Weakly coupled two slow- two fast systems, folded node and mixed mode oscillationsM

We study Mixed Mode Oscillations (MMOs) in systems of two weakly coupled slow/fast oscillators. We focus on the existence and properties of a folded singularity called FSN II that allows the emergence of MMOs in the presence of a suitable global return mechanism. As FSN II corresponds to a transcritical bifurcation for a desingularized reduced system, we prove that, under certain non-degeneracy conditions, such a transcritical bifurcation exists. We then apply this result to the case of two coupled systems of FitzHugh- Nagumo type. This leads to a non trivial condition on the coupling that enables the existence of MMOs

Emergence of Cooperation in Non-scale-free Networks

Evolutionary game theory is one of the key paradigms behind many scientific disciplines from science to engineering. Previous studies proposed a strategy updating mechanism, which successfully demonstrated that the scale-free network can provide a framework for the emergence of cooperation. Instead, individuals in random graphs and small-world networks do not favor cooperation under this updating rule. However, a recent empirical result shows the heterogeneous networks do not promote cooperation when humans play a Prisoner's Dilemma. In this paper, we propose a strategy updating rule with payoff memory. We observe that the random graphs and small-world networks can provide even better frameworks for cooperation than the scale-free networks in this scenario. Our observations suggest that the degree heterogeneity may be neither a sufficient condition nor a necessary condition for the widespread cooperation in complex networks. Also, the topological structures are not sufficed to determine the level of cooperation in complex networks.Comment: 6 pages, 5 figure

Fence-sitters Protect Cooperation in Complex Networks

Evolutionary game theory is one of the key paradigms behind many scientific disciplines from science to engineering. In complex networks, because of the difficulty of formulating the replicator dynamics, most of previous studies are confined to a numerical level. In this paper, we introduce a vectorial formulation to derive three classes of individuals' payoff analytically. The three classes are pure cooperators, pure defectors, and fence-sitters. Here, fence-sitters are the individuals who change their strategies at least once in the strategy evolutionary process. As a general approach, our vectorial formalization can be applied to all the two-strategies games. To clarify the function of the fence-sitters, we define a parameter, payoff memory, as the number of rounds that the individuals' payoffs are aggregated. We observe that the payoff memory can control the fence-sitters' effects and the level of cooperation efficiently. Our results indicate that the fence-sitters' role is nontrivial in the complex topologies, which protects cooperation in an indirect way. Our results may provide a better understanding of the composition of cooperators in a circumstance where the temptation to defect is larger.Comment: an article with 6 pages, 3 figure

On a coupled time-dependent SIR models fitting with New York and New-Jersey states COVID-19 data

This article describes a simple Susceptible Infected Recovered (SIR) model fitting with COVID-19 data for the month of march 2020 in New York (NY) state. The model is a classical SIR, but is non-autonomous; the rate of susceptible people becoming infected is adjusted over time in order to fit the available data. The death rate is also secondarily adjusted. Our fitting is made under the assumption that due to limiting number of tests, a large part of the infected population has not been tested positive. In the last part, we extend the model to take into account the daily fluxes between New Jersey (NJ) and NY states and fit the data for both states. Our simple model fits the available data, and illustrates typical dynamics of the disease: exponential increase, apex and decrease. The model highlights a decrease in the transmission rate over the period which gives a quantitative illustration about how lockdown policies reduce the spread of the pandemic. The coupled model with NY and NJ states shows a wave in NJ following the NY wave, illustrating the mechanism of spread from one attractive hot spot to its neighbor.

A method to discriminate between localized and chaotic quantum systems

We derive a criterion that distinguishes whether a generic isolated quantum system initially set out of equilibrium can be considered as localized close to its initial state, or chaotic. Our approach considers the time evolution in the Lanczos basis, which maps the system's dynamics onto that of a particle moving in a one-dimensional lattice where both the energy in the lattice sites and the tunneling from one lattice site to the next are inhomogeneous. We infer a criterion that allows distinguishing localized from chaotic systems. This criterion involves the coupling strengths between Lanczos states and their expectation energy fluctuations. We verify its validity by inspecting three cases, corresponding to Anderson localization as a function of dimension, the out-of-equilibrium dynamics of a many-body dipolar spin system, and integrable systems. We finally show that our approach provides a justification for the Wigner surmise and the eigenstate thermalization hypothesis, which have both been proposed to characterize quantum chaotic systems. Indeed, our criterion for a system to be chaotic implies the level repulsion (also known as spectral rigidity) of eigenenergies, which is characteristic of the Wigner-Dyson distribution; and we also demonstrate that in the chaotic regime, the expectation value of any local observable only weakly varies as a function of eigenstates. Our demonstration allows to define the class of operators to which the eigenstate thermalization applies, as the ones that connect states that are coupled at weak order by the Hamiltonian.Comment: 15 pages, 6 figure