Ultimately, the designed controller guarantees the synchronization error converges to a small region around the origin, along with the uniform, semiglobal ultimate boundedness of all signals, thereby mitigating Zeno behavior. Lastly, two numerical simulations are carried out to demonstrate the robustness and precision of the proposed scheme.
Natural spreading processes are more accurately depicted by epidemic spreading processes on dynamic multiplex networks compared to those occurring on single-layered networks. To evaluate the effects of individuals in the awareness layer on epidemic dissemination, we present a two-layered network model that includes individuals who disregard the epidemic, and we analyze how differing individual traits in the awareness layer affect the spread of diseases. The two-layered network model's structure is partitioned into an information transmission component and a disease spread component. Nodes in each layer signify individual entities, with their interconnections differing from those in other layers. The probability of infection in individuals with a strong understanding of infection prevention is lower than that of individuals with limited awareness of transmission risks, aligning with the practical implementation of infection-prevention measures. Our proposed epidemic model's threshold is analytically determined through the application of the micro-Markov chain approach, demonstrating the awareness layer's influence on the disease spread threshold. The impact of individuals with differing traits on the disease spreading dynamics is explored through extensive Monte Carlo numerical simulations thereafter. The transmission of infectious diseases is notably curtailed by individuals with high centrality within the awareness network. In addition, we offer conjectures and interpretations regarding the roughly linear relationship between individuals with low centrality in the awareness layer and the number of infected individuals.
This study investigated the Henon map's dynamics with information-theoretic quantifiers, comparing the results with experimental data from brain regions known for chaotic behavior. Replicating chaotic brain dynamics in Parkinson's and epilepsy patients using the Henon map as a model was the intended goal. The dynamic attributes of the Henon map were evaluated against data obtained from the subthalamic nucleus, medial frontal cortex, and a q-DG model of neuronal input-output. This model, allowing for easy numerical simulations, was chosen to replicate the local behavior within a population. The temporal causality within the time series was a key consideration when utilizing information theory tools, Shannon entropy, statistical complexity, and Fisher's information for analysis. To accomplish this objective, multiple windows spanning the time series were investigated. The study's conclusions highlighted the inability of both the Henon map and the q-DG model to perfectly capture the observed dynamics of the scrutinized brain regions. However, by paying close attention to the parameters, scales, and sampling procedures utilized, they were able to develop models exhibiting certain aspects of neural activity patterns. These outcomes imply a more multifaceted and complex range of normal neural dynamics within the subthalamic nucleus, existing across the complexity-entropy causality plane, exceeding the explanatory scope of chaotic models. These systems' dynamic behavior, as revealed through the use of these tools, is markedly dependent on the investigated temporal scale. The rising scale of the sample set scrutinized leads to a more substantial dissimilarity between the Henon map's dynamics and those of organic and artificial neural networks.
We utilize computer-assisted analytical tools to examine the two-dimensional neuron model put forward by Chialvo in 1995, which appears in Chaos, Solitons Fractals, volume 5, pages 461-479. Utilizing a set-theoretic topological framework, as pioneered by Arai et al. in 2009 [SIAM J. Appl.], we employ a stringent global dynamic analysis methodology. Sentences are dynamically listed here. The system's task involves generating and returning a list of diverse sentences. Sections 8, 757-789 served as the initial foundation, which was later developed and extended. In addition, we've developed a new algorithm for analyzing the time it takes to return within a chain recurrent set. Lapatinib clinical trial Using the results of this analysis, combined with the size of the chain recurrent set, a new technique was developed to identify parameter subsets which may display chaotic behavior. Various dynamical systems benefit from this approach, and we examine some of its practical facets.
The process of reconstructing network connections from quantifiable data enhances our comprehension of the interplay between nodes. Still, the nodes of immeasurable magnitude, further distinguished as hidden nodes, introduce novel obstacles to the reconstruction of real-world networks. Several procedures for detecting hidden nodes have been introduced, however, many face limitations due to the characteristics of the computational model, network layout, and other environmental variables. Using the random variable resetting method, this paper proposes a general theoretical approach to detect hidden nodes. Lapatinib clinical trial A new time series, comprising hidden node information and generated from random variable resetting reconstruction, is constructed. This time series' autocovariance is subsequently analyzed theoretically, culminating in a quantitative measure for identifying hidden nodes. To understand the influence of key factors, our method is numerically simulated across discrete and continuous systems. Lapatinib clinical trial Our theoretical derivation is validated and the robustness of the detection method, across diverse conditions, is illustrated by the simulation results.
A method for quantifying the sensitivity of a cellular automaton (CA) to variations in its starting configuration involves adapting the Lyapunov exponent, a concept originally developed for continuous dynamical systems, to CAs. Currently, these endeavors are circumscribed by a CA having only two states. Their applicability is significantly constrained by the fact that numerous CA-based models necessitate three or more states. We broadly generalize the prior approach for N-dimensional, k-state cellular automata, enabling the application of either deterministic or probabilistic update rules. Our proposed extension elucidates the distinctions between different types of defects that propagate, and the paths along which they spread. In addition, to fully grasp the stability of CA, we introduce supplementary concepts, comprising the average Lyapunov exponent and the correlation coefficient of the difference pattern's growth. Our methodology is illustrated with intriguing examples of three-state and four-state rules, and further demonstrated through a cellular automata-based forest fire model. Our extension, besides improving the generalizability of existing approaches, permits the identification of behavioral traits that distinguish Class IV CAs from Class III CAs, a previously challenging undertaking under Wolfram's classification.
PiNNs, a recently developed powerful solver, have effectively tackled a considerable assortment of partial differential equations (PDEs) under numerous initial and boundary conditions. This paper introduces trapz-PiNNs, a physics-informed neural network implementation combining a modified trapezoidal rule for accurate fractional Laplacian calculations, enabling the solution of space-fractional Fokker-Planck equations in both two and three spatial dimensions. We meticulously examine the modified trapezoidal rule, validating its second-order accuracy. Employing a spectrum of numerical examples, we highlight the considerable expressive potential of trapz-PiNNs, evident in their ability to forecast solutions with remarkably low L2 relative error. We further our analysis with local metrics, such as point-wise absolute and relative errors, to pinpoint areas requiring optimization. A method for improving trapz-PiNN's performance, focusing on local metrics, is detailed, provided that physical observations or accurate high-fidelity simulations of the true solution exist. PDEs on rectangular domains, incorporating fractional Laplacians with arbitrary (0, 2) exponents, find solutions using the trapz-PiNN framework. The potential for broader application, including higher dimensional settings or other confined areas, also exists.
A mathematical model of sexual response is derived and analyzed in this paper. Our initial analysis focuses on two studies that theorized a connection between the sexual response cycle and a cusp catastrophe. We then address the invalidity of this connection, but show its analogy to excitable systems. This forms the foundation from which a phenomenological mathematical model of sexual response is derived, with variables representing levels of physiological and psychological arousal. To discern the stability characteristics of the model's equilibrium state, bifurcation analysis is employed, while numerical simulations are conducted to showcase the diverse behaviors predicted by the model. The Masters-Johnson sexual response cycle's dynamics are manifested in canard-like trajectories that initially adhere to an unstable slow manifold, then making a considerable phase space excursion. A stochastic version of the model is also investigated, with the analytical determination of the spectrum, variance, and coherence of stochastic oscillations around a stable deterministic steady state, which permits the computation of confidence regions. The methods of large deviation theory are used to scrutinize stochastic escape from the area surrounding a deterministically stable steady state; this is supplemented by the use of action plot and quasi-potential methodologies to calculate the most probable escape paths. Considering the implications for a deeper understanding of human sexual response dynamics and improving clinical methodology, we discuss our findings.