Linear Chaotic Systems Research Articles

Predicting the Emergence of Multistability in a Monoparametric PWL System

One of the main problems in the study of dynamical systems is to explore the asymptotic behavior of the model when a parameter varies continuously. When these variations lead to the appearance of coexisting states, the study of the global properties of the system becomes an even more complex task, since it is almost impossible to predict the stability change. In this paper, we present a simple method for characterizing qualitative changes in the dynamics of a family of Piece-Wise Linear (PWL) chaotic systems, that transit from monostable to multistable behavior by a single bifurcation parameter. By characterizing the magnitude of the stable and unstable manifolds associated with the eigendirections, it is possible to analytically find tipping points in the linear model that are consistent with the occurrence of coexisting states in the dynamics. The results show agreement between the bifurcation diagrams of the linear operator, the bifurcation diagrams of the PWL system, and the multistability phenomenon validation in analog electronics. The presented work makes it possible to know the mechanism by which the system exhibits the break of its stability and the corresponding basin of attraction. This introduces a new methodology for the analysis of dynamical systems in search of dynamical changes such as coexisting attractors.

Chaotic Mapping-Based Anti-Sorting Radio Frequency Stealth Signals and Compressed Sensing-Based Echo Signal Processing Technology.

Radio frequency (RF) stealth anti-sorting technology can improve the battlefield survival rate of radar and is one of the research hotspots in the radar field. In this study, the signal design principle of anti-sequential difference histogram (SDIF) sorting was explored for the main sorting algorithm of the SDIF. Furthermore, we designed a piecewise linear chaotic system with interval number parameterization based on random disturbance and proposed a method to modulate the repetition period of widely spaced signal pulses using a chaotic system. Then, considering the difficulty of the traditional signal processing method to measure the velocity of the highly random anti-sorting signals designed in this paper, we used compressed sensing (CS) technology to process the echoes of the signals to solve the velocity and distance of the detection targets. Finally, simulation verification was performed from the correctness of the signal design principle, the performance of the chaotic system, the anti-sorting performance of the designed signals and the recovery and reconstruction performance of the signals by CS. The results show that: (a) the signal design principle presented in this paper can guide the signal design correctly; (b) the performance of the piecewise linear chaotic system with interval number parameterization is better than that of the classical one-dimensional chaotic system; (c) the anti-sorting signal modulated by the chaotic system can achieve anti-SDIF sorting, and the anti-sorting signals designed in this paper can be processed to obtain the velocity and distance of the targets.

Proposing and Dynamical Analysis of a Hyperjerk Piecewise Linear Chaotic System with Offset Boostable Variable and Hidden Attractors

Designing chaotic systems with different properties helps to increase our knowledge about real-world chaotic systems. In this article, a piecewise linear (PWL) term is employed to modify a simple chaotic system and obtain a new chaotic model. The proposed model does not have any equilibrium for different values of the control parameters. Therefore, its attractor is hidden. It is shown that the PWL term causes an offset boostable variable. This feature provides more flexibility and controllability in the designed system. Numerical analyses show that periodic and chaotic attractors coexist in some fixed values of the parameters, indicating multistability. Also, the feasibility of the system is approved by designing field programmable gate arrays (FPGA).

A New 4D Piecewise Linear Multiscroll Chaotic System with Multistability and Its FPGA-Based Implementation

Due to the complex behavior of a multiscroll chaotic system, it is a good candidate for the secure communications. In this paper, by adding an additional variable to the modified Lorenz-type system, a new chaotic system that includes only linear and piecewise items but can generate 4n + 4 scroll chaotic attractors via choosing the various values of natural number n is proposed. Its dynamics including bifurcation, multistability, and symmetric coexisting attractors, as well as various chaotic and periodic behaviors, are analyzed by means of attraction basin, bifurcation diagram, dynamic map, phase portrait, Lyapunov exponent spectrum, and C0 complexity in detail. The mechanism of the occurrence for generating multiscroll chaotic attractors is presented. Finally, this multiscroll chaotic system is implemented by using the Altera Cyclone IV EP4CE10F17C8 FPGA. It is found that this FPGA-based design has an advantage of requiring less resources for 0% of the embedded multipliers and 0% of the PLLs of this FPGA are occupied.

Privacy protection for fog computing and the internet of things data based on blockchain

With the development of the Internet of Things (IoT) field, more and more data are generated by IoT devices and transferred over the network. However, a large amount of IoT data is sensitive, and the leakage of such data is a privacy breach. The security of sensitive IoT data is a big issue, as the data is shared over an insecure network channel. Current solutions include symmetric encryption and access controls to secure the data transfer, but they have some drawbacks such as a single point of failure. Blockchain is a promising distributed ledger technology that can prevent the malicious tampering of data, offering reliable data storage. This paper proposes a distributed access control system based on blockchain technology to secure IoT data. The proposed mechanism is based on fog computing and the concept of the alliance chain. This method uses mixed linear and nonlinear spatiotemporal chaotic systems (MLNCML) and the least significant bit (LSB) to encrypt the IoT data on an edge node and then upload the encrypted data to the cloud. The proposed mechanism can solve the problem of a single point of failure of access control by providing the dynamic and fine-grained access control for IoT data. The experimental results of this method demonstrated that it can protect the privacy of IoT data efficiently.

An efficient batch images encryption method based on DNA encoding and PWLCM

With the rapid development of the Internet, there is a pressing requirement for data security protection in Internet. However, in some applications, resource constrained devices in Internet requires that the algorithm for data encryption have high reliability and availability, as well as extremely low latency, and strong robustness in harsh environment. In this work, we propose an efficient batch images encryption/decryption scheme based on DNA encoding and piecewise linear chaotic mapping system (PWLCM). Different from all existing methods, the proposed method can encrypt/decrypt any number of images with different sizes via only single encryption computation, and the computing cost is equivalent to that of encrypting a single image. Experimental results demonstrate that the proposed method has strong robustness to resist the interference of harsh environment, such as noise contamination, JPEG compression, and image content loss. We also investigate the computing cost, encryption effect and the security. Comparing with the state-of-the-art methods, the proposed method has high encryption efficiency, stronger robustness and security.

Parallelizable cipher of color image based on two-dimensional chaotic system

This paper aims to present a parallel implementation based color image encryption using non -linear chaotic system. The adopted chaotic system was suggested and approved in our previous work [1] which generates key streams with chaotic behavior. In this paper, pixel level permutation algorithm based on chaotic map generation is investigated and analyzed. The encryption–decryption schemes are achieved in parallel and composed of three main phases: chaotic keys generation, pixel-level permutation and bit-level diffusion phase. Both permutation and diffusion processes are achieved according to the chaotic keys. The parallel implementation of the proposed image encryption system is realized and inspired with parallel computing library offered by Matlab 2018, which equips highly performance than the pipeline ones and would be helpful to utilize in image encryption/decryption for real time application. Security and statistical analysis in addition to the main differential attacks analysis are specified to evaluate the performance of the proposed image encryption algorithm with parallel implementation. From the experimental results, the output image of the encryption task shows a higher randomness of the encrypted image which can be effectively resistant to attacker. Furthermore, the run time of encryption process is faster than other research works.

Detecting Causality in Multivariate Time Series via Non-Uniform Embedding

Causal analysis based on non-uniform embedding schemes is an important way to detect the underlying interactions between dynamic systems. However, there are still some obstacles to estimating high-dimensional conditional mutual information and forming optimal mixed embedding vector in traditional non-uniform embedding schemes. In this study, we present a new non-uniform embedding method framed in information theory to detect causality for multivariate time series, named LM-PMIME, which integrates the low-dimensional approximation of conditional mutual information and the mixed search strategy for the construction of the mixed embedding vector. We apply the proposed method to simulations of linear stochastic, nonlinear stochastic, and chaotic systems, demonstrating its superiority over partial conditional mutual information from mixed embedding (PMIME) method. Moreover, the proposed method works well for multivariate time series with weak coupling strengths, especially for chaotic systems. In the actual application, we show its applicability to epilepsy multichannel electrocorticographic recordings.

Scattering statistics in nonlinear wave chaotic systems.

The Random Coupling Model (RCM) is a statistical approach for studying the scattering properties of linear wave chaotic systems in the semi-classical regime. Its success has been experimentally verified in various over-moded wave settings, including both microwave and acoustic systems. It is of great interest to extend its use in nonlinear systems. This paper studies the impact of a nonlinear port on the measured statistical electromagnetic properties of a ray-chaotic complex enclosure in the short wavelength limit. A Vector Network Analyzer is upgraded with a high power option, which enables calibrated scattering (S) parameter measurements up to +43dBm. By attaching a diode to the excitation antenna, amplitude-dependent S-parameters and Wigner reaction matrix (impedance) statistics are observed. We have systematically studied how the key components in the RCM are affected by this nonlinear port, including the radiation impedance, short ray orbit corrections, and statistical properties. By applying the newly developed radiation efficiency extension to the RCM, we find that the diode admittance increases with the excitation amplitude. This reduces the amount of power entering the cavity through the port so that the diode effectively acts as a protection element. As a result, we have developed a quantitative understanding of the statistical scattering properties of a semi-classical wave chaotic system with a nonlinear coupling channel.

Dynamics Models of Synchronized Piecewise Linear Discrete Chaotic Systems of High Order

This paper deals with the methods for investigating the nonlinear dynamics of discrete chaotic systems (DCS) applied to piecewise linear systems of the third order. The paper proposes an approach to the analysis of the systems under research and their improvement. Thus, effective and mathematically sound methods for the analysis of nonlinear motions in the models under consideration are proposed. It makes it possible to obtain simple calculated relations for determining the basic dynamic characteristics of systems. Based on these methods, the authors developed algorithms for calculating the dynamic characteristics of discrete systems, i.e. areas of the existence of steady-state motion, areas of stability, capture band, and parameters of transients. By virtue of the developed methods and algorithms, the dynamic modes of several models of discrete phase synchronization systems can be analyzed. They are as follows: Pulsed and digital different orders, dual-ring systems of various types, including combined ones, and systems with cyclic interruption of auto-tuning. The efficiency of various devices for information processing, generation and stabilization could be increased by using the mentioned discrete synchronization systems on the grounds of the results of the analysis. We are now developing original software for analyzing the dynamic characteristics of various classes of discrete phase synchronization systems, based on the developed methods and algorithms.

Blended Chaos Control of Permanent Magnet Linear Synchronous Motor

Aiming at the chaos problem of permanent magnet linear synchronous motor in direct-drive wave power generation system, based on the ideal chaotic model with increased rotor edge effect, the permanent magnet linear synchronous motor system chaotic system was analyzed and a new sliding mode compound chaos control strategy was proposed. This strategy compensates for the shortcomings of the sliding mode control lag and chattering by using the complementary advantages of compound control, and can correct system parameters in real time. By using Lyapunov stability criterion, it was proved that the global convergence of the system is consistent. The simulation results show that the composite chaos control strategy can rapidly disengage the chaotic state of the motor system, suppress system chattering, weaken the control overshoot phenomenon, and has strong robustness and high control precision.

Comparison of the influences of nodal dynamics on network synchronized regions

A new piecewise linear unified chaotic (PLUC) system is firstly presented, and then its fundamental dynamical behaviors are analyzed. This modified chaotic system, as well as the unified chaotic (UC) one, is taken as network nodal oscillators for investigating the difference of influences of nodal dynamics on the bifurcation of network synchronized regions. It is found that beyond the greatly similar bifurcation modes between PLUC and UC networks, the synchronized regions in PLUC networks are far narrower at almost each parameter a than those in UC networks for most of inner coupling matrices, indicating the PLUC node makes the network more difficult to synchronization. Our numerical investigations show that this phenomenon is closely related with nodal dynamical properties, such as the boundary of attractors, the largest Lyapunov exponent and Lyapunov dimension.

Nonlinear wave chaos: statistics of second harmonic fields.

Concepts from the field of wave chaos have been shown to successfully predict the statistical properties of linear electromagnetic fields in electrically large enclosures. The Random Coupling Model (RCM) describes these properties by incorporating both universal features described by Random Matrix Theory and the system-specific features of particular system realizations. In an effort to extend this approach to the nonlinear domain, we add an active nonlinear frequency-doubling circuit to an otherwise linear wave chaotic system, and we measure the statistical properties of the resulting second harmonic fields. We develop an RCM-based model of this system as two linear chaotic cavities coupled by means of a nonlinear transfer function. The harmonic field strengths are predicted to be the product of two statistical quantities and the nonlinearity characteristics. Statistical results from measurement-based calculation, RCM-based simulation, and direct experimental measurements are compared and show good agreement over many decades of power.

Electronic implementation of a practical matched filter for a chaos-based communication system

Presented is an electronic implementation of a matched filter intended for chaos-based communication systems. While implementing the transmitter side of such systems is relatively trivial, the receiver has proven challenging to develop. Most chaotic systems lack a known fixed basis function, making it difficult to develop a matched filter for them. Instead, their receivers rely on more complicated or less effective techniques to compensate for the presence of noise. However, a previously developed manifold piecewise linear chaotic system has been shown to have an exact analytic solution. This solution has enabled the development of a matched filter for use in any communication system based on this chaotic system. The original communication system operated at a fundamental frequency of 84 Hz, much too low for any practical applications. Therefore, newer communication systems have been designed to operate at higher frequencies. In this work, the performance of this matched filter has been evaluated with a 18.4 kHz chaotic oscillator.

Modified Function Projective Synchronization for a Partially Linear and Fractional-Order Financial Chaotic System with Uncertain Parameters

This paper investigates the modified function projective synchronization between fractional-order chaotic systems, which are partially linear financial systems with uncertain parameters. Based on the stability theory of fractional-order systems and the Lyapunov matrix equation, a controller is obtained for the synchronization between fractional-order financial chaotic systems. Using the controller, the error systems converged to zero as time tends to infinity, and the uncertain parameters were also estimated so that the phenomenon of parameter distortion was effectively avoided. Numerical simulations demonstrate the validity and feasibility of the proposed method.

Generation Method of Multipiecewise Linear Chaotic Systems Based on the Heteroclinic Shil’nikov Theorem and Switching Control

Based on the heteroclinic Shil’nikov theorem and switching control, a kind of multipiecewise linear chaotic system is constructed in this paper. Firstly, two fundamental linear systems are constructed via linearization of a chaotic system at its two equilibrium points. Secondly, a two-piecewise linear chaotic system which satisfies the Shil’nikov theorem is generated by constructing heteroclinic loop between equilibrium points of the two fundamental systems by switching control. Finally, another multipiecewise linear chaotic system that also satisfies the Shil’nikov theorem is obtained via alternate translation of the two fundamental linear systems and heteroclinic loop construction of adjacent equilibria for the multipiecewise linear system. Some basic dynamical characteristics, including divergence, Lyapunov exponents, and bifurcation diagrams of the constructed systems, are analyzed. Meanwhile, computer simulation and circuit design are used for the proposed chaotic systems, and they are demonstrated to be effective for the method of chaos anticontrol.

Complete Synchronization of Two Bidirectionally Coupled Piecewise Linear Chaotic Systems Perturbed by Poisson Noise

We establish the sufficient conditions of complete synchronization with probability 1 in the coupled bidirectional piecewise linear chaotic systems with common or different Poisson noises and the positive effects of Poisson noises on the complete synchronization will be found based on the proposed theoretical results. We compare the effects of Poisson noise on the complete synchronization with the case of Gaussian white noise , and then we claim Poisson noise can induce complete synchronization easier than Gaussian white noise in the sense of noise strength. Furthermore, a numerical example is presented to show the excellent agreement between theoretical and numerical analysis.

An information theoretic alternative to model a natural system using observational information alone

[1] How to define a system? This is a problem faced routinely in science and engineering, with solutions developed from our understanding of the processes inherent, to assessing the underlying structure based on observational evidence alone. In general, system specification involves identifying a few meaningful predictors (from a large enough set that is plausibly related to the response) and formulating a relation between them and the system response being modeled. For systems where physical relationships are less apparent, and sufficient observational records exist, a range of statistical alternatives have been investigated as a possible way of specifying the underlying form. Here we introduce partial information (PI) as a new means for specifying the system, its key advantage being the relative lack of major assumptions about the processes being modeled in order to characterize the complete system. In addition to PI which offers a means of identifying the system predictors of interest, we also introduce the concept of partial weights (PWs) which uses the identified predictors to formulate a predictive model that acknowledges the relative contributions, predictor variables make to the prediction of the response. We assess the utility of the PI-PW framework using synthetically generated data sets from known linear, nonlinear, and high-dimensional dynamic yet chaotic systems and demonstrate the efficacy of the procedure in ascertaining the underlying true system with varying extents of observational evidence available. We highlight how this framework can be invaluable in formulating prediction models for natural systems which are modeled using empirical or semiempirical alternatives and discuss current limitations that still need to be overcome.

A novel image encryption scheme based on a linear hyperbolic chaotic system of partial differential equations

Compared with general chaotic systems, a linear hyperbolic chaotic system of partial differential equations with nonlinear boundary conditions has larger parameter space, stronger sensitivity to initial condition and control parameter, better random-like behavior and so on, but it has not been employed in cryptography so far. Then using its significant properties, we present a new cryptosystem with coupled map lattices and time-varying delay. The proposed image encryption algorithm with permutation–diffusion architecture can overcome some drawbacks in the existing methods, because the sum of pixel value of original image is used for determining the permutation parameters and the previous cipher image information is utilized in the next diffusion. Theoretical analysis and computer experiments confirm that the new algorithm is efficient, practicable, and reliable, with high potential to be adopted for network security and secure communications.

Modified function projective synchronization for a class of partially linear fractional order chaotic systems

In this paper, a method of modified function projective synchronization for a class of partially linear fractional order chaotic systems is proposed. This kind of system can be constructed by single variable coupling response system. The modified function projective synchronization controller is designed by Routh-Hurwitz conditions. Theoretical analyses and numerical simulations of two chaotic systems verify the effectiveness of the proposed method.