Hierarchy of rational order families of chaotic maps with an invariant measure

We introduce an interesting hierarchy of rational order chaotic maps that possess an invariant measure. In contrast to the previously introduced hierarchy of chaotic maps [1–5], with merely entropy production, the rational order chaotic maps can simultaneously produce and consume entropy. We compute the Kolmogorov-Sinai entropy of these maps analytically and also their Lyapunov exponent numerically, where the obtained numerical results support the analytical calculations.     

Hierarchy of Chaotic Maps with an Invariant Measure and their Compositions

Abstract

We give a hierarchy of many-parameter families of maps of the interval [0, 1] with an invariant measure and using the measure, we calculate Kolmogorov–Sinai entropy of these maps analytically. In contrary to the usual one-dimensional maps these maps do not possess period doubling or period-n-tupling cascade bifurcation to chaos, but they have single fixed point attractor at certain region of parameters space, where they bifurcate directly to chaos without having period-n-tupling scenario exactly at certain values of the parameters.     

Algebraic Entropy of Birational Maps with Invariant Curves

The relevance of the algebraic entropy in the study of birational discrete time dynamical systems highlights the need to relate it to other characteristics of these systems. In this Letter, two complementary proofs are given that the foliation of the space by invariant curves implies that the algebraic entropy is zero.     

An Improved Calculation Formula of the Extended Entropic Chaos Degree and Its Application to Two-Dimensional Chaotic Maps

The Lyapunov exponent is primarily used to quantify the chaos of a dynamical system. However, it is difficult to compute the Lyapunov exponent of dynamical systems from a time series. The entropic chaos degree is a criterion for quantifying chaos in dynamical systems through information dynamics, which is directly computable for any time series. However, it requires higher values than the Lyapunov exponent for any chaotic map. Therefore, the improved entropic chaos degree for a one-dimensional chaotic map under typical chaotic conditions was introduced to reduce the difference between the Lyapunov exponent and the entropic chaos degree. Moreover, the improved entropic chaos degree was extended for a multidimensional chaotic map. Recently, the author has shown that the extended entropic chaos degree takes the same value as the total sum of the Lyapunov exponents under typical chaotic conditions. However, the author has assumed a value of infinity for some numbers, especially the number of mapping points. Nevertheless, in actual numerical computations, these numbers are treated as finite. This study proposes an improved calculation formula of the extended entropic chaos degree to obtain appropriate numerical computation results for two-dimensional chaotic maps.     

Analysis of Chaotic Dynamics by the Extended Entropic Chaos Degree

The Lyapunov exponent is the most-well-known measure for quantifying chaos in a dynamical system. However, its computation for any time series without information regarding a dynamical system is challenging because the Jacobian matrix of the map generating the dynamical system is required. The entropic chaos degree measures the chaos of a dynamical system as an information quantity in the framework of Information Dynamics and can be directly computed for any time series even if the dynamical system is unknown. A recent study introduced the extended entropic chaos degree, which attained the same value as the total sum of the Lyapunov exponents under typical chaotic conditions. Moreover, an improved calculation formula for the extended entropic chaos degree was recently proposed to obtain appropriate numerical computation results for multidimensional chaotic maps. This study shows that all Lyapunov exponents of a chaotic map can be estimated to calculate the extended entropic chaos degree and proposes a computational algorithm for the extended entropic chaos degree; furthermore, this computational algorithm was applied to one and two-dimensional chaotic maps. The results indicate that the extended entropic chaos degree may be a viable alternative to the Lyapunov exponent for both one and two-dimensional chaotic dynamics.     

Scaling Equation for Invariant Measure

An iterated function system (IFS) is constructed. It is shown that the invariant measure of IFS satisfies the same equation as scaling equation for wavelet transform (WT). Obviously, IFS and scaling equation of WT both have contraction mapping principle.     

Controlling Chaotic Behavior in a Bose-Einstein Condensate with Linear Feedback

The spatial structure of a Bose-Einstein condensate （BEC） loaded into an optical lattice potential is investigated and the spatially chaotic distributions of the condensates are revealed. A method of chaos control with linear feedback is presented in this paper. By using the method, we propose a scheme of controlling chaotic behavior in a BEC with atomic mirrors. The results of the computer simulation show that controlling the chaos into the stable states could be realized by adjusting the coefficient of feedback only if the maximum Lyapunov exponent of the system is negative.     

Controlling Chaotic Behavior in a Bose-Einstein Condensate with Linear Feedback

The spatial structure of a Bose-Einstein condensate (BEC) loaded into an optical lattice potential is investigated and the spatially chaotic distributions of the condensates are revealed. A method of chaos control with linear feedback is presented in this paper. By using the method, we propose a scheme of controlling chaotic behavior in a BEC with atomic mirrors. The results of the computer simulation show that controlling the chaos into the stable states could be realized by adjusting the coefficient of feedback only if the maximum Lyapunov exponent of the system is negative.     

DC-DC变换器的信息熵分析

为确定不同反馈系数k下DC-DC变换器系统的行为,结合系统处于周期状态时的稳定性和系统处于混沌时不会重复经过每一点的特点,提出了一种采用极限思想和信息熵来估计DC-DC变换器非线性行为的方法.该方法能准确分析系统处于周期状态和混沌状态的熵值,量化了DC-DC变换器倍周期分叉和混沌行为.以一阶电压反馈DCM Boost变换器和DCM Buck变换器为例进行仿真.研究结果表明,所提出的信息熵可以准确反映分叉点、周期数及产生混沌的位置,完善了该类变换器非线性动力学分析的理论和方法.     

连续混沌调频信号的动力学设计与分析

提出了基于连续混沌时间尺度变换的连续混沌调频信号动力学设计理论,并以Colpitts电路为例给出了具体的实现模型.通过吸引子重构和Lyapunov指数分析,阐述了连续混沌调频信号的动力学特性.结果显示连续混沌调频信号的最大Lyapunov指数与混沌信号的最大Lyapunov指数相同;连续混沌调频信号的动力学设计实现方式简单且系统参数易于控制,同时信号形式与数学意义上的连续混沌调频信号保持一致.     

地球变化磁场Z分量的混沌动力学特性分析

对地球变化磁场进行了混沌动力学特性分析,主要进行了以下两个方面的工作:1)运用多种方法对多个地磁台站同一时段变化磁场分量数据(分为磁扰较小和磁扰较大时段)进行了混沌特性分析,主要是为了以更充分的判据来验证地球变化磁场是否具有混沌特性;2)同一地磁台站不同时段变化磁场分量数据的混沌特性分析,主要是验证地球变化磁场是否具有变参数混沌特性.分析结果表明:1)变化磁场时间序列确实具有混沌特性;2)从混沌时间序列相空间重构的角度可初步认为模拟地球变化磁场动力学系统所必须的独立变量应该为6左右;3)变化磁场时间序列不仅具有混沌特性而且其参数也是在缓慢变化着的,具有变参数混沌系统的特性.     

Chaotic phenomena in Josephson circuits coupled quantum cellular neural networks

In this paper the nonlinear dynamical behaviour of a quantum cellular neural network (QCNN) by coupling Josephson circuits was investigated and it was shown that the QCNN using only two of them can cause the onset of chaotic oscillation. The theoretical analysis and simulation for the two Josephson-circuits-coupled QCNN have been done by using the amplitude and phase as state variables. The complex chaotic behaviours can be observed and then proved by calculating Lyapunov exponents. The study provides valuable information about QCNNs for future application in high-parallel signal processing and novel chaotic generators.     

Measure Synchronization on Symplectic Map

Measure synchronization in coupled Hamiltonian systems is a novel synchronization phenomenon. The measure synchronization on symplectic map is observed numerically, for identical coupled systems with different parameters. We have found the properties of the characteristic frequency and the amplitude of phase locking in regular motion when the measure synchronization of coupled systems is obtained. The relations between the change of the largest Lyapunov exponent and the course of phase desynchronization are also discussed in coupled systems, some useful results are obtained. A new approach is proposed for describing the measure synchronization of coupled systems numerically, which is advantage in judging the measure synchronization, especially for the coupled systems in nonregular region.     

Measures with infinite Lyapunov exponents for the periodic Lorentz gas

We study invariant measures for the periodic Lorentz gas which are supported on the set of points with infinite Lyapunov exponents. We construct examples of such measures which are measures of maximal entropy and ones which are not.     

Note on the Kaplan–Yorke Dimension and Linear Transport Coefficients

A number of new relations between the Kaplan–Yorke dimension, phase space contraction, transport coefficients and the maximal Lyapunov exponents are given for dissipative thermostatted systems, subject to a small but non-zero external field in a nonequilibrium stationary state. A condition for the extensivity of phase space dimension reduction is given. A new expression for the linear transport coefficients in terms of the Kaplan–Yorke dimension is derived. Alternatively, the Kaplan–Yorke dimension for a dissipative macroscopic system can be expressed in terms of the linear transport coefficients of the system. The agreement with computer simulations for an atomic fluid at small shear rates is very good.     

Nonlinear Full Invariant of Compact Banach-Space Maps

We characterize a nonlinear full invariant of compact Banach-space maps: Let (X, ‖.‖) and (Y, ‖.‖) be two Banach spaces and P_C(X, Y) be all compact maps which map (X, ‖.‖) to (Y, ‖.‖). Then each weak operator-topology subseries-convergent series ∑_i P_i in P_c(X, Y) is also uniform-topology subseries-convergent iff each bounded map from (X, ‖.‖) to (l¹, ‖.‖₁) is a compact map. The necessary condition for each weak operator-topology subseries-convergent series ∑_i P_i in P_C(X, Y) to be also uniform-topology subseries-convergent is that (X, ‖.‖) and (X′, ‖.‖) both contain no copy of c₀. This necessary condition is not sufficient.PACS: 02.10 By, 02.10 Gd     

Chaotic oscillations and noise transformations in a simple dissipative system with delayed feedback

We analyze the statistical behavior of signals in nonlinear circuits with delayed feedback in the presence of external Markovian noise. For the special class of circuits with intense phase mixing we develop an approach for the computation of the probability distributions and multitime correlation functions based on the random phase approximation. Both Gaussian and Kubo-Andersen models of external noise statistics are analyzed and the existence of the stationary (asymptotic) random process in the long-time limit is shown. We demonstrate that a nonlinear system with chaotic behavior becomes a noise amplifier with specific statistical transformation properties.     

一类自治混沌系统的动力学分析与电路实现

分析讨论一类新的三维二次自治混沌系统.通过理论分析、数值仿真,计算系统的Lyapunov指数谱和Lyapunov维数,以及系统的分岔图等,对系统的基本动力学特性进行分析.设计实现该系统的混沌电路,建立系统参数和电路参数的对应关系,运用Pspice软件对电路进行仿真实验,证实分析的正确性.所讨论的系统不同于Lorenz系统、Rössler系统、Chen系统和Lü系统等,且电路实现很简单实用.     

Every Continuous Operator Has an Invariant Measure

We prove the existence of an invariant measure for a large class of random processes with discrete time without assuming their linearity. Our main examples are “processes with variable length”, in which components may appear and disappear in the course of functioning. One of these examples displays non-uniqueness of invariant measure in a 1-D process.