Discontinuous bifurcation and coexistence of attractors in a piecewise linear map with a gap

Coexistence of attractors with striking characteristics is observed in this work, where a stable period-5 attractor coexists successively with chaotic band-ll, period-6, chaotic band-12 and band-6 attractors. They are induced by dif- ferent mechanisms due to the interaction between the discontinuity and the non-invertibility. A characteristic boundary collision bifurcation, is observed. The critical conditions are obtained both analytically and numerically.

Noise destroys the coexistence of periodic orbits of a piecewise linear map

The effects of Gaussian white noise and Gaussian colored noise on the periodic orbits of period-5（P-5） and period-6（P-6） in their coexisting domain of a piecewise linear map are investigated numerically.The probability densities of some orbits are calculated.When the noise intensity is D = 0.0001,only the orbits of P-5 exist,and the coexisting phenomenon is destroyed.On the other hand,the self-correlation time τ of the colored noise also affects the coexisting phenomenon.When τc〈τ〈τc,only the orbits of P-5 appear,and the stability of the orbits of P-5 is enhanced.However,when τ〉τc（τc and τc are critical values）,only the orbits of P-6 exist,and the stability of the P-6 orbits is enhanced greatly.When τ〈τc,the orbits of P-5 and P-6 coexist,but the stability of the P-5 orbits is enhanced and that of P-6 is weakened with τ increasing.

Phase order in one-dimensional piecewise linear discontinuous map

The phase order in a one-dimensional（1 D） piecewise linear discontinuous map is investigated. The striking feature is that the phase order may be ordered or disordered in multi-band chaotic regimes, in contrast to the ordered phase in continuous systems. We carried out an analysis to illuminate the underlying mechanism for the emergence of the disordered phase in multi-band chaotic regimes, and proved that the phase order is sensitive to the density distribution of the trajectories of the attractors. The scaling behavior of the net direction phase at a transition point is observed. The analytical proof of this scaling relation is obtained. Both the numerical and analytical results show that the exponent is 1, which is controlled by the feature of the map independent on whether the system is continuous or discontinuous. It extends the universality of the scaling behavior to systems with discontinuity. The result in this work is important to understanding the property of chaotic motion in discontinuous systems.

A Chaotic Block Encryption Scheme Based on the Piecewise Nonlinear Map

In order to improve communication security, proposed a chaotic block cryptographic scheme based on the coupled piecewise nonlinear map. Using the coupled chaotic systems to generate random binary sequences, a key-dependent shill approach permated the plain-text block and then using the classical chaotic masking technique encrypted the plain-text block. Simulation results show that the proposed algorithm has excellent cryptographic properties such as diffusion and confusion properties and it can resist the know plaintext attacks and chosen plain-text attacks.

融合边界处理机制的学习型麻雀搜索算法

为改善麻雀搜索算法(SSA)初始化阶段种群分布不充分,寻优过程中容易受到局部最优解干扰的不足,提出融合边界处理机制的学习型麻雀搜索算法(HSSA)。使用Piecewise map初始化种群,提高种群的分散程度;使用排序配对学习与竞争学习策略分别更新跟随者和警戒者,确保各代的最优解信息能够引导下一代的位置更新;自适应的警戒者数量使得警戒者作用被强调,提供灵活的应变机制;根据不同阶段的寻优特点制定多策略边界处理机制,保留住种群数量的同时,为超出边界的个体提供更加合理的搜索位置。经过12个基准函数的仿真实验,并借助消融实验、Wilcoxon秩和检验等证明了HSSA在收敛速度上的稳定性和寻优的高效性。

Symbolic dynamics of Belykh-type maps

The symbolic dynamics of a Belykh-type map （a two-dimensional discon- tinuous piecewise linear map） is investigated. The admissibility condition for symbol sequences named the pruning front conjecture is proved under a hyperbolicity condition. Using this result, a symbolic dynamics model of the map is constructed according to its pruning front and primary pruned region. Moreover, the boundary of the parameter region in which the map is chaotic of a horseshoe type is given.

Admissibility Conditions for Symbolic Sequences in Dynamics of Digital Filter with Two's Complement Arithmetic

In this paper, we discuss a class of piecewise linear hyperbolic maps on the 2-torus. These maps arise in the second-order digital fdter with two＇ s complement arithmetic. By introducing codings underlying the map operations, we give some admissibility conditions for symbolic sequences and find some periodic properties of these symbolic sequences. Then we use these conditions to check the admissibility of periodic symbol sequences.

Intermittent Behaviors in Coupled Piecewise Expanding Map Lattices

In this paper,we propose a new method to study intermittent behaviors of coupled piecewise-expanding map lattices.We show that the successive transition between ordered and disordered phases occurs for almost every orbit when the coupling is small.That is,lim inf n→∞∑1≤i,j≤m|x_(i)(n)−x_(j)(n)|=0,lim sup n→∞∑1≤i,j≤m|x_(i)(n)−x_(j)(n)|≥c_(0)>0,where xi(n)correspond to the coordinates of m nodes at the iterative step n.Moreover,when the uncoupled system is generated by the tent map and the lattice consists of two nodes,we prove a phase transition occurs between synchronization and intermittent behaviors.That is,limn→∞|x_(1)(n)−x_(2)(n)|=0 for c−1/2<1/4 and intermittent behaviors occur for|c−1/2|>1/4,where 0≤c≤1 is the coupling.

The Inherent Law of the Unpredictability of Financial Asset Price Fluctuations: Multistability and Chaos

This paper aims at understanding the price dynamics generated by the interaction of traders relying on heterogeneous expectations in an asset pricing model.In the present work the authors analyze a financial market populated by five types of boundedly rational speculators-two types of fundamentalists,two types of chartists and trend followers which submit buying/selling orders according to different trading rules.The authors formulate a stock market model represented as a 2 dimensional piecewise linear discontinuous map.The proposed contribution to the existing financial literature is two aspects.First,the authors perform study of the model involving a 2 dimensional piecewise linear discontinuous map through a combination of qualitative and quantitative methods.The authors focus on the existence conditions of chaos and the multi-stability regions in parameter plane.Related border collision bifurcation curves and basins of multi-attractors are also given.The authors find that chaos or quasi-period exists only in the case of fixed point being a saddle(regular or flip)and that the coexistence of multiple attractors may exist when the fixed point is an attractor,but it is common for spiral and flip fixed points.