Farrukh Mukhamedov, José F. F. Mendes
In the paper we describe basin of attraction $p$-adic dynamical system $G(x)=(ax)^2(x+1)$. Moreover, we also describe the Siegel discs of the system, since the structure of the orbits of the system is related to the geometry of the $p$-adic Siegel discs.
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In our previous investigations, we have developed the renormalization group method to $p$-adic models on Cayley trees, this method is closely related to the investigation of $p$-adic dynamical systems associated with a given model. In this paper, we study chaotic behavior of the Potts-Bethe mapping. We point out that a similar kind of result is not known in the case of real numbers (with rigorous proofs).
In the present paper, by conducting research on the dynamics of the $p$-adic generalized Ising mapping corresponding to renormalization group associated with the $p$-adic Ising-Vannemenus model on a Cayley tree, we have determined the existence of the fixed points of a given function. Simultaneously, the attractors of the dynamical system have been found. We have come to a conclusion that the considered mapping is topologically conjugate to the symbolic shift which implies its chaoticity and as an application, we have established the existence of periodic $p$-adic Gibbs measures for the $p$-ad…
This paper deals with the problem of forecast the Logistic Chaotic Map using Fuzzy Times Series (FTS). Chaotic Systems are very sensible to changes in its parameters and in the initial conditions, turning them into hard systems to model and forecast. In this case, we relay in the robustness of Fuzzy Time Series to model and forecast the logistic map. We use the Akaike Information Criterion (AIC) as an index to determine the number of sub intervals for the definition of the fuzzy set.
In our previous investigations, we have developed the renormalization group method to $p$-adic $q$-state Potts model on the Cayley tree of order $k$. This method is closely related to the examination of dynamical behavior of the $p$-adic Potts-Bethe mapping which depends on parameters $q,k$. In \cite{MFKh18} we have considered the case when $q$ is not divisible by $p$, and under some conditions it was established that the mapping is conjugate to the full shift. The present paper is a continuation of the mentioned paper, but here we investigate the case when $q$ is divisible by $p$ and $k$ is a…
Resonance properties and chaotic dynamics of a three-dimensional discrete logistic ecological system within the neighborhoods of bifurcation points
本文研究三维离散逻辑斯蒂生态模型在分岔点附近的动力学，利用完全判别理论、分岔与规范形理论分析余维1和余维2分岔及Marotto混沌，并通过数值模拟验证理论结果。
Two symmetrically coupled logistic equations are proposed to mimic the competitive interaction between two species. The phenomena of coexistence, oscillations and chaos are present in this cubic discrete system. This work, together with two other similar ones recently published by the authors, completes a triptych dedicated to the two-species relationships present in Nature, namely the symbiosis, the predator-prey and the competition. These models can be used as basic ingredients to build up more complex interactions in the ecological networks.
A cubic discrete coupled logistic equation is proposed to model the predator-prey problem. The coupling depends on the population size of both species and on a positive constant $λ$, which could depend on the prey reproduction rate and on the predator hunting strategy. Different dynamical regimes are obtained when $λ$ is modified. For small $λ$, the species become extinct. For a bigger $λ$, the preys survive but the predators extinguish. Only when the prey population reaches a critical value then predators can coexist with preys. For increasing $λ$, a bistable regime appears where the populati…
A route to chaos is studied in 3-dimensional maps of logistic type. Mechanisms of period doubling for invariant closed curves (ICC) are found for specific 3-dimensional maps. These bifurcations cannot be observed for ICC in the 2-dimensional case. When the parameter of the system is modified, localized oscillations occur on the ICC that give rise to weakly chaotic rings, then to chaotic attractors, which finally disappear by contact bifurcations. These maps can be considered as models for the symbiotic interaction of three species.
In this paper, a rigorous analysis of the behavior of the standard logistic map, Logistic Tent system (LTS), Logistic-Sine system (LSS) and Tent-Sine system (TSS) is performed using 0-1 test and three state test (3ST). In this work, it has been proved that the strength of the chaotic behavior is not uniform. Through extensive experiment and analysis, the strong and weak chaotic regions of LTS, LSS and TSS have been identified. This would enable researchers using these maps, to have better choices of control parameters as key values, for stronger encryption. In addition, this paper serves as a …
Rational map associated with the Sigmoid Beverton-Holt model on the projective line over $\mathbb{Q}_p$
本文利用 $p$-进多项式最小分解及 $p$-进排斥子混沌理论，刻画了 $\n\b\mathbb{Q}_p$ 射影直线上 Sigmoid Beverton-Holt 模型对应有理映射的动力学结构。
In this paper, we show our discovery that state-transition networks in several chaotic dynamical systems are "scale-free networks," with a technique to understand a dynamical system as a whole, which we call the analysis for "Discretized-State Transition" (DST) networks; This scale-free nature is found universally in the logistic map, the sine map, the cubic map, the general symmetric map, the sine-circle map, the Gaussian map, and the delayed logistic map. Our findings prove that there is a hidden order in chaos, which has not detected yet. Furthermore, we anticipate that our study opens up a…
On the Performance of Linear Adaptive Filters driven by the Ergodic Chaotic Logistic Map
本文证明了逻辑斯蒂映射在参数 λ=4 时生成的零均值序列不相关，并据此分析了 FIR 数字自适应滤波器的性能，推导出 LMS 方案中阻尼参数的最优值，从而为信道估计提供最大性能且保证稳定性的理论依据。
In this work, we present an algorithm for the design of $n\times n$-bits substitution boxes (S-boxes) based on time series of a discrete dynamical system with chaotic behavior. The elements of a $n\times n$-bits substitution box are given by binary sequences generated by time series with uniform distribution. Particularly, time series with uniform distribution are generated via two lag time chaotic series of the logistic map. The aim of using these two lag time sequences is to hide the map used, and thus U-shape distribution of the logistic map is avoided and uncorrelated S-box elements are ob…
In this paper a new type of chaotic system based on sin and logistic systems is introduced. Also the behavior of this new system is studied by using various tests. The results of these tests indicate the appropriate behavior for the proposed new system.
We present an introduction to the study of chaos in discrete and continuous dynamical systems using the CAS Maxima. These notes are intended to cover the standard topics and techniques: discrete and continuous logistic equation to model growth population, staircase plots, bifurcation diagrams and chaos transition, nonlinear continuous dynamics (Lorentz system and Duffing oscillator), Lyapunov exponents, Poincaré sections, fractal dimension and strange attractors. The distinctive feature here is the use of free software with just one ingredient: the CAS Maxima. It is cross-platform and have ext…
Chaotic attractors, chaotic saddles and periodic orbits are examples of chain-recurrent sets. Using arbitrary small controls, a trajectory starting from any point in a chain-recurrent set can be steered to any other in that set. The qualitative behavior of a dynamical system can be encapsulated in a graph. Its nodes are chain-recurrent sets. There is an edge from node A to node B if, using arbitrary small controls, a trajectory starting from any point of A can be steered to any point of B. We discuss physical systems that have infinitely many disjoint coexisting nodes. Such infinite collection…
A symmetrical cubic discrete coupled logistic equation is proposed to model the symbiotic interaction of two isolated species. The coupling depends on the population size of both species and on a positive constant $λ$, named the mutual benefit. Different dynamical regimes are obtained when the mutual benefit is modified. For small $λ$, the species become extinct. For increasing $λ$, the system stabilizes in a synchronized state or oscillates in a 2 periodic orbit. For the greatest permitted values of $λ$, the dynamics evolves into a quasiperiodic, into a chaotic scenario or into extinction. Th…
We use standard deep neural networks to classify univariate time series generated by discrete and continuous dynamical systems based on their chaotic or non-chaotic behaviour. Our approach to circumvent the lack of precise models for some of the most challenging real-life applications is to train different neural networks on a data set from a dynamical system with a basic or low-dimensional phase space and then use these networks to classify univariate time series of a dynamical system with more intricate or high-dimensional phase space. We illustrate this generalisation approach using the log…
Coupled map lattices (CMLs) are prototypical dynamical systems on networks/graphs. They exhibit complex patterns generated via the interplay of diffusive/Laplacian coupling and nonlinear reactions modelled by a single iterated map at each node; the maps are often taken as unimodal, e.g., logistic or tent maps. In this letter, we propose a class of higher-order coupled dynamical systems involving the hypergraph Laplacian, which we call coupled hypergraph maps (CHMs). By combining linearized (in-)stability analysis of synchronized states, hypergraph spectral theory, and numerical methods, we det…
We study the evolution of the probability density of ensembles of iterates of the logistic map that advance towards and finally remain at attractors of representative dynamical regimes. We consider the mirror families of superstable attractors along the period-doubling cascade, and of chaotic-band attractors along the inverse band-splitting cascade. We examine also their common aperiodic accumulation point. The iteration time progress of the densities of trajectories is determined via the action of the Frobenius-Perron (FP) operator. As a difference with the study of individual orbits, the ana…
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