One-dimensional Maps Research Articles

Bifurcation Analysis of Reduced Network Model of Coupled Gaussian Maps for Associative Memory

This paper proposes an associative memory model based on a coupled system of Gaussian maps. A one-dimensional Gaussian map describes a discrete-time dynamical system, and the coupled system of Gaussian maps can generate various phenomena including asymmetric fixed and periodic points. The Gaussian associative memory can effectively recall one of the stored patterns, which were triggered by an input pattern by associating the asymmetric two-periodic points observed in the coupled system with the binary values of output patterns. To investigate the Gaussian associative memory model, we formed its reduced model and analyzed the bifurcation structure. Pseudo-patterns were observed for the proposed model along with other conventional associative memory models, and the obtained patterns were related to the high-order or quasi-periodic points and the chaotic trajectories. In this paper, the structure of the Gaussian associative memory and its reduced models are introduced as well as the results of the bifurcation analysis are presented. Furthermore, the output sequences obtained from simulation of the recalling process are presented. We discuss the mechanism and the characteristics of the Gaussian associative memory based on the results of the analysis and the simulations conducted.

Equality of kolmogorov-sinai and permutation entropy for one-dimensional maps consisting of countably many monotone parts

In this paper, we show that, under some technical assumptions, the Kolmogorov-Sinai entropy and the permutation entropy are equal for one-dimensional maps if there exists a countable partition of the domain of definition into intervals such that the considered map is monotone on each of those intervals. This is a generalization of a result by Bandt, Pompe and G. Keller, who showed that the above holds true under the additional assumptions that the number of intervals on which the map is monotone is finite and that the map is continuous on each of those intervals.

An Elementary Study of Chaotic Behaviors in 1-D Maps

In this article, we have discussed basic concepts of one-dimensional maps like Cubic map, Sine map and analyzed their chaotic behaviors in several senses in the unit interval. We have mainly focused on Orbit Analysis, Time Series Analysis, Lyapunov Exponent Analysis, Sensitivity to Initial Conditions, Bifurcation Diagram, Cobweb Diagram, Histogram, Mathematical Analysis by Newton’s Iteration, Trajectories and Sensitivity to Numerical Inaccuracies of the said maps. We have tried to make decision about these mentioned maps whether chaotic or not on a unique interval of parameter value. We have performed numerical calculations and graphical representations for all parameter values on that interval and have tried to find if there is any single value of parameter for which those maps are chaotic. In our calculations we have found there are many values for which those maps are chaotic. We have showed numerical calculations and graphical representations for single value of the parameter only in this paper which gives a clear visualization of chaotic dynamics. We performed all graphical activities by using Mathematica and MATLAB.

Analyzing Devaney Chaos of a Sine–Cosine Compound Function System

The one-dimensional Sine map and Chebyshev map are classical chaotic maps, which have clear chaotic characteristics. In this paper, we establish a chaotic framework based on a Sine–Cosine compound function system by analyzing the existing one-dimensional Sine map and Chebyshev map. The sensitive dependence on initial conditions, topological transitivity and periodic-point density of this chaotic framework is proved, showing that the chaotic framework satisfies Devaney’s chaos definition. In order to illustrate the chaotic behavior of the chaotic framework, we propose three examples, called Cosine–Polynomial (C–P) map, Sine–Tangent (S–T) map and Sine–Exponent (S–E) map, respectively. Then, we evaluate the chaotic behavior with Sine map and Chebyshev map by analyzing bifurcation diagrams, Lyapunov exponents, correlation dimensions, Kolmogorov entropy and [Formula: see text] complexity. Experimental results show that the chaotic framework has better unpredictability and more complex chaotic behaviors than the classical Sine map and Chebyshev map. The results also verify the effectiveness of the theoretical analysis of the proposed chaotic framework.

ON SOLUTION OF AMPLITUDE-PHASE PROBLEM

The paper describes an unconventional method of solving the amplitude-phase problem. The main properties of the Hilbert transform in the discrete and continual cases for one-dimensional and two-dimensional mappings are considered. These transformations are widely used to solve amplitude-phase problem. A numerical method for solving of two-dimensional amplitudephase problem is proposed. Preliminary information about the zeros of the Fourier transform of the initial signal is not required for this method. The method is based on the apparatus of nonlinear singular integral equations. Computational schemes for solving the corresponding nonlinear singular integral equations are developed. An algorithm for finding initial values for realization of iterative methods is proposed. Quadrature formulas of the calculation of singular integrals are proposed.

On symmetric color image encryption system with permutation-diffusion simultaneous operation

In the traditional image encryption systems, the permutation-diffusion operation is often conducted in two essentially separable or independent steps, which appears not only to be inefficient but also vulnerable in being attacked since each step can be targeted separately. In this paper, we propose a new plaintext-related cryptosystem in which the permutation-diffusion operation is performed in a simultaneous manner. More specifically, the simultaneous permutation-diffusion operation (PDSO) leads to the confusion and diffusion interacted each other, and thus generates better security level as well as higher efficiency of encryption process. The proposed encryption scheme only requires a single PDSO in the corresponding encryption process, in which the first encrypted pixel’s position in cipher image is randomly chosen and its value is determined by both pixels in the original image and the elements in key-streams obtained by three one-dimensional chaotic mappings, and the current encrypting pixel’s value is set to be associated with the previous encrypted pixel. Under our framework, the proposed encryption scheme is highly sensitive to minor differences in both secret key and plain image, resulting in resisting chosen/known plaintext attacks effectively. Simulation results show that the proposed scheme demonstrates excellent security capacity with high efficiency, comparing with existing approaches in literature.

A Nonlinearly Modulated Logistic Map with Delay for Image Encryption

Considering that a majority of the traditional one-dimensional discrete chaotic maps have disadvantages including a relatively narrow chaotic range, smaller Lyapunov exponents, and excessive periodic windows, a new nonlinearly modulated Logistic map with delay model (NMLD) is proposed. Accordingly, a chaotic map called a first-order Feigenbaum-Logistic NMLD (FL-NMLD) is proposed. Simulation results demonstrate that FL-NMLD has a considerably wider chaotic range, larger Lyapunov exponents, and superior ergodicity compared with existing chaotic maps. Based on FL-NMLD, we propose a new image encryption algorithm that joins the pixel plane and bit-plane shuffle (JPB). The simulation and test results confirm that JPB has higher security than simple pixel-plane encryption and is faster than simple bit-plane encryption. Moreover, it can resist the majority of attacks including statistical and differential attacks.

A Piecewise Linear Maximum Entropy Method for Invariant Measures of Random Maps with Position-Dependent Probabilities

We present a numerical method for the approximation of absolutely continuous invariant measures of one-dimensional random maps, based on the maximum entropy principle and piecewise linear moment functions. Numerical results are also presented to show the convergence of the algorithm.

Polynomial maps with hidden complex dynamics

The dynamics of a class of one-dimensional polynomial maps are studied, and interesting dynamics are observed under certain conditions: the existence of periodic points with even periods except for one fixed point; the coexistence of two attractors, an attracting fixed point and a hidden attractor; the existence of a double period-doubling bifurcation, which is different from the classical period-doubling bifurcation of the Logistic map; the existence of Li-Yorke chaos. Furthermore, based on this one-dimensional map, the corresponding generalized Henon map is investigated, and some interesting dynamics are found for certain parameter values: the coexistence of an attracting fixed point and a hidden attractor; the existence of Smale horseshoe for a subshift of finite type and also Li-Yorke chaos.

Noise-induced multistability in the square root map

The effects of small-amplitude additive Gaussian white noise on the one-dimensional square root map are investigated. In particular, the focus is on the unexpected effects noise of varying amplitudes has on the system for parameter regions just outside intervals of multistability. It is shown that in these regions periodic behaviour that is unstable in the deterministic system can be effectively stabilised by the addition of noise of an appropriate amplitude. Features of noise-induced transitions from stable to stabilised unstable periodic behaviour are highlighted, and it is shown how these features can be understood by examining relative levels of expansion and contraction in the deterministic map.

Infinite mixing for one-dimensional maps with an indifferent fixed point

We study the properties of ‘infinite-volume mixing’ for two classes of intermittent maps: expanding maps with an indifferent fixed point at 0 preserving an infinite, absolutely continuous measure, and expanding maps with an indifferent fixed point at preserving the Lebesgue measure. All maps have full branches. While certain properties are easily adjudicated, the so-called global-local mixing, namely the decorrelation of a global and a local observable, is harder to prove. We do this for two subclasses of systems. The first subclass includes, among others, the Farey map. The second class includes the standard Pomeau–Manneville map mod 1. Morevoer, we use global-local mixing to prove certain limit theorems for our intermittent maps.

Cryptanalysis and improvement of an image encryption scheme using combination of one-dimensional chaotic maps

Cryptanalysis and improvement of an image encryption scheme using combination of one-dimensional chaotic maps

Encryption of image data using compressive sensing and chaotic system

An image cryptosystem using chaotic compressive sensing is designed to achieve simultaneous compression - encryption. Compressive sensing requires a measurement matrix to compressively sample a sparse signal and to guarantee its recovery at the receiver. In this paper, a new one-dimensional chaotic map is proposed which is used to construct the chaotic measurement matrix. Performance analysis demonstrates that the proposed chaotic map is highly chaotic, ergodic, highly sensitive to the initial conditions and suitable for chaotic compressive sensing. The parameters of the chaotic system are used as the secret key in the construction of measurement matrix and also the masking matrix. The sparse representation of the image is obtained using discrete wavelet transform. The sparse coefficients are then compressively sampled and encrypted using the chaotic measurement matrix and masking matrix. A parallel compressive sensing framework is employed which greatly improves the efficiency of the proposed chaotic compressive sensing scheme. Simulation results shows that the proposed scheme has good security performance against various attacks and better reconstruction performance, when compared with the commonly used random measurement matrix.

Describing dynamics of driven multistable oscillators with phase transfer curves.

Phase response curve is an important tool in the studies of stable self-sustained oscillations; it describes a phase shift under action of an external perturbation. We consider multistable oscillators with several stable limit cycles. Under a perturbation, transitions from one oscillating mode to another one may occur. We define phase transfer curves to describe the phase shifts at such transitions. This allows for a construction of one-dimensional maps that characterize periodically kicked multistable oscillators. We show that these maps are good approximations of the full dynamics for large periods of forcing.

Delay-Independent Design for Synchronization in Delayed-Coupled One-Dimensional Map Networks

Delay-Independent Design for Synchronization in Delayed-Coupled One-Dimensional Map Networks

Parameter estimation for 1D PWL chaotic maps using noisy dynamics

Many physical situations involve chaotic systems implemented in hardware. Among them one-dimensional piecewise linear maps are popular candidates for such applications because of their property of generating robust chaos. In physical implementations, the control parameter of these maps may deviate from its ideal value due to hardware imprecision. Since the dynamics of a chaotic map is completely defined by its control parameter, one needs to know the value of the parameter in a hardware realisation. In this paper, we show that it is possible to determine the parameter, through the realisation of the unstable fixed point of the map, by utilising noise that is always present in the system. We present this in the form of an algorithm and demonstrate its efficacy through simulated results. We also determine the bounds on the signal-to-noise ratio required for successful parameter estimation. The proposed approach is expected to be beneficial to the existing noise reduction techniques and time series recovery algorithms that require a reasonably accurate knowledge of the map.

Phase order in one-dimensional piecewise linear discontinuous map**Project supported by the National Natural Science Foundation of China (Grant No. 11645005) and the Interdisciplinary Incubation Project of Shaanxi Normal University (Grant No. 5).

The phase order in a one-dimensional (1D) 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.

Some characteristics of the M function methodology to describe the reinjection process in chaotic intermittency

The M function methodology allows calculating the statistical properties of intermittency in a broad class of one-dimensional maps. It has shown to be very accurate in type I, II, III and V intermittencies; and it also includes the uniform reinjection as a particular case. This paper studies some properties of the M function methodology. We establish the conditions that a reinjection probability density function must verify to obtain and we describe new pathological cases where the reinjection is not uniform, but the characteristic relation is the same that for uniform reinjection.

The variation of invariant graphs in forced systems.

In skew-product systems with contractive factors, all orbits asymptotically approach the graph of the so-called sync function; hence, the corresponding regularity properties primarily matter. In the literature, sync function Lipschitz continuity and differentiability have been proved to hold depending on the derivative of the base reciprocal, if not on its Lyapunov exponent. However, forcing topological features can also impact the sync function regularity. Here, we estimate the total variation of sync functions generated by one-dimensional Markov maps. A sharp condition for bounded variation is obtained depending on parameters, which involves the Markov map topological entropy. The results are illustrated with examples.

Investigation of the Class of One-Dimensional Unimodal Mappings Obtained in the Modeling of the Lemming Population

A model that describes periodic changes in the lemming population size as a function of the physical characteristics of ecologically significant parameters of the habitat is analyzed. Earlier studies on modeling of the population size dynamics for tundra animals substantiated a new class of one-dimensional unimodal mappings of a state space into itself. The dependence of trajectory stability on parameter variation and the bifurcation and asymptotic properties of the trajectories are in the focus of attention when this class of mappings is studied. Methods for approximating implicitly defined sets are used for numerical analysis of the maps: trajectory tubes, attractors, and bifurcation diagrams are constructed. An example of a bifurcation diagram construction along the contour in the parameter space is given for this class of mappings; the diagram enables the analysis of the dependence of trajectory behavior on changes of the modeling map parameters. Thus, the use of the mapping of the type considered for the model description of the lemming population enabled the study of the effect of long-term biospheric rhythms, which include time intervals with extreme living conditions. The study showed that such impacts do not cause degeneration of the population: the periods with chaotic dynamics that emerge under extreme conditions are replaced by well-ordered behavior in the form of small-period population cycles.