{C}\cup\{\infty\}$ with weight $l\in\mathbb{N}\cup\{0\}\cup\{\infty\}$, if $E_{f}(S,l)=E_{g}(S,l)$ where $$E_{f}(S,l)=\bigcup\limits_{a \in S}\{(z,t) \in \mathbb{C}\times\mathbb{N}~ |~ f(z)=a ~\text{with~ multiplicity}~ p\},$$ where $t=p$ if $p\leq l$ and $t=p+1$ if $p>l$. In this paper, we improve and supplement the result of L. W. Liao and C. C. Yang (On the cardinality of the unique range sets for meromorphic and entire functions, Indian J. Pure appl. Math., 31 (2000), no. 4, 431-440) by showing that there exist a finite set $S$ with cardinality $\geq 13$ such that $E_{f}(S,1)=E_{g}(S,1)$ implies $f\equiv g$., Comment: 10 pages. arXiv admin note: text overlap with arXiv:1711.08808, arXiv:1608.02125

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., Comment: 11 pages, 7 figure-panels

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., Comment: Accepted in Chaos

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 $\epsilon_\omega\exp(-C(k/k_d)^2)$ and $C\epsilon_\omega^{2/3}k^{-1}\exp(-C(k/k_d)^2)$ respectively; for this 2D flow, $\epsilon_\omega$ 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: $E(k)=C\epsilon_\omega^{2/3}k^{-3}\exp(-C(k/k_d)^2)$. This model of the energy spectrum is based on wavenumber-dependent enstrophy flux, and it deviates significantly from power law energy spectrum reported earlier.

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., Comment: 9 pages, 4 figures

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., Comment: 11 pages, 1 figure, Accepted in MNRAS

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: $B(t)\propto|J_0(\omega_\gamma t)|$ for a two-level system (atom or molecule) in thermal radiation field. Here $\omega_\gamma$ 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 $\omega_\gamma$---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., Comment: 12 pages and 5 figures. Major revision of the 2nd version. Accepted for publication in Journal of Statistical Mechanics: Theory and Experiment

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., Comment: Published in Chaos. This version has some typos corrected

In this paper we investigate the convection phenomenon in the intracluster medium (the weakly-collisional magnetized inhomogeneous plasma permeating galaxy clusters) where the concentration gradient of the Helium ions is not ignorable. To this end, we build upon the general machinery employed to study the salt finger instability found in the oceans. The salt finger instability is a form of double diffusive convection where the diffusions of two physical quantities---heat and salt concentrations---occur with different diffusion rates. The analogous instability in the intracluster medium may result owing to the magnetic field mediated anisotropic diffusions of the heat and the Helium ions (in the sea of the Hydrogen ions and the free electrons). These two diffusions have inherently different diffusion rates. Hence the convection caused by the onset of this instability is an example of double diffusive convection in the astrophysical settings. A consequence of this instability is the formation of the vertical filamentary structures having more concentration of the Helium ions with respect to the immediate neighbourhoods of the filaments. We term these structures as Helium fingers in analogy with the salt fingers found in the case of the salt finger instability. Here we show that the width of a Helium finger scales as one-fourth power of the radius of the inner region of the intracluster medium in the supercritical regime. We also determine the explicit mathematical expression of the criterion for the onset of the heat-flux-driven buoyancy instability modified by the presence of inhomogeneously distributed Helium ions., Comment: 7 pages, 1 figure

Synchronization is the process of achieving identical dynamics among coupled identical units. If the units are different from each other, their dynamics cannot become identical; yet, after transients, there may emerge a functional relationship between them -- a phenomenon termed "generalized synchronization." Here, we show that the concept of transient uncoupling, recently introduced for synchronizing identical units, also supports generalized synchronization among nonidentical chaotic units. Generalized synchronization can be achieved by transient uncoupling even when it is impossible by regular coupling. We furthermore demonstrate that transient uncoupling stabilizes synchronization in the presence of common noise. Transient uncoupling works best if the units stay uncoupled whenever the driven orbit visits regions that are locally diverging in its phase space. Thus, to select a favorable uncoupling region, we propose an intuitive method that measures the local divergence at the phase points of the driven unit's trajectory by linearizing the flow and subsequently suppresses the divergence by uncoupling.

Nonlinear time series analysis has been widely used to search for signatures of low-dimensional chaos in light curves emanating from astrophysical bodies. A particularly popular example is the microquasar GRS 1915+105, whose irregular but systematic X-ray variability has been well studied using data acquired by the Rossi X-ray Timing Explorer. With a view to building simpler models of X-ray variability, attempts have been made to classify the light curves of GRS 1915+105 as chaotic or stochastic. Contrary to some of the earlier suggestions, after careful analysis, we find no evidence for chaos or determinism in any of the GRS 1915+105 classes. The dearth of long and stationary data sets representing all the different variability classes of GRS 1915+105 makes it a poor candidate for analysis using nonlinear time series techniques. We conclude that either very exhaustive data analysis with sufficiently long and stationary light curves should be performed, keeping all the pitfalls of nonlinear time series analysis in mind, or alternative schemes of classifying the light curves should be adopted. The generic limitations of the techniques that we point out in the context of GRS 1915+105 affect all similar investigations of light curves from other astrophysical sources., Comment: 14 pages, 6 figures; minor corrections; slightly shorter version published in ApJ

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., Comment: 12 pages, 1 figure

We use the machinery usually employed for studying the onset of Rayleigh--B\'enard convection in hydro- and magnetohydro-dynamic settings to address the onset of convection induced by the magnetothermal instability and the heat-flux-buoyancy-driven-instability in the weakly-collisional magnetized plasma permeating the intracluster medium. Since most of the related numerical simulations consider the plasma being bounded between two `plates' on which boundary conditions are specified, our strategy provides a framework that could enable a more direct connection between analytical and numerical studies. We derive the conditions for the onset of these instabilities considering the effects of induced magnetic tension resulting from a finite plasma beta. We provide expressions for the Rayleigh number in terms of the wave vector associated with a given mode, which allow us to characterize the modes that are first to become unstable. For both the heat-flux-buoyancy-driven-instability and the magnetothermal instability, oscillatory marginal stable states are possible., Comment: 10 pages, 1 figure

We show that the equivalent linearization technique, when used properly, enables us to calculate frequency corrections of weakly nonlinear oscillators beyond the first order in nonlinearity. We illustrate the method by applying it to the conservative anharmonic oscillators and the nonconservative van der Pol oscillator that are respectively paradigmatic systems for modeling center-type oscillatory states and limit cycle type oscillatory states. The choice of these systems is also prompted by the fact that first order frequency corrections may vanish for both these types of oscillators, thereby rendering the calculation of the higher order corrections rather important. The method presented herein is very general in nature and, hence, in principle applicable to any arbitrary periodic oscillator., Comment: 4 pages, Accepted in EPJ B

We have performed systemmatic local linear stability analysis on a radially stratified infinite self-gravitating cylinder of rotating plasma under the influence of magnetic field. In order to render the system analytically tractable, we have focussed solely on the axisymmetric modes of perturbations. Using cylindrical coordinate system, we have derived the critical linear mass density of a non-rotating filament required for gravitational collapse to ensue in the presence of azimuthal magnetic field. Moreover, for such filaments threaded by axial magnetic field, we show that the growth rates of the modes having non-zero radial wavenumber are reduced more strongly by the magnetic field than that of the modes having zero radial wavenumber. More importantly, our study contributes to the understanding of the stability property of rotating astrophysical filaments that are more often than not influenced by magnetic fields. In addition to complementing many relevant numerical studies reported the literature, our results on filaments under the influence of magnetic field generalize some of the very recent analytical works (e.g.,~\citet{jog2014}, etc.). For example, here we prove that even a weak magnetic field can play a dominant role in determining stability of the filament when the rotation timescale is larger than the free fall timescale. A filamentary structure with faster rotation is, however, comparatively more stable for the same magnetic field. The results reported herein, due to strong locality assumption, are strictly valid for the modes for which one can ignore the radial variations in the density and the magnetic field profiles., Comment: Accepted in MNRAS

In this paper, we show how to use canonical perturbation theory for dissipative dynamical systems capable of showing limit cycle oscillations. Thus, our work surmounts the hitherto perceived barrier for canonical perturbation theory that it can be applied only to a class of conservative systems, viz.,~Hamiltonian systems. In the process, we also find Hamiltonian structure for an important subset of Li\'enard system--- a paradigmatic system for modeling isolated and asymptotic oscillatory state. We discuss the possibility of extending our method to encompass even wider range of non-conservative systems., Comment: 8 Pages, Published in Physical Review E (2015)

Finding conditions that support synchronization is a fertile and active area of research with applications across multiple disciplines. Here we present and analyze a scheme for synchronizing chaotic dynamical systems by transiently uncoupling them. Specifically, systems coupled only in a fraction of their state space may synchronize even if fully coupled they do not. Although, for many standard systems, coupling strengths need to be bounded to ensure synchrony, transient uncoupling removes this bound and thus enables synchronization in an infinite range of effective coupling strengths. The presented coupling scheme thus opens up the possibility to induce synchrony in (biological or technical) systems whose parameters are fixed and cannot be modified continuously., Comment: 5 pages, 6 figures

Over the last decade, substantial efforts have been devoted to understanding the stability properties, transport phenomena, and long-term evolution of weakly-collisional, magnetized plasmas which are stratified in temperature. These studies have improved our understanding of the physics governing the intra-cluster medium (ICM), but assumed that ICM is a homogeneous. This, however, might not be a good approximation if heavy elements sediment in the inner region of the galaxy cluster. In this paper, we analyze the stability of a weakly-collisional, magnetized plane-parallel atmosphere which is stratified in both temperature and composition. This allows us to discuss for the first time the dynamics of weakly-collisional environments where heat conduction, momentum transport, and ion-diffusion are anisotropic with respect to the direction of the magnetic field. We show that, depending on the relative signs and magnitudes of the gradients in the temperature and the mean molecular weight, the plasma can be subject to a wide variety of unstable modes which include modifications to the magnetothermal instability (MTI), the heat-flux-driven buoyancy instability (HBI), and overstable gravity modes previously studied in homogeneous media. We discuss the astrophysical implications of our findings for a representative galaxy cluster where helium has sedimented. Our findings suggest that the core insulation that results from the magnetic field configurations that arise as a natural consequence of the HBI, which would be MTI stable in a homogeneous medium, could be alleviated if the mean molecular weight gradient is steep enough, i.e., $(\nabla \mu)/\mu > (\nabla T)/T$. This study constitutes a first step toward understanding the interaction between magnetic turbulence and the diffusion of heavy elements in the ICM. (abridged), Comment: 16 pages, 7 figures. This article supersedes arXiv:1111.3372 (5 pages, 3 figures). The present version of this article is published in ApJ

We discover presence of a hitherto unexplored type of resonance in a parametrically excited Van der Pol oscillator. The oscillator also possesses a state of anti-resonance. In the weak non-linear limit, we explain how to practically get a complete picture of different states of limiting oscillations present in the oscillator when the non-linear term therein is excited by an arbitrary $2\pi$ periodic function of time. We also illustrate how two such oscillators can be coupled to behave like a two-state switch allowing an sharp change of value of amplitude for stable oscillations from one constant to another., Comment: Constructive comments are most welcome