Chaos, Solitons and Fractals: X最新文献

Some ideas are presented about a geometric motivation of the apparent capacity of generalized logistic equations to describe the outbreak of quite many epidemics, possibly including that of the COVID-19 infection. This interpretation pivots on the complex, possibly fractal, structure of the locus describing the “contagion event set”, and on what can be learnt from the models of trophic webs with “herd behaviour”.

Under the hypothesis that the total number of cases, as a function of time, is fitted by a solution of the Generalized Richards Model, it is argued that the exponents appearing in that differential equation, usually determined empirically, represent the geometric signature of the non-space filling, network-like locus on which contagious contacts take place.

Chaotic entanglement is a new method used to deliver chaotic physical process, as suggested in this work. Primary rationale is to entangle more than two mathematical product stationery linear schemes by means of entanglement functions to make a chaotic system that develops in a chaotic manner.Existence of Hopf bifurcation is looked into by selecting the set aside bifurcation parameter. More accurately, we consider the stableness and bifurcations of sense of equilibrium in the modern chaotic system. In addition, there is involvement of chaos in mathematical systems that have one positive Lyapunov exponent. Furthermore, there are four requirements that are needed to achieve chaos entanglement. In that way through dissimilar linear schemes and dissimilar entanglement functions, a collection of fresh chaotic attractors has been created and abundant coordination compound dynamics are exhibited. The breakthrough suggests that it is not difficult any longer to construct new obviously planned chaotic systems/networks for applied science practical application such as chaos-based secure communication.

Chaotic system has been widely used in the design of pseudorandom sequence generators. However, the performance of pseudorandom sequence generators is greatly affected by the chaotic degradation, which is caused by computational accuracy. In this paper, we propose a self-perturbed pseudorandom sequence generator based on hyper-chaotic system to overcome this problem. A novel hyper-chaotic system is constructed to achieve a complex dynamic behavior. One of the feedback controllers is used to disturb the other dimensions so that the short period can be avoided. Based on the proposed hyper-chaotic system, a pseudorandom number generator is designed. The dynamic behavior of the hyper-chaotic system is analyzed. The randomness and the security of the pseudorandom sequence generated by the proposed scheme are tested and analyzed. The results show that this scheme has a larger key space and a higher level of randomness. It is suitable to be used in privacy encryption and secure communication.

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.