Multi-scroll Chaotic System Research Articles

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.

A Novel Nonideal Flux-Controlled Memristor Model for Generating Arbitrary Multi-Double-Scroll and Multi-Double-Wing Attractors

This paper proposes a novel nonideal flux-controlled memristor model with a multipiecewise linear memductance function, which can be used to construct a memristive multi-scroll or multi-wing chaotic system. Importantly, arbitrary multi-double-scroll and multi-double-wing attractors can be generated depending on this memristor model directly and without the need to change the original nonlinear terms of the system. Another highlight is that the odd or even number of the double-scroll and double-wing attractors can also be freely controlled by the memristor model. To further illustrate these unique features, by introducing the memristor model into two classical chaotic systems, i.e. Jerk system and Lorenz system, multi-double-scroll and multi-double-wing chaotic attractors are obtained respectively. The formation mechanism of the multi-double-wing and multi-double-scroll attractors is also discussed. Moreover, the randomness of the chaotic binary sequences generated by the proposed memristor model is tested by the National Institute of Standards and Technology test suite. The tested results are better than those of the well-known Lorenz system. Furthermore, the corresponding circuits are constructed. The experimental results and the numerical simulations coincide well with each other, showing the effectiveness and feasibility of the proposed memristor model.

Investigating the complex behaviour of multi-scroll chaotic system with Caputo fractal-fractional operator

Abstract A new set of differential and integral operators has recently been proposed by Abdon et al., merging the fractional derivative and the fractal derivative, taking into account non-locality, memory and fractal effects. These operators have demonstrated the complex behaviour of many physical phenomena, which generally does not predict in ordinary operators or sometimes also in fractional operators. In this article, we investigate a three dimensional quadratic multi scroll chaotic dynamical system under Caputo fractal-fractional operator. We study the proposed model by replacing the fractional derivative by fractal-fractional derivatives based on Caputo. Through Schauder’s fixed point theorem, we establish existence theory to ensure that the model possesses at least one solution. Also, Banach fixed theorem guarantees the uniqueness of solution of the proposed model. By mean of non-linear functional analysis, we derive that the proposed model is Ulam-Hyres stable under the new fractal-fractional derivative. We establish the numerical scheme of the considered model through Lagrangian piece-wise interpolation. For the different values of fractional order and fractal dimension, we present the complex behaviour of the proposed model.

Dynamic Characteristics Analysis of a Multi-scroll Conservative Chaotic System with Sinusoidal Nonlinearity

Dynamic Characteristics Analysis of a Multi-scroll Conservative Chaotic System with Sinusoidal Nonlinearity

Scaling of Attractors of a Multiscroll Memristive Chaotic System and its Generalized Synchronization with Sliding Mode Control

Memristor can greatly enhance the complexity of a chaotic system because of its nonlinear characteristics. In this paper, three different memristor models are introduced to the Yang system. The chaotic attractors with single scroll and double scrolls can be obtained by adjusting the action intensities of three memristors and all the attractors inherit the scaling property of attractors of the Yang system. By employing the complex polynomials transformation method in the chaotic system to expand the number of scrolls of the system, the ring-shaped multiscroll attractors are generated, and the number of scrolls can be changed by adjusting the powers of complex polynomials, which show that the memristive system has flexible scalability. Next, a synchronization method for the multiscroll chaotic system is proposed. The generalized synchronization controller and parameter adaptive law are designed by employing sliding mode control. The sufficient conditions for synchronization are given by Lyapunov stability theory. This method can be applied to the synchronization of multiscroll systems generated by means of changing the state variables of the original system by function transformation and then adding the transformation matrix to the system. Compared with the existing synchronization method, this method has a wider scope of application, and it can synchronize two multiscroll chaotic systems with greater difference. In addition, the conditions to be satisfied in this method are simpler. Finally, the method proposed above is applied to the synchronization between a chaotic system with a ring-shaped eight-scroll attractor and a grid-shaped [Formula: see text]-scroll attractor chaotic system with interference signals. The numerical simulation results verify the effectiveness of the method.

Physical Layer Encryption in DMT Based on Digital Multi-Scroll Chaotic System

Physical layer encryption based on digital chaos emerged as a promising methodology for securing data transmission. Discrete multi-tone (DMT) is a widely investigated advanced modulation format in intensity modulation and direct detection (IM/DD) systems, because of its high spectral efficiency, low cost and high dispersion tolerance. In this letter, we successfully performed a physical layer chaotic encryption method on an IM/DD system and experimentally demonstrated the transmission of 100 Gb/s encrypted 32QAM-DMT and 125 Gb/s pre-equalized encrypted 32QAM-DMT signals over 2-km standard single mode fiber (SSMF) with bit error rate (BER) under the hard decision forward error correction (HD-FEC) threshold of 3.8×10 -3 . The proposed encryption algorithm utilizes three independent pseudo-random sequences generated by the multi-scroll chaotic system for secure data transmission, and has no substantial impact on the transmission performance.

Multi-Scroll Chaotic System Model and Its Cryptographic Application

This paper introduces a new chaotic system model that could generate multi-direction multi-scroll (MDMS) chaotic attractors. The fundamental dynamics characteristics of this model are investigated, and the feasibility of the proposed method is confirmed by numerical simulation based on MATLAB. Additionally, the new chaotic system is implemented in FPGA-based platform. Implementation results are the same in MATLAB and FPGA, which indicates that they are suitable for practical applications. To investigate the cryptographic application of the new chaotic system, the chaotic sequences generated by the proposed system are used to encrypt images. The key space of the algorithm as well as key sensitivity, plain image sensitivity, plaintext histogram, correlation and information entropy are simulated and analyzed. In contrast to several encryption schemes, the proposed algorithm is very sensitive to plaintext and the key.

Analysis and implementation of new fractional-order multi-scroll hidden attractors* *Project supported by the National Natural Science Foundation of China (Grant Nos. 61561022 and 61672226).

To improve the complexity of chaotic signals, in this paper we first put forward a new three-dimensional quadratic fractional-order multi-scroll hidden chaotic system, then we use the Adomian decomposition algorithm to solve the proposed fractional-order chaotic system and obtain the chaotic phase diagrams of different orders, as well as the Lyaponov exponent spectrum, bifurcation diagram, and SE complexity of the 0.99-order system. In the process of analyzing the system, we find that the system possesses the dynamic behaviors of hidden attractors and hidden bifurcations. Next, we also propose a method of using the Lyapunov exponents to describe the basins of attraction of the chaotic system in the matlab environment for the first time, and obtain the basins of attraction under different order conditions. Finally, we construct an analog circuit system of the fractional-order chaotic system by using an equivalent circuit module of the fractional-order integral operators, thus realizing the 0.9-order multi-scroll hidden chaotic attractors.

Image encryption application in a system for compounding self-excited and hidden attractors

Abstract Nowadays, chaos-based encryption systems implemented by hardware attracted highly attention. While chaotic systems, encryption algorithms and hardware platform are the core of the chaos-based image encryption system. In this paper, an image encryption system is developed. The proposed image encryption scheme employs a multi-scroll chaotic system and the Arnold map as entropy sources. The images are preprocessed by XOR operation between chaotic sequences and image pixels; Arnold map is used to change the position of the processed image pixels for scrambling the image. The chaotic sequences are generated by the multi-scroll chaotic system which is constructed by introducing Piecewise (PW) functions into Lorenz-like system. The modified Lorenz-like system produces a novel composite chaotic attractor consisting of a pair of different magnitude self-excited attractors and a pair of different magnitude hidden attractors. Through adjusting the parameters of PW functions, the system can also produce novel grid multi-scroll attractors. The hardware implementation of the image encryption scheme has been realized by using FPGA XC7A35TCPG236 and VGA monitor. The proposed encryption system consumes low power (248 mW) and has high throughput (4 bit/us) with an encryption time of 237 ms for encrypting a 256 × 256 plain image. Then, some basically tests are proved our image encryption system effectively.

Electronic implementation dataset to monoparametric control the number of scrolls generated.

The increasing usage of chaotic systems in the development of security systems, from mobile surveillance devices to the implementation of secure communication systems, leads to devising analog electronic implementations of research. This article presents the electronic implementation dataset from a multi-scroll chaotic system capable of generates until 9-scrolls throughout a monoparametric control based on a Saturated Non-Lineal Function (SNLF). The implemented system controls the number of generated scrolls being capable of producing attractors of single scroll and attractors with 3, 5, 7, and 9-scrolls. Moreover, the implemented system produces a family of bistable behaviors. The authors report the phenomenon in [1], where a complete analysis in the system has been carried out.

A Multiscroll Chaotic Attractors with Arrangement of Saddle-Shapes and Its Field Programmable Gate Array (FPGA) Implementation

Based on the step function and signum function, a chaotic system which can generate multiscroll chaotic attractors with arrangement of saddle-shapes is proposed and the stability of its equilibrium points is analyzed. The under mechanism for the generation of multiscroll chaotic attractors and the reason for the arrangement of saddle shapes and being symmetric abouty-axis are presented, and the rule for controlling the number of scroll chaotic attractors with saddle shapes is designed. Based on the core chips including Altera Cyclone IV EP4CE10F17C8 Field Programmable Gate Array and Digital to Analog Converter chip AD9767, the peripheral circuit and the Verilog Hardware Description Language program for realization of the proposed multiscroll chaotic system is constructed and some experimental results are presented for confirmation. The research result shows that the occupation of multipliers and Phase-Locked Loops in Field Programmable Gate Array is zero.

Hardware Design of a Kind of Grid Multi-Scroll Chaotic System Based on a MSP430F169 Chip

Based on only MSP430F169 chip, a digital chaotic generator for a grid-like JERK chaotic system is designed, from which grid-like [Formula: see text] chaotic attractors are generated. By analyzing the nonlinear functions of the gird-like chaotic system, the saddle focal balance points are respectively extended in [Formula: see text],[Formula: see text] direction. According to the hardware requirements of MSP430F169 chip, using Euler algorithm to discretize the chaotic system, the values of 36 saddle focal balance points need to be recalculated. The numerical values of the focal balance points which are involving iterative operations are given by the proposed comparative piece-wise method, their analysis and numerical simulations are also performed. The software and hardware design ideas are given for implementation, the experiment results are verification of the feasibility of the scheme.

Memristive multi-scroll chaotic system and its synchronization under sliding mode control

Memristive multi-scroll chaotic system and its synchronization under sliding mode control

A new random diffusion algorithm based on the multi-scroll Chua's chaotic circuit system

In this paper, a new image encryption algorithm was proposed based on the Chua's multi-scroll chaotic circuit system. In comparison with other classical discrete systems, the new multi-scroll system has higher complexity, thus it was more suitable for the design of the image encryption algorithm. The correlation of the adjacent pixels was greatly eliminated by using the Arnold cat transformation. According to the scroll number of the multi-scroll chaotic system on x-y direction, parameter p and q of the Arnold transformation can be determined. Especially, the pixel value of the ciphered image of the new random diffusion algorithm depends on two random non-adjacent pixels and the interference value of the multi-scroll chaotic system, it significantly reduced the correlation between the adjacent pixels, and it greatly increases the ability to resist deciphering. The experimental results show that the encryption effect of the ciphered image were significantly higher than that of other encryption algorithms. It provided a data support and theoretical direction for the research of the related cryptography fields.

Hyperjerk multiscroll oscillators with megastability: Analysis, FPGA implementation and a novel ANN-ring-based True Random Number Generator

Metastable oscillators are a special case of chaotic attractors with infinite coexisting attractors. In this paper we propose a class of hyperjerk chaotic oscillators which exhibits megastability with and without forcing term. As the systems have infinite equilibrium points, all these coexisting attractors falls in the special category of hidden attractors. We have considered the unforced hyperjerk system to show the multiscrolls and also to investigate the system bifurcations. Then, the proposed hyperjerk multiscroll chaotic system has been designed as ANN-based on FPGA. TanSig activation function, used in the design of ANN-based chaotic oscillator, has been modeled using CORDIC-LUT approximation. To improve the chaos-based engineering applications, this paper presents a new dual entropy core hybrid TRNG having high speed by using the structures of ANN-based chaotic system and ring oscillator. The designs have been coded in VHDL with respect to IEEE-754-1985 number standard. The implemented new proposed TRNG unit has been synthesized for Virtex-6 FPGA chip. After the Place-Route process, FPGA chip statistics and maximum operating frequency have been given as 167.4 MHz. 1 million bit stream generated by the proposed TRNG has been subjected to NIST-800-22 randomness tests and it has successfully passed all of the randomness tests.

Multi-scroll hidden attractors with two stable equilibrium points.

Multiscroll hidden attractors have attracted extensive research interest in recent years. However, the previously reported multiscroll hidden attractors belong to only one category of hidden attractors, namely, the hidden attractors without equilibrium points. Up to now, multiscroll hidden attractors with stable equilibrium points have not been reported. This paper proposes a multiscroll chaotic system with two equilibrium points. The number of scrolls can be increased by adding breakpoints of a nonlinear function. Moreover, the two equilibrium points are stable node-foci equilibrium points. According to the classification of hidden attractors, the multiscroll attractors generated by a novel system are the hidden attractors with stable equilibrium points. The dynamical characteristics of the novel system are studied using the spectrum of Lyapunov exponents, a bifurcation diagram, and a Poincaré map. Furthermore, the novel system is implemented by electronic circuits. The hardware experiment results are consistent with the numerical simulations.

Multiscroll chaotic system with sigmoid nonlinearity and its fractional order form with synchronization application

In this paper, a multiscroll snap oscillator with hyperbolic tangent function is proposed. There is no limitation in the number of scrolls and it can be increased by proper choice of a specific function. The Lyapunov exponents of the proposed system are obtained to testify the chaotic behavior of the system. Fractional order multiscroll system is derived from its integer order model by using the Adams–Bashforth–Moulton algorithm. A new scheme is applied in order to investigate the synchronization of the multiscroll systems. The main objective of the paper is to propose a multiscroll attractor and show that the number of scrolls can be controlled by the only nonlinear function. Such systems are less investigated in the literatures and has many real time applications like image and voice encryption, random number generators, chaos based communication systems and so on.

FPGA Implementation of the Fractional Order Integrator/Differentiator: Two Approaches and Applications

Exploring the use of fractional calculus is essential for it to be used properly in various applications. Implementing the fractional operator $D^\alpha $ in FPGA is an important research topic in fractional calculus; in the literature, only a few FPGA implementations have been proposed due to the memory dependence of the fractional order systems. In this paper, FPGA implementations of fractional order integrator/differentiator based on the Grunwald–Letnikov (GL) operator are proposed. Two algorithms are developed based on look-up table and quadratic and piece-wise linear approximation approaches to realize the GL operator and achieve high accuracy, better performance, and efficient usage of FPGA resources and to reduce the memory dependence of fractional order systems. The proposed work shows a maximum absolute error reduction of 91% compared with previous approaches. The proposed building blocks are used to implement a fractional order transfer functions for applications, including a Heavisides inductor-terminated lossy line system, a damped oscillator, a fractional order controller, and a fractional order V-shape multi-scroll chaotic system.

A Novel Multi-Attractor Period Multi-Scroll Chaotic Integrated Circuit Based on CMOS Wide Adjustable CCCII

This paper proposes a multi-attractor period multi-scroll chaotic system and a second-generation current-controlled current conveyor (CCCII) with a wide tunable range of intrinsic resistance. The newly proposed multi-attractor period multi-scroll chaotic system is constructed by this CCCII. The proposed chaotic system can generate single-attractor period, double-attractor period, three-attractor period, and even more attractor period multi-scroll. The four-dimensional state equation of the chaotic system contains four nonlinear functions for generating multi-attractor period multi-scroll chaos. Since the intrinsic resistance of the CCCII can be adjusted by external voltages or currents, the multi-attractor period multi-scroll chaotic integrated circuit based on the proposed CCCII have a tunable characteristic. It is worth noting that a chaotic integrated circuit has been implemented completely using current signals instead of voltage signals, and more interestingly, it does not contain any passive resistor. The dynamic characteristics of the multi-attractor period multi-scroll chaotic system are given. The circuit designs of the CMOS CCCII and the corresponding chaotic circuit are made in detail. We conduct the numerical and circuit simulations of the proposed chaotic system and the integrated circuit implementation of the chaotic circuit based on the CMOS CCCII. The correctness of the CCCII circuit and the chaotic system are proved through hardware experiments. The comparison between the implementation scheme of the chaotic circuit in this paper and several reported chaotic circuits is conducted.

EMI Suppression of High-Frequency Isolated Quasi Z-Source Inverter Based on Multi-Scroll Chaotic PWM Modulation

The high frequency isolated quasi Z-source inverter is a new type of inverter, which is suitable for photovoltaic generation system because of its high lift to voltage ratio, transient bridge direct access and electrical isolation.With the increase of its power density, the working frequency of its semiconductor switching devices is increasing. Switching devices in high-frequency states generate a large amount of electromagnetic noise, affecting the normal operation of surrounding electrical equipment and threatening the stability of the power grid. For the optimization of electromagnetic compatibility of high frequency isolation quasi Z-source inverter, using pulse width modulation (PWM) chaotic modulation technology, put forward using Chen multi-scroll chaotic system and traditional PWM are combined, using the new PWM control technology can inhibit electromagnetic interference (EMI) from the noise source, effectively reduce the high frequency isolation quasi Z-source switch frequency and its harmonics noise power, and optimize the total harmonic distortion (THD) of the output current by analyzing the chaotic modulation coefficient. Finally, the correctness of the theory is verified by Saber simulation and circuit test. This paper can provide guidelines for the electro magnetic compatibility (EMC) design of the high frequency isolated quasi Z-source inverter and provide the theoretical basis for the EMI optimization design of the power electronic system.