Synthesizing chaotic maps with prescribed invariant densities

The Inverse Frobenius–Perron Problem (IFPP) concerns the creation of discrete chaotic mappings with arbitrary invariant densities. In this Letter, we present a new and elegant solution to the IFPP, based on positive matrix theory. Our method allows chaotic maps with arbitrary piecewise-constant invariant densities, and with arbitrary mixing properties, to be synthesized.     

Harmonic Averages and New Explicit Constants for Invariant Densities of Piecewise Expanding Maps of the Interval

The statistical behavior of families of maps is important in studying the stability properties of chaotic maps. For a piecewise expanding map τ whose slope >2 in magnitude, much is known about the stability of the associated invariant density. However, when the map has slope magnitude ≤2 many different behaviors can occur as shown in (Keller in Monatsh. Math. 94(4): 313–333, 1982) for W maps. The main results of this note use a harmonic average of slopes condition to obtain new explicit constants for the upper and lower bounds of the invariant probability density function associated with the map, as well as a bound for the speed of convergence to the density. Since these constants are determined explicitly the results can be extended to families of approximating maps.     

The transition to chaos for a special solution of the area-preserving quadratic map

Following previous work on chaotic boundaries of half-plane Hamiltonian maps a special solution of the area-preserving quadratic map is introduced and investigated. The breakdown of regular bounded motion on invariant curves is found from the radius of convergence of a power series whose successive terms oscillate wildly due to the presence of small divisors. Previous techniques for taming such series are found to be insufficient and new ones are introduced.It is found that half-plane Hamiltonian maps appear to have certain universal features and that the chaotic boundary has similarities to the boundaries of Siegel domains of complex conformal maps.The chaotic boundary function α_c(ν) has some interesting new features which are not fully understood.     

Lyapunov exponent for chaotic 1D maps with uniform invariant distribution

Some properties of iterative functions of 1D chaotic maps that provide uniform invariant distribution are formulated. A method for synthesizing strictly nonlinear maps with uniform invariant distribution is demonstrated. The Lyapunov exponents for such maps are analyzed and it is shown that, among the maps with a specified number of full branches, piecewise linear maps with branches characterized by equal moduli of angular coefficients have the maximum Lyapunov exponent.     

A novel scheme for image encryption based on 2D piecewise chaotic maps

In this paper, a hierarchy of two-dimensional piecewise nonlinear chaotic maps with an invariant measure is introduced. These maps have interesting features such as invariant measure, ergodicity and the possibility of K-S entropy calculation. Then by using significant properties of these chaotic maps such as ergodicity, sensitivity to initial condition and control parameter, one-way computation and random like behavior, we present a new scheme for image encryption. Based on all analysis and experimental results, it can be concluded that, this scheme is efficient, practicable and reliable, with high potential to be adopted for network security and secure communications. Although the two-dimensional piecewise nonlinear chaotic maps presented in this paper aims at image encryption, it is not just limited to this area and can be widely applied in other information security fields.     

Infinite invariant density determines statistics of time averages for weak chaos

Weakly chaotic nonlinear maps with marginal fixed points have an infinite invariant measure. Time averages of integrable and nonintegrable observables remain random even in the long time limit. Temporal averages of integrable observables are described by the Aaronson-Darling-Kac theorem. We find the distribution of time averages of nonintegrable observables, for example, the time average position of the particle, x[over ˉ]. We show how this distribution is related to the infinite invariant density. We establish four identities between amplitude ratios controlling the statistics of the problem.     

Chaotic boundary of a Hamiltonial map

The half-plane maps are a class of complex Hamiltonial maps whose invariant curves at most fixed irrational frequencies can be obtained as a convergent Taylor series expansion. For these maps the boundary between regular ordered motion on invariant curves and irregular chaotic motion is given by the radius of convergence of the series. The successive terms of the series oscillate wildly, due to the presence of small divisors. Methods are presented for taming the series, based on the conversion of the convergenceexponentC = -lnα_c into the integral of a continuous but nondifferentiable lambda function, whose graph whows a similarity structure on small scales. Self-similarity properties are illustrated for the chaotic boundary function α_c, where v is the frequency.     

Persistence of order and structure in chaos

In this paper some results are presented concerning one-dimensional chaotic maps with arbitrarily many critical points. Let f be a chaotic map belonging to some suitable class of C¹ maps from a nontrivial interval X into itself.

Assuming that f is of class C¹⁺ for some > 0, we have that the set of aperiodic points for f has Lebesgue measure zero; further, if f(X) is bounded then there exists a positive integer p such that almost every point in the interval is asymptotically periodic with period p. Moreover, it will turn out that this asymptotically periodic behaviour in the complicated dynamics of f is persistent under small smooth perturbations.

The topological structure of the nonwandering set of f will be described, and this structure is invariant under small C¹ perturbations of the map f.

Assuming that f is of class C², the map f is C² structurally stable provided that f satisfies some suitable conditions.

Finally, it will turn out that maps with a negative Schwarzian derivative belong to the suitable class of maps mentioned above.     

Two-dimensional solvable chaos

Two methods are proposed to construct two-dimensional chaotic maps. Several examples of exactly solvable chaotic maps and their invariant measures are obtained. They are isomorphic maps of square to square, plane to plane and circle to circle having various symmetry such as uniform, rotational and the quartic rotational symmetry.     

Classification of one-dimensional chaotic models

Classifying chaotic maps using the relation between a map and its conjugate basic map with uniform invariant distribution is suggested. It is shown that every symmetric one-dimensional chaotic map with two monotonic branches is topologically equivalent to a tent map or a Bernoulli shift. An algorithm for finding a function conjugating two maps is formulated.     

Noisy one-dimensional maps near a crisis. I. Weak Gaussian white and colored noise

We study one-dimensional single-humped maps near the boundary crisis at fully developed chaos in the presence of additive weak Gaussian white noise. By means of a new perturbation-like method the quasi-invariant density is calculated from the invariant density at the crisis in the absence of noise. In the precritical regime, where the deterministic map may show periodic windows, a necessary and sufficient condition for the validity of this method is derived. From the quasi-invariant density we determine the escape rate, which has the form of a scaling law and compares excellently with results from numerical simulations. We find that deterministic transient chaos is stabilized by weak noise whenever the maximum of the map is of orderz>1. Finally, we extend our method to more general maps near a boundary crisis and to multiplicative as well as colored weak Gaussian noise. Within this extended class of noises and for single-humped maps with any fixed orderz>0 of the maximum, in the scaling law for the escape rate both the critical exponents and the scaling function are universal.     

Fully developed chaotic 1−d maps

Single-hump 1–d maps are investigated which generate ergodic process on an interval mapped everywhere two-to-one onto itself. Introducing a new transformation transverse to conjugation it is shown that such maps are related by smooth transformations to each other. It is found that each of the families consisting of conjugate maps contains a map everywhere expanding and producing ergodic iterations according to the uniform probability density. The general framework is used to construct maps together with their probability density functions. Quantities characterizing the dynamics are calculated and their parameter dependence while maintaining the fully developed chaotic state is studied. Furthermore, universal maps exhibiting fully developed chaos are considered.     

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.     

Topological invariants in the study of a chaotic food chain system

The study of ecological systems has generated deep interest in exploring the complexity of chaotic food chains. The role of chaos in ecosystems is not entirely understood. One approach to have a better comprehension of ecological chaos is by analyzing it in mathematical models of basic food chains. In this article it is considered a classical chaotic food chain model from the literature. We use the theory of symbolic dynamics to study the topological entropy and the parameter space ordering of kneading sequences associated with one-dimensional maps that reproduce significant aspects of the model dynamics. The topological entropy allows us to distinguish different chaotic states in some realistic system parameter region. Another numerical invariant is introduced in order to characterize isentropic dynamics. Studying a set of maps with the same topological entropy, we exhibit numerical results about the relation between the second topological invariant and each of the control parameters in consideration. This work provides an illustration of how our understanding of ecological models can be enhanced by the theory of symbolic dynamics.     

Quantisations of Piecewise Parabolic Maps on the Torus and their Quantum Limits

For general quantum systems the semiclassical behaviour of eigenfunctions in relation to the ergodic properties of the underlying classical system is quite difficult to understand. The Wignerfunctions of eigenstates converge weakly to invariant measures of the classical system, the so-called quantum limits, and one would like to understand which invariant measures can occur that way, thereby classifying the semiclassical behaviour of eigenfunctions. We introduce a class of maps on the torus for whose quantisations we can understand the set of quantum limits in great detail. In particular we can construct examples of ergodic maps which have singular ergodic measures as quantum limits, and examples of non-ergodic maps where arbitrary convex combinations of absolutely continuous ergodic measures can occur as quantum limits. The maps we quantise are obtained by cutting and stacking.     

Evaluation of channel coding and decoding algorithms using discrete chaotic maps

In this paper we address the design of channel encoding algorithms using one-dimensional nonlinear chaotic maps starting from the desired invariant probability density function (pdf) of the data sent to the channel. We show that, with some simple changes, it is straightforward to make use of a known encoding framework based upon the Bernoulli shift map and adapt it readily to carry the information bit sequence produced by a binary source in a practical way. On the decoder side, we introduce four already known decoding algorithms and compare the resulting performance of the corresponding transmitters. The performance in terms of the bit error rate shows that the most important design clue is related not only to the pdf of the data produced by the chosen discrete map: the own dynamics of the maps is also of the highest importance and has to be taken into account when designing the whole transmitting and receiving system. We also show that a good performance in such systems needs the extensive use of all the evidence stored in the whole chaotic sequence.     

Dynamics of the Chaplygin ball on a rotating plane

This paper addresses the problem of the Chaplygin ball rolling on a horizontal plane which rotates with constant angular velocity. In this case, the equations of motion admit area integrals, an integral of squared angular momentum and the Jacobi integral, which is a generalization of the energy integral, and possess an invariant measure. After reduction the problem reduces to investigating a three-dimensional Poincaré map that preserves phase volume (with density defined by the invariant measure). We show that in the general case the system’s dynamics is chaotic.     

基于复合非线性数字滤波器的Hash函数构造

在对多个满足Kelber条件的滤波器组成的复合系统进行初步分析的基础上，提出了一个基于复合非线性数字滤波器的带密钥的Hash算法.算法首先构建能产生高维混沌序列的复合滤波器系统，然后在明文作用的复合序列控制下随机选择滤波器子系统，并以复合系统的初态作为密钥，以粗粒化的量化迭代轨迹作为明文的Hash值.讨论了复合系统实现Hash函数的不可逆性、防伪造性、初值敏感性等特点.研究结果表明：基于复合非线性数字滤波器的Hash算法简单快速，比基于单一混沌映射的Hash算法有着更高的安全性，同时滤波器结构中没有复杂的浮点运算，比一般复合混沌系统更易于软硬件实现. 关键词： Hash 函数 混沌 非线性自回归数字滤波器     

Anticipating the dynamics of chaotic maps

We study the regime of anticipated synchronization in unidirectionally coupled chaotic maps such that the slave map has its own output re-injected after a certain delay. For a class of simple maps, we give analytic conditions for the stability of the synchronized solution, and present results of numerical simulations of coupled 1D Bernoulli-like maps and 2D Baker maps, that agree well with the analytic predictions.     

Properties of fully developed chaos in one-dimensional maps

We consider single-humped symmetric one-dimensional maps generating fully developed chaotic iterations specified by the property that on the attractor the mapping is everywhere two to one. To calculate the probability distribution function, and in turn the Lyapunov exponent and the correlation function, a perturbation expansion is developed for the invariant measure. Besides deriving some general results, we treat several examples in detail and compare our theoretical results with recent numerical ones. Furthermore, a necessary condition is deduced for the probability distribution function to be symmetric and an effective functional iteration method for the measure is introduced for numerical purposes.