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A significant advantage of this framework is its flexibility to adapt to various problems with minimal changes in coding. Also, once the NN is trained, it gives us an analytical representation of the solution at any desired instant in time outside the initial discretization. Learning stiff ODEs opens up possibilities of using X-TFC in applications with large time ranges, such as chemical dynamics in energy conversion, nuclear dynamics systems, life sciences, and environmental engineering.We study thermalization of weakly nonintegrable nonlinear unitary lattice dynamics. We identify two distinct thermalization regimes close to the integrable limits of either linear dynamics or disconnected lattice dynamics. For weak nonlinearity, the almost conserved actions correspond to extended observables which are coupled into a long-range network. For weakly connected lattices, the corresponding local observables are coupled into a short-range network. We compute the evolution of the variance σ ( T ) of finite time average distributions for extended and local observables. We extract the ergodization time scale T which marks the onset of thermalization, and determine the type of network through the subsequent decay of σ ( T ). We use the complementary analysis of Lyapunov spectra [M. Malishava and S. Flach, Phys. Rev. Lett. 128, 134102 (2022)] and compare the Lyapunov time T with T. We characterize the spatial properties of the tangent vector and arrive at a complete classification picture of weakly nonintegrable macroscopic thermalization dynamics.The slogan "nobody is safe until everybody is safe" is a dictum to raise awareness that in an interconnected world, pandemics, such as COVID-19, require a global approach. Motivated by the ongoing COVID-19 pandemic, we model here the spread of a virus in interconnected communities and explore different vaccination scenarios, assuming that the efficacy of the vaccination wanes over time. We start with susceptible populations and consider a susceptible-vaccinated-infected-recovered model with unvaccinated ("Bronze"), moderately vaccinated ("Silver"), and very-well-vaccinated ("Gold") communities, connected through different types of networks via a diffusive linear coupling for local spreading. We show that when considering interactions in "Bronze"-"Gold" and "Bronze"-"Silver" communities, the "Bronze" community is driving an increase in infections in the "Silver" and "Gold" communities. This shows a detrimental, unidirectional effect of non-vaccinated to vaccinated communities. Regarding the interactions between a network can increase significantly the rate of infected population in other communities. This reveals the necessity of a global effort to facilitate access to vaccines for all communities.Active soft materials exhibit various dynamics ranging from boat pulsation to thin membrane deformation. In the present work, in situ prepared ethanol-containing chitosan gels propel in continuous and intermittent motion. The active life of the organic material loaded to the constant fuel level follows a linear scaling, and its maximal velocity and projection area decrease steeply with chitosan concentration. A thin propelling platelet forms at low polymer content, leading to the suppression of intermittent motion. Moreover, the fast accelerating thin gels can split into a crescent and circular-like shape or fission into multiple asymmetric fragments.We consider a variant of the mean-field model of coupled phase oscillators with uniform distribution of natural frequencies. By establishing correlations between the quenched disorder of intrinsic frequencies and coupling strength with both in- and out-coupling heterogeneities, we reveal a generic criterion for the onset of partial locking that takes place in a domain with the coexistence of phase-locked oscillators and drifters. The critical points manifesting the instability of the stationary states are obtained analytically. In particular, the bifurcation mechanism of the equilibrium states is uncovered by the use of frequency-dependent version of the Ott-Antonsen reduction consistently with the analysis based on the self-consistent approach. We demonstrate that both the manner of coupling heterogeneity and correlation exponent have influence on the emergent patterns of partial locking. Our research could find applicability in better understanding the phase transitions and related collective phenomena involving synchronization control in networked systems.Public goods games are widely used to model social dilemmas involving multiple agents. Though defection is the only rational choice for an individual in a public goods game, cooperative behavior is observed in a variety of social dilemmas, which is the subject of our research. Punishing defectors has been shown to be an effective mechanism for promoting cooperation, but it relies on the third-party umpire being fair. In this article, an umpire intervention model with corruption is proposed to explore the impact of corruption on punishment mechanisms. [https://www.selleckchem.com/products/e7766-diammonium-salt.html E7766 purchase] In our model, players and umpires operate in a multilayer network. The players play public goods games, which are overseen by umpires. Fair umpires punish defectors, whereas corrupt umpires take bribes from defectors rather than meting out a punishment. We separately explore the effects of the fraction of fair umpires ρ, the spatial distribution, and the fine cost α and bribe cost β. Our Monte Carlo simulation shows that the above factors have a significant impact on cooperation. Intervention by an umpire always improves social efficiency, even for an entirely corrupt system. Moreover, relatively developed systems can resist corruption. Staggered and centralized distributions always have opposite effects on cooperative behavior, and these effects depend on ρ and r. We also find that whether cooperators fully occupy the player layer depends only on whether β reaches a certain threshold.Contributions of various natural and anthropogenic factors to trends of surface air temperatures at different latitudes of the Northern and Southern hemispheres on various temporal horizons are estimated from climate data since the 19th century in empirical autoregressive models. Along with anthropogenic forcing, we assess the impact of several natural climate modes including Atlantic Multidecadal Oscillation, El-Nino/Southern Oscillation, Interdecadal Pacific Oscillation, Pacific Decadal Oscillation, and Antarctic Oscillation. On relatively short intervals of the length of two or three decades, contributions of climate variability modes are considerable and comparable to the contributions of greenhouse gases and even exceed the latter. On longer intervals of about half a century and greater, the contributions of greenhouse gases dominate at all latitudinal belts including polar, middle, and tropical ones.Previous studies on network robustness mainly concentrated on hub node failures with fully known network structure information. However, hub nodes are often well protected and not accessible to damage or malfunction in a real-world networked system. In addition, one can only gain insight into limited network connectivity knowledge due to large-scale properties and dynamic changes of the network itself. In particular, two different aggression patterns are present in a network attack memory based attack, in which failed nodes are not attacked again, or non-memory based attack; that is, nodes can be repeatedly attacked. Inspired by these motivations, we propose an attack pattern with and without memory based on randomly choosing n non-hub nodes with known connectivity information. We use a network system with the Poisson and power-law degree distribution to study the network robustness after applying two failure strategies of non-hub nodes. Additionally, the critical threshold 1 - p and the size of the giant component S are determined for a network configuration model with an arbitrary degree distribution. The results indicate that the system undergoes a continuous second-order phase transition subject to the above attack strategies. We find that 1 - p gradually tends to be stable after increasing rapidly with n. Moreover, the failure of non-hub nodes with a higher degree is more destructive to the network system and makes it more vulnerable. Furthermore, from comparing the attack strategies with and without memory, the results highlight that the system shows better robustness under a non-memory based attack relative to memory based attacks for n &gt; 1. Attacks with memory can block the system's connectivity more efficiently, which has potential applications in real-world systems. Our model sheds light on network resilience under memory and non-memory based attacks with limited information attacks and provides valuable insights into designing robust real-world systems.We propose a definition of the asymptotic phase for quantum nonlinear oscillators from the viewpoint of the Koopman operator theory. The asymptotic phase is a fundamental quantity for the analysis of classical limit-cycle oscillators, but it has not been defined explicitly for quantum nonlinear oscillators. In this study, we define the asymptotic phase for quantum oscillatory systems by using the eigenoperator of the backward Liouville operator associated with the fundamental oscillation frequency. By using the quantum van der Pol oscillator with a Kerr effect as an example, we illustrate that the proposed asymptotic phase appropriately yields isochronous phase values in both semiclassical and strong quantum regimes.In this work, we consider the nonparametric estimation problem of the drift function of stochastic differential equations driven by the α-stable Lévy process. We first optimize the Kullback-Leibler divergence between the path probabilities of two stochastic differential equations with different drift functions. We then construct the variational formula based on the stationary Fokker-Planck equation using the Lagrangian multiplier. Moreover, we apply the empirical distribution to replace the stationary density, combining it with the data information, and we present the estimator of the drift function from the perspective of the process. In the numerical experiment, we investigate the effect of the different amounts of data and different α values. The experimental results demonstrate that the estimation result of the drift function is related to both and that the exact drift function agrees well with the estimated result. The estimation result will be better when the amount of data increases, and the estimation result is also better when the α value increases.Recently, extracting data-driven governing laws of dynamical systems through deep learning frameworks has gained much attention in various fields. Moreover, a growing amount of research work tends to transfer deterministic dynamical systems to stochastic dynamical systems, especially those driven by non-Gaussian multiplicative noise. However, many log-likelihood based algorithms that work well for Gaussian cases cannot be directly extended to non-Gaussian scenarios, which could have high errors and low convergence issues. In this work, we overcome some of these challenges and identify stochastic dynamical systems driven by α-stable Lévy noise from only random pairwise data. Our innovations include (1) designing a deep learning approach to learn both drift and diffusion coefficients for Lévy induced noise with α across all values, (2) learning complex multiplicative noise without restrictions on small noise intensity, and (3) proposing an end-to-end complete framework for stochastic system identification under a general input data assumption, that is, an α-stable random variable.