One-dimensional Piecewise Linear Map Research Articles

Degenerate Period Adding Bifurcation Structure of One-Dimensional Bimodal Piecewise Linear Maps

Motivated by a problem in the management of ecological populations, we study the bifurcation structure known as period adding structure for a family of one-dimensional bimodal piecewise linear maps...

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.

A financial market model with two discontinuities: Bifurcation structures in the chaotic domain.

We continue the investigation of a one-dimensional piecewise linear map with two discontinuity points. Such a map may arise from a simple asset-pricing model with heterogeneous speculators, which can help us to explain the intricate bull and bear behavior of financial markets. Our focus is on bifurcation structures observed in the chaotic domain of the map's parameter space, which is associated with robust multiband chaotic attractors. Such structures, related to the map with two discontinuities, have been not studied before. We show that besides the standard bandcount adding and bandcount incrementing bifurcation structures, associated with two partitions, there exist peculiar bandcount adding and bandcount incrementing structures involving all three partitions. Moreover, the map's three partitions may generate intriguing bistability phenomena.

1次元区分的1次分数写像の真軌道生成法とその安全性検証(応用, 応用カオス研究部会)

1次元区分的1次分数写像の真軌道生成法とその安全性検証(応用, 応用カオス研究部会)

Bifurcation structure in the skew tent map and its application as a border collision normal form

The goal of the present paper is to collect the results related to dynamics of a one-dimensional piecewise linear map widely known as the skew tent map. These results may be useful for the researchers working on theoretical and applied problems in the field of nonsmooth dynamical systems. In particular, we propose the complete description of the bifurcation structure of the parameter space of the skew tent map, providing the related proofs. It is also shown how these results can be used to classify border collision bifurcations in one-dimensional piecewise smooth maps.

Symmetry breaking in a bull and bear financial market model

We investigate bifurcation structures in the parameter space of a one-dimensional piecewise linear map with two discontinuity points. This map describes endogenous bull and bear market dynamics arising from a simple asset-pricing model. An important feature of our model is that some speculators only enter the market if the price is sufficiently distant to its fundamental value. Our analysis starts with the investigation of a particular case in which the map is symmetric with respect to the origin, associated with equal market entry thresholds in the bull and bear market. We then generalize our analysis by exploring how novel bifurcation structures may emerge when the map’s symmetry is broken.

Comparison of random and deterministic characteristics of chaotic signals issued from a one-dimensional piecewise linear map

This paper presents the characteristics of the power spectral density (PSD) of chaotic signals generated by a one-dimensional piecewise linear map. The majority of previous research on chaos show that chaotic signals are rather broadband with impulsive auto-correlation sequence (ACS). However, recent studies of the skew tent map [4, 5] have shown that the PSD and ACS are modi ed according to certain values of the bifurcations. We propose to extend this work to other maps and to study relations between bandwidth and Lyapunov exponents.

Border collision bifurcations and power spectral density of chaotic signals generated by one-dimensional discontinuous piecewise linear maps

Recently, many papers have appeared which study the power spectral density (PSD) of signals issued from some specific maps. This interest in the PSD is due to the importance of frequency in the telecommunications and transmission security. With the large number of wireless systems, the availability of frequencies for transmission and reception is increasingly uncommon for wireless communications. Also, guided media have limitations related to the bandwidth of a signal. In this paper, we investigate some properties associated to the border-collision bifurcations in a one-dimensional piecewise-linear map with three slopes and two parameters. We derive analytical expressions for the autocorrelation sequence, power spectral density (PSD) of chaotic signals generated by our piecewise-linear map. We prove the existence of strong relation between different types of the power spectral density (low-pass, high-pass or band-stop) and the parameters. We also find a relation between the type of spectrum and the order of attractive cycles which are located after the border collision bifurcation between chaos and cycles.

Computation of true chaotic orbits using cubic irrationals

We introduce a method that enables us to generate long true orbits of discrete-time dynamical systems defined by one-dimensional piecewise linear fractional maps with integer coefficients. The method uses cubic irrationals to represent numbers and involves only integer arithmetic to compute true orbits. By applying the method to the Bernoulli map and a modified Bernoulli map, we show that it successfully generates true chaotic and intermittent orbits, respectively, in contrast with conventional simulation methods. We demonstrate through simulations concerning invariant measures and the power spectrum that the statistical properties of the true orbits generated agree well with those of typical orbits of the two maps.

Analytical expressions for power spectral density issued from one-dimensional continuous piecewise linear maps with three slopes

During the two last decades, chaotic signals have been increasingly considered in telecommunications, signal processing or secure transmissions. Despite the importance of frequency in the telecommunications and transmission security there are few works that explore the spectrum of chaotic signals by considering the bandwidths. Previous works have concerned skew tent maps. In this paper, we derive analytical expressions for the autocorrelation sequence (ACS) and power spectral density (PSD) of chaotic signals generated by one-dimensional continuous piecewise linear maps with three slopes. We obtain similar results to those obtained using the skew-tent map. Our results permit to conjecture that the bandwidth of chaotic signals obtained from piecewise linear continuous map can have the same behaviour whatever be the number of slopes in the map.

One-dimensional maps with two discontinuity points and three linear branches: mathematical lessons for understanding the dynamics of financial markets

We develop a simple financial market model with heterogeneous interacting speculators. The dynamics of our model is driven by a one-dimensional discontinuous piecewise linear map, having two discontinuity points and three linear branches. On the one hand, we study this map analytically and numerically to advance our knowledge about such dynamical systems. In particular, not much is known about discontinuous maps involving three branches. On the other hand, we seek to improve our understanding of the functioning of financial markets. We find, for instance, that such maps can generate complex bull and bear market dynamics.

NUMERICAL ANALYSIS OF TRANSIENT AND PERIODIC DYNAMICS IN SINGLE AND COUPLED NAGUMO–SATO MODELS

The Nagumo–Sato (NS) model is a one-dimensional piecewise linear map that describes simplified dynamics of a single neuron. The NS model and its network extension, coupled Nagumo–Sato models, exhibit complex behavior both in their transient dynamics and after converging to periodic orbits. However, the way the period and the transient length change against the parameters is not completely understood. In this study, we numerically investigate the transient and periodic dynamics in single and coupled NS models. Simulation results indicate the following observations. (1) The period of a single NS model shows layered structures associated with the Farey sequence. (2) Two coupled NS models show discontinuity in the transient length, even though the period does not change. (3) In the case of an associative memory model consisting of NS models, there exists a small parameter region where both the period and the transient length increase considerably. The dynamics within the region is much more complex than that outside the region.

PERIOD ADDING IN PIECEWISE LINEAR MAPS WITH TWO DISCONTINUITIES

In this work we consider the border collision bifurcations occurring in a one-dimensional piecewise linear map with two discontinuity points. The map, motivated by an economic application, is written in a generic form and considered in the stable regime, with all slopes between zero and one. We prove that the period adding structures occur in maps with more than one discontinuity points and that the Leonov's method to calculate the bifurcation curves forming these structures is applicable also in this case. We demonstrate the existence of particular codimension-2 bifurcation (big-bang bifurcation) points in the parameter space, from which infinitely many bifurcation curves are issuing associated with cycles involving several partitions. We describe how the bifurcation structure of a map with one discontinuity is modified by the introduction of a second discontinuity point, which causes orbits to appear located on three partitions and organized again in a period-adding structure. We also describe particular codimension-2 bifurcation points which represent limit sets of doubly infinite sequences of bifurcation curves and appear due to the existence of two discontinuities.

Border collision bifurcations in a two-dimensional piecewise smooth map from a simple switching circuit

In recent years, the study of chaotic and complex phenomena in electronic circuits has been widely developed due to the increasing number of applications. In these studies, associated with the use of chaotic sequences, chaos is required to be robust (not occurring only in a set of zero measure and persistent to perturbations of the system). These properties are not easy to be proved, and numerical simulations are often used. In this work, we consider a simple electronic switching circuit, proposed as chaos generator. The object of our study is to determine the ranges of the parameters at which the dynamics are chaotic, rigorously proving that chaos is robust. This is obtained showing that the model can be studied via a two-dimensional piecewise smooth map in triangular form and associated with a one-dimensional piecewise linear map. The bifurcations in the parameter space are determined analytically. These are the border collision bifurcation curves, the degenerate flip bifurcations, which only are allowed to occur to destabilize the stable cycles, and the homoclinic bifurcations occurring in cyclical chaotic regions leading to chaos in 1-piece.

Border collision bifurcation curves and their classification in a family of 1D discontinuous maps

In this paper we consider a one-dimensional piecewise linear discontinuous map in canonical form, which may be used in several physical and engineering applications as well as to model some simple financial markets. We classify three different kinds of possible dynamic behaviors associated with the stable cycles. One regime (i) is the same existing in the continuous case and it is characterized by periodicity regions following the period increment by 1 rule. The second one (ii) is the regime characterized by periodicity regions of period increment higher than 1 (we shall see examples with 2 and 3), and by bistability. The third one (iii) is characterized by infinitely many periodicity regions of stable cycles, which follow the period adding structure, and multistability cannot exist. The analytical equations of the border collision bifurcation curves bounding the regions of existence of stable cycles are determined by using a new approach.

Characterization of intermittency in renewal processes: Application to earthquakes

We construct a one-dimensional piecewise linear intermittent map from the interevent time distribution for a given renewal process. Then, we characterize intermittency by the asymptotic behavior near the indifferent fixed point in the piecewise linear intermittent map. Thus, we provide a framework to understand a unified characterization of intermittency and also present the Lyapunov exponent for renewal processes. This method is applied to the occurrence of earthquakes using the Japan Meteorological Agency and the National Earthquake Information Center catalog. By analyzing the return map of interevent times, we find that interevent times are not independent and identically distributed random variables but that the conditional probability distribution functions in the tail obey the Weibull distribution.

Geometry of repeated measurements in chaotic systems

We use joint probability matrices for measurements at different times to describe chaotic systems. By coarse graining the range of the measured variable into uniformly sized bins we can generate matrices that contain both topological and metric information about the systems being studied. Armed with this tool we examine two extreme families of chaotic systems. In the case of one-dimensional piecewise linear maps, we can construct transfer matrices that depend on the map and partition used, and which allow us to generate the respective joint probability matrices for all times as well as the exact time evolution of the mutual information function. We find that the mutual information decays linearly or exponentially depending on whether the second-largest eigenvalue of the transfer matrix is zero or not. In the case of three-dimensional, continuous-time chaotic systems we generate the joint probability matrices directly from numerical data. We show that these matrices directly provide attractor reconstructions with information about the attractor's probability measure.

DESIGN OF CHAOTIC PSEUDO-RANDOM BIT GENERATOR AND ITS APPLICATIONS IN STREAM-CIPHER CRYPTOGRAPHY

Because of the sensitivity of chaotic systems on initial conditions/control parameters, when chaotic systems are realized in a discrete space with finite states, the dynamical properties will be far different from the ones described by the continuous chaos theory, and some degradation will arise. This problem will cause the chaotic trajectory eventually periodic. In order to solve the problem, a new binary stream-cipher algorithm based on one-dimensional piecewise linear chaotic map is proposed in this paper. In the process of encryption and decryption, we employ a secret variant to perturb the chaotic trajectory and the control parameter to lengthen the cycle-length of chaotic trajectory. In addition, we design a nonlinear principle to generate a pseudo-random chaotic bit sequence as key stream. Cryptanalysis shows that the cryptosystem is of high security.

Features analysis of a parametric PWL chaotic map and its utilization for secure transmissions

In this paper a complete analysis of a chaotic function is provided. The chaotic map is a parametric one-dimensional non invertible piecewise linear (PWL) map which is used to generate binary sequences. Implementation, key management and security associated to this chaotic generator are discussed. Then, we propose to use this chaotic map and its sequences in a communication system. The proposed communication system relies on the CSK principle and allows to transmit a binary information by switching between orthogonal binary sequences. The system is then analyzed by providing the bit-error rates derivation both theoretically and by computer simulations.