Search Results (11)

Mathematical models for infectious diseases, particularly autonomous ODE models, are generally known to possess simple dynamics, often converging to stable disease-free or endemic equilibria. This paper investigates the dynamic consequences of a crucial, yet often overlooked, component of pandemic response: the saturation of public health testing. We extend the standard SIR model to include compartments for ‘Confirmed’ (C) and ‘Monitored’ (M) individuals, resulting in a new SICMR model. By fitting the model to U.S. COVID-19 pandemic data (specifically the Omicron wave of late 2021), we demonstrate that capacity constraints in testing destabilize the testing-free endemic equilibrium ($E_1$). This equilibrium becomes an unstable saddle-focus. The instability is driven by a sociological feedback loop, where the rise in confirmed cases drive testing effort, modeled by a nonlinear Holling Type II functional response. We explicitly verify that the eigenvalues for the best-fit model satisfy the Shilnikov condition (${\lambda }_u>{\lambda }_s$), demonstrating the system possesses the necessary ingredients for complex, chaotic-like dynamics. Furthermore, we employ Stochastic Differential Equations (SDEs) to show that intrinsic noise interacts with this instability to generate ’noise-induced bursting,’ replicating the complex wave-like patterns observed in empirical data. Our results suggest that public health interventions, such as testing, are not merely passive controls but active dynamical variables that can fundamentally alter the qualitative stability of an epidemic. Full article

Circuit implementation is a widely accepted method for validating theoretical insights observed in chaotic systems. It also serves as a basis for numerous chaos-based engineering applications, including data encryption, random number generation, secure communication, neuromorphic computing, and so forth. To get feasible, compact, and cost-effective circuit implementations of chaotic systems, the underlying mathematical model may be simplified while preserving all rich nonlinear behaviors. In this framework, this manuscript presents a simplified Hopfield Neural Network (HNN) capable of generating a broad spectrum of complex behaviors using a minimal number of electronic elements. Based on Shil’nikov’s theorem for heteroclinic orbits, the number of non-zero synaptic connections in the matrix weights is reduced, while simultaneously using only one nonlinear activation function. As a result of these simplifications, we obtain the most compact electronic implementation of a tri-neuron HNN with the lowest component count but retaining complex dynamics. Comprehensive theoretical and numerical analyses by equilibrium points, density-colored continuation diagrams, basin of attraction, and Lyapunov exponents, confirm the presence of periodic oscillations, spiking, bursting, and chaos. Such chaotic dynamics range from single-scroll chaotic attractors to double-scroll chaotic attractors, as well as coexisting attractors to transient chaos. A brief security application of an S-Box utilizing the presented HNN is also given. Finally, a physical implementation of the HNN is given to confirm the proposed approach. Experimental observations are in good agreement with numerical results, demonstrating the usefulness of the proposed approach. Full article

Consider the dynamical system x ˙ = f ( x ) , where x ∈ R n is the state vector, x ˙ is the time or spatial derivative, and f is the system model. We wish to identify unknown f from its time-series or spatial data. For this, we propose a Bayesian framework based on the maximum a posteriori (MAP) point estimate, to give a generalized Tikhonov regularization method with the residual and regularization terms identified, respectively, with the negative logarithms of the likelihood and prior distributions. As well as estimates of the model coefficients, the Bayesian interpretation provides access to the full Bayesian apparatus, including the ranking of models, the quantification of model uncertainties, and the estimation of unknown (nuisance) hyperparameters. For multivariate Gaussian likelihood and prior distributions, the Bayesian formulation gives a Gaussian posterior distribution, in which the numerator contains a Mahalanobis distance or “Gaussian norm”. In this study, two Bayesian algorithms for the estimation of hyperparameters—the joint maximum a posteriori (JMAP) and variational Bayesian approximation (VBA)—are compared to the popular SINDy, LASSO, and ridge regression algorithms for the analysis of several dynamical systems with additive noise. We consider two dynamical systems, the Lorenz convection system and the Shil’nikov cubic system, with four choices of noise model: symmetric Gaussian or Laplace noise and skewed Rayleigh or Erlang noise, with different magnitudes. The posterior Gaussian norm is found to provide a robust metric for quantitative model selection—with quantification of the model uncertainties—across all dynamical systems and noise models examined. Full article

We present a novel set of quantitative measures for “likeness” (error function) designed to alleviate the time-consuming and subjective nature of manually comparing biological recordings from electrophysiological experiments with the outcomes of their mathematical models. Our innovative “blended” system approach offers an objective, high-throughput, and computationally efficient method for comparing biological and mathematical models. This approach involves using voltage recordings of biological neurons to drive and train mathematical models, facilitating the derivation of the error function for further parameter optimization. Our calibration process incorporates measurements such as action potential (AP) frequency, voltage moving average, voltage envelopes, and the probability of post-synaptic channels. To assess the effectiveness of our method, we utilized the sea slug Melibe leonina swim central pattern generator (CPG) as our model circuit and conducted electrophysiological experiments with TTX to isolate CPG interneurons. During the comparison of biological recordings and mathematically simulated neurons, we performed a grid search of inhibitory and excitatory synapse conductance. Our findings indicate that a weighted sum of simple functions is essential for comprehensively capturing a neuron’s rhythmic activity. Overall, our study suggests that our blended system approach holds promise for enabling objective and high-throughput comparisons between biological and mathematical models, offering significant potential for advancing research in neural circuitry and related fields. Full article

In the process of aerospace service, circular mesh antennas generate large nonlinear vibrations under an alternating thermal load. In this paper, the Smale horseshoe and Shilnikov-type multi-pulse chaotic motions of the six-dimensional non-autonomous system for circular mesh antennas are first investigated. The Poincare map is generalized and applied to the six-dimensional non-autonomous system to analyze the existence of Smale horseshoe chaos. Based on the topological horseshoe theory, the three-dimensional solid torus structure is mapped into a logarithmic spiral structure, and the original structure appears to expand in two directions and contract in one direction. There exists chaos in the sense of a Smale horseshoe. The nonlinear equations of the circular mesh antenna under the conditions of the unperturbed and perturbed situations are analyzed, respectively. For the perturbation analysis of the six-dimensional non-autonomous system, the energy difference function is calculated. The transverse zero point of the energy difference function satisfies the non-degenerate conditions, which indicates that the system exists Shilnikov-type multi-pulse chaotic motions. In summary, the researches have verified the existence of chaotic motion in the six-dimensional non-autonomous system for the circular mesh antenna. Full article

From the perspectives of the thermodynamics of irreversible processes and the theory of complex systems, a characterization of longevity and aging and their relationships with the emergence and evolution of cancer was carried out. It was found that: (1) the rate of entropy production could be used as an index of the robustness, plasticity, and aggressiveness of cancer, as well as a measure of biological age; (2) the aging process, as well as the evolution of cancer, goes through what we call a “biological phase transition”; (3) the process of metastasis, which occurs during the epithelial–mesenchymal transition (EMT), appears to be a phase transition that is far from thermodynamic equilibrium and exhibits Shilnikov chaos-like dynamic behavior, which guarantees the robustness of the process and, in turn, its unpredictability; (4) as the ferroptosis process progresses, the complexity of the dynamics that are associated with the emergence and evolution of cancer decreases. The theoretical framework that was developed in this study could contribute to a better understanding of the biophysical and chemical phenomena of longevity and aging and their relationships with cancer. Full article

The dynamical evolution of electrical discharge machining (EDM) has drawn immense research interest. Previous research on mechanism analysis has discussed the deterministic nonlinearity of gap states at pulse-on discharging duration, while describing the pulse-off deionization process separately as a stochastic evolutionary process. In this case, the precise model describing a complete machining process, as well as the optimum performance parameters of EDM, can hardly be determined. The main purpose of this paper is to clarify whether the EDM system can maintain consistency in dynamic characteristics within a discharge interval. A nonlinear self-maintained equivalent model is first established, and two threshold conditions are obtained by the Shilnikov theory. The theoretical results prove that the EDM system could lead to chaos without external excitation. The time series of the deionization process recorded in the EDM experiments are then analyzed to further validate this theoretical conclusion. Qualitative chaotic analyses verify that the autonomous EDM process has chaotic characteristics. Quantitative methods are used to estimate the chaotic feature of the autonomous EDM process. By comparing the quantitative results of the autonomous EDM process with the non-autonomous EDM process, a deduction is further made that the EDM system will evolve towards steady chaos under an autonomous state. Full article

This article raises the issue of decision support system (DSS) development in enterprises concerning the compensation system (CS). The topic is relevant as the CS is one of the main components in human resource management in business. A key element of such DSSs is CS models that provide predictive analytics. Such models are able to give information about how a particular CS affects output, product quality, employee satisfaction, and wage fund. Thus, the main goal of this article is to obtain a CS statistical model and its formulas for determining the probability densities of resultant indicators. To achieve this goal, the authors conducted several blocks of research. Firstly, mathematical formalization of CS functionality was described. Secondly, a statistical model of CS was built. Thirdly, calculations of CS result indicators were made. Reliable scientific methods were used: black box modeling and statistical modeling. This article proposes a statistical and analytical model. As an example, a piecework-bonus system statistical model is demonstrated. The discussion derives formulas of integral estimations showing the probability density of the resulting CS indicators and the related statistical characteristics. These results can be used to predict the behavior of the workforce. This constitutes the scientific novelty of the study, which will establish significant advances in the development of DSSs in the field of labor economics and HR management. Full article

In this paper, the dynamics of a 3D autonomous dissipative nonlinear system of ODEs-Rössler prototype-4 system, was investigated. Using Lyapunov-Andronov theory, we obtain a new analytical formula for the first Lyapunov’s (focal) value at the boundary of stability of the corresponding equilibrium state. On the other hand, the global analysis reveals that the system may exhibit the phenomena of Shilnikov chaos. Further, it is shown via analytical calculations that the considered system can be presented in the form of a linear oscillator with one nonlinear automatic regulator. Finally, it is found that for some new combinations of parameters, the system demonstrates chaotic behavior and transition from chaos to regular behavior is realized through inverse period-doubling bifurcations. Full article

In this study we extend a model, proposed by Dendrinos, which describes dynamics of change of influence in a social system containing a public sector and a private sector. The novelty is that we reconfigure the system and consider a system consisting of a public sector, a private sector, and a non-governmental organizations (NGO) sector. The additional sector changes the model’s system of equations with an additional equation, and additional interactions must be taken into account. We show that for selected values of the parameters of the model’s system of equations, chaos of Shilnikov kind can exist. We illustrate the arising of the corresponding chaotic attractor and discuss the obtained results from the point of view of interaction between the three sectors. Full article