Chaos, Solitons and Fractals: X最新文献

We provide an analytic estimate for the size of the bulbs adjoining the main cardioid of the Mandelbrot set. The bulbs are approximate circles, and are associated with the stability regions in the complex parameter μ-space of period-q orbits of the underlying map $z\to z^2-\mu$. For the (p, q) orbit with winding number p/q, the associated stability bulb is an approximate circle with radius $\frac{1}{q^2}\sin \frac{\pi p}{q}$.

Bipartite graphs are often found to represent the connectivity between the components of many systems such as ecosystems. A bipartite graph is a set of n nodes that is decomposed into two disjoint subsets, having m and $n-m$ vertices each, such that there are no adjacent vertices within the same set. The connectivity between both sets, which is the relevant quantity in terms of connections, can be quantified by a parameter α ∈ [0, 1] that equals the ratio of existent adjacent pairs over the total number of possible adjacent pairs. Here, we study the spectral and localization properties of such random bipartite graphs. Specifically, within a Random Matrix Theory (RMT) approach, we identify a scaling parameter ξ ≡ ξ(n, m, α) that fixes the localization properties of the eigenvectors of the adjacency matrices of random bipartite graphs. We also show that, when ξ < 1/10 (ξ > 10) the eigenvectors are localized (extended), whereas the localization–to–delocalization transition occurs in the interval 1/10 < ξ < 10. Finally, given the potential applications of our findings, we round off the study by demonstrating that for fixed ξ, the spectral properties of our graph model are also universal.

To evaluate the long-range cross-correlation in non-stationary bi-variate time-series, detrending-operation-based analysis methods such as the detrending moving-average cross-correlation analysis (DMCA), are widely used. However, its mathematical foundation has not been well established. In this paper, we propose a generalized theory to form the foundation of DMCA-type methods and introduce the higher-order DMCA in which Savitzky-Golay filters are employed as the detrending operator. Using this theory, we can understand the mathematical basis of DMCA-type methods. Our theory establishes a rigorous relationship between the DMCA-type analysis, the cross-correlation function analysis, and the cross-power spectral analysis. Based on the mathematical validity, we provide a practical guide for the use of higher-order DMCA. Additionally, we present illustrative results of a numerical and real-world analysis. To achieve reliable and accurate detection of the long-range cross-correlation, we emphasize the importance of time-lag estimation and time scale correction in DMCA, which has not been pointed out in the previous studies.

We explore simple models aimed at the study of social contagion, in which contagion proceeds through two stages. When coupled with demographic turnover, we show that two-stage contagion leads to nonlinear phenomena which are not present in the basic ‘classical’ models of mathematical epidemiology. These include: bistability, critical transitions, endogenous oscillations, and excitability, suggesting that contagion models with stages could account for some aspects of the complex dynamics encountered in social life. These phenomena, and the bifurcations involved, are studied by a combination of analytical and numerical means.

We study linear and nonlinear fractional differential equations of order 0 < α < 1, involving the Atangana–Baleanu fractional derivative. We establish existence and uniqueness results to the linear and nonlinear problems using Banach fixed point theorem. We then develop a numerical technique based on the Chebyshev collocation method to solve the problem. As an important application we consider the fractional Riccati equation. Two examples are presented to test the efficiency of the proposed technique, where a notable agreement between the approximate and the exact solutions is obtained. Also, the approximate solutions approach to the exact solutions of the corresponding ordinary differential equations as the fractional derivative approaches 1.

A mathematical model providing an asymptotic description of macro-economic system is considered in this work. The system in general deals with performance, behaviour, decision-making of an economy as a whole and also the structure. Due to the complexities of this system, a more complex mathematical model is requested. In this work, we considered the extension of the model using some non-local differential operators and the stochastic approach where the given parameters are converted to normal distributions. We have presented the conditions of existence of uniquely exact solutions of the system using the fixed-point theorem approach. Each model is solved numerical via a newly introduced modified Adams-Bashforth for fractional differential equations. We presented numerical simulations for different values of fractional order. The models with the Atangana-Baleanu and Caputo differential operators provided us with new attractors.

Nowadays, with more and more people choosing energy-saving product, green supply chain is increasingly becoming popular. In this paper, we build a dynamic Stackelberg game model that contains government, two green innovation enterprise which is a leader and produces two kinds of green product with similar function but different quality and a retailer selling two kinds of green product meanwhile. We use classical backward induction to solve the model. Price decision is considered first and is divided into two stages. We analyze the equilibrium price of green innovation enterprise and retailer and the stable region. Next, we analyze energy-saving index of two kinds of green product. Through numerical simulation, we further analyze system's stability conditions and system's dynamic evolution process when different parameter is adopted. In the end, we consider green products’ baseline energy-saving index made by government and get the optimal baseline energy-saving index by numerical simulation. We find that too stringent subsidy standards may can't induce manufacturer to invest greener technology. Besides, we use the delay feedback control to effectively control the chaotic phenomenon.

In this work we build a convolutional neural network capable of identifying individual birds by their songs. Since the actual data available from each individual is very limited, we use a dynamical system capable of synthesizing realistic songs, to generate surrogate-training data. The different synthetic songs are the result of integrating the dynamical system with slightly varied parameters. We show that a data set built in this way allows us to train the network to successfully identify the different individuals in our study. In this way, we present a novel way to perform data augmentation using dynamical systems.

A linear partial differential equation is shown to exhibit three properties often used to define chaotic dynamics. The system comprises a one-dimensional wave equation with gain that operates on a semi-infinite line. A boundary condition enforces that the waves remain finite. It is shown that the resulting solution set is dense with periodic orbits, contains transitive orbits, and exhibits extreme sensitivity to initial conditions. Definitions of chaos are considered in light of such linear chaos.