Chaos: An Interdisciplinary Journal of Nonlinear Science

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ISSN / EISSN : 1054-1500 / 1089-7682
Published by: AIP Publishing (10.1063)
Total articles ≅ 6,424
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Chaos: An Interdisciplinary Journal of Nonlinear Science, Volume 32; https://doi.org/10.1063/5.0080843

Abstract:
In network control theory, driving all the nodes in the Feedback Vertex Set (FVS) by node-state override forces the network into one of its attractors (long-term dynamic behaviors). The FVS is often composed of more nodes than can be realistically manipulated in a system; for example, only up to three nodes can be controlled in intracellular networks, while their FVS may contain more than 10 nodes. Thus, we developed an approach to rank subsets of the FVS on Boolean models of intracellular networks using topological, dynamics-independent measures. We investigated the use of seven topological prediction measures sorted into three categories—centrality measures, propagation measures, and cycle-based measures. Using each measure, every subset was ranked and then evaluated against two dynamics-based metrics that measure the ability of interventions to drive the system toward or away from its attractors: To Control and Away Control. After examining an array of biological networks, we found that the FVS subsets that ranked in the top according to the propagation metrics can most effectively control the network. This result was independently corroborated on a second array of different Boolean models of biological networks. Consequently, overriding the entire FVS is not required to drive a biological network to one of its attractors, and this method provides a way to reliably identify effective FVS subsets without the knowledge of the network dynamics.
Peter T. Haugen, Andrew D. P. Smith,
Chaos: An Interdisciplinary Journal of Nonlinear Science, Volume 32; https://doi.org/10.1063/5.0076147

Abstract:
We analyze the rotational dynamics of six magnetic dipoles of identical strength at the vertices of a regular hexagon with a variable-strength dipole in the center. The seven dipoles spin freely about fixed axes that are perpendicular to the plane of the hexagon, with their dipole moments directed parallel to the plane. Equilibrium dipole orientations are calculated as a function of the relative strength of the central dipole. Small-amplitude perturbations about these equilibrium states are calculated in the absence of friction and are compared with analytical results in the limit of zero and infinite central dipole strength. Normal modes and frequencies are presented. Bifurcations are seen at two critical values of the central dipole strength, with bistability between these values.
Suchithra K. S., Gopalakrishnan E. A., ,
Chaos: An Interdisciplinary Journal of Nonlinear Science, Volume 32; https://doi.org/10.1063/5.0093450

Abstract:
Renewable energy sources in modern power systems pose a serious challenge to the power system stability in the presence of stochastic fluctuations. Many efforts have been made to assess power system stability from the viewpoint of the bifurcation theory. However, these studies have not covered the dynamic evolution of renewable energy integrated, non-autonomous power systems. Here, we numerically explore the transition phenomena exhibited by a non-autonomous stochastic bi-stable power system oscillator model. We use additive white Gaussian noise to model the stochasticity in power systems. We observe that the delay in the transition observed for the variation of mechanical power as a function of time shows significant variations in the presence of noise. We identify that if the angular velocity approaches the noise floor before crossing the unstable manifold, the rate at which the parameter evolves has no control over the transition characteristics. In such cases, the response of the system is purely controlled by the noise, and the system undergoes noise-induced transitions to limit-cycle oscillations. Furthermore, we employ an emergency control strategy to maintain the stable non-oscillatory state once the system has crossed the quasi-static bifurcation point. We demonstrate an effective control strategy that opens a possibility of maintaining the stability of electric utility that operates near the physical limits.
, , A. V. Nguyen,
Chaos: An Interdisciplinary Journal of Nonlinear Science, Volume 32; https://doi.org/10.1063/5.0066066

Abstract:
Many natural systems exhibit chaotic behavior, including the weather, hydrology, neuroscience, and population dynamics. Although many chaotic systems can be described by relatively simple dynamical equations, characterizing these systems can be challenging due to sensitivity to initial conditions and difficulties in differentiating chaotic behavior from noise. Ideally, one wishes to find a parsimonious set of equations that describe a dynamical system. However, model selection is more challenging when only a subset of the variables are experimentally accessible. Manifold learning methods using time-delay embeddings can successfully reconstruct the underlying structure of the system from data with hidden variables, but not the equations. Recent work in sparse-optimization based model selection has enabled model discovery given a library of possible terms, but regression-based methods require measurements of all state variables. We present a method combining variational annealing—a technique previously used for parameter estimation in chaotic systems with hidden variables—with sparse-optimization methods to perform model identification for chaotic systems with unmeasured variables. We applied the method to ground-truth time-series simulated from the classic Lorenz system and experimental data from an electrical circuit with Lorenz-system like behavior. In both cases, we successfully recover the expected equations with two measured and one hidden variable. Application to simulated data from the Colpitts oscillator demonstrates successful model selection of terms within nonlinear functions. We discuss the robustness of our method to varying noise.
Chaos: An Interdisciplinary Journal of Nonlinear Science, Volume 32; https://doi.org/10.1063/5.0090150

Abstract:
We investigate the possibility of avoiding the escape of chaotic scattering trajectories in two-degree-of-freedom Hamiltonian systems. We develop a continuous control technique based on the introduction of coupling forces between the chaotic trajectories and some periodic orbits of the system. The main results are shown through numerical simulations, which confirm that all trajectories starting near the stable manifold of the chaotic saddle can be controlled. We also show that it is possible to jump between different unstable periodic orbits until reaching a stable periodic orbit belonging to a Kolmogorov–Arnold–Moser island.
, Flavio H. Fenton,
Chaos: An Interdisciplinary Journal of Nonlinear Science, Volume 32; https://doi.org/10.1063/5.0087812

Abstract:
Computational modeling and experimental/clinical prediction of the complex signals during cardiac arrhythmias have the potential to lead to new approaches for prevention and treatment. Machine-learning (ML) and deep-learning approaches can be used for time-series forecasting and have recently been applied to cardiac electrophysiology. While the high spatiotemporal nonlinearity of cardiac electrical dynamics has hindered application of these approaches, the fact that cardiac voltage time series are not random suggests that reliable and efficient ML methods have the potential to predict future action potentials. This work introduces and evaluates an integrated architecture in which a long short-term memory autoencoder (AE) is integrated into the echo state network (ESN) framework. In this approach, the AE learns a compressed representation of the input nonlinear time series. Then, the trained encoder serves as a feature-extraction component, feeding the learned features into the recurrent ESN reservoir. The proposed AE-ESN approach is evaluated using synthetic and experimental voltage time series from cardiac cells, which exhibit nonlinear and chaotic behavior. Compared to the baseline and physics-informed ESN approaches, the AE-ESN yields mean absolute errors in predicted voltage 6–14 times smaller when forecasting approximately 20 future action potentials for the datasets considered. The AE-ESN also demonstrates less sensitivity to algorithmic parameter settings. Furthermore, the representation provided by the feature-extraction component removes the requirement in previous work for explicitly introducing external stimulus currents, which may not be easily extracted from real-world datasets, as additional time series, thereby making the AE-ESN easier to apply to clinical data.
Iván León,
Chaos: An Interdisciplinary Journal of Nonlinear Science, Volume 32; https://doi.org/10.1063/5.0093001

Abstract:
The dynamics of ensembles of phase oscillators are usually described considering their infinite-size limit. In practice, however, this limit is fully accessible only if the Ott–Antonsen theory can be applied, and the heterogeneity is distributed following a rational function. In this work, we demonstrate the usefulness of a moment-based scheme to reproduce the dynamics of infinitely many oscillators. Our analysis is particularized for Gaussian heterogeneities, leading to a Fourier–Hermite decomposition of the oscillator density. The Fourier–Hermite moments obey a set of hierarchical ordinary differential equations. As a preliminary experiment, the effects of truncating the moment system and implementing different closures are tested in the analytically solvable Kuramoto model. The moment-based approach proves to be much more efficient than the direct simulation of a large oscillator ensemble. The convenience of the moment-based approach is exploited in two illustrative examples: (i) the Kuramoto model with bimodal frequency distribution, and (ii) the “enlarged Kuramoto model” (endowed with nonpairwise interactions). In both systems, we obtain new results inaccessible through direct numerical integration of populations.
Chaos: An Interdisciplinary Journal of Nonlinear Science, Volume 32; https://doi.org/10.1063/5.0082002

Abstract:
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 “Gold,” “Silver,” and “Bronze” communities in a network, we find that two factors play a central role: the coupling strength in the dynamics and network density. When considering the spread of a virus in Barabási–Albert networks, infections in “Silver” and “Gold” communities are lower than in “Bronze” communities. We find that the “Gold” communities are the best in keeping their infection levels low. However, a small number of “Bronze” communities are enough to give rise to an increase in infections in moderately and well-vaccinated communities. When studying the spread of a virus in dense Erdős–Rényi and sparse Watts–Strogatz and Barabási–Albert networks, the communities reach the disease-free state in the dense Erdős–Rényi networks, but not in the sparse Watts–Strogatz and Barabási–Albert networks. However, we also find that if all these networks are dense enough, all types of communities reach the disease-free state. We conclude that the presence of a few unvaccinated or partially vaccinated communities in 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.
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