Multi-scroll Chaotic Attractors Research Articles

Improved Design and Numerical Calculation of Chaotic Circuit in Jerk System

Chaotic secure communication has always been a hot topic in nonlinear circuit research. In this paper, an improved jerk chaotic circuit is designed by using cell circuits and nonlinear function circuits. Simulation results show that the circuit can generate multi-scroll chaotic attractors. The position and number of chaotic attractors can be controlled by adding or deleting the number of nonlinear function circuit or by adjusting its parameters. Theoretical analysis and numerical calculation show that the circuit can describe jerk system, and the calculation results are consistent with the circuit simulation results. The circuit is not only simple in structure, easy to operate and high in experimental accuracy, but also easy to control chaos, which is widely used in secure communication.

Coexistence of multi-scroll chaotic attractors for a three-dimensional quadratic autonomous fractional system with non-local and non-singular kernel

In this paper, we present a numerical scheme and mathematical analysis for the famous three-dimensional quadratic autonomous self-govern system that happens to be chaotic with the coexistence of multi-scroll attractors. The scheme is based on the Atangana-Baleanu fractional derivative in the Caputo sense. The formulation of these schemes introduces the non-local and non-singular kernel to the fractional derivatives. The fractional derivative is then approximated using the family of the Adams–Bashforth schemes. The results are presented in both numerical and graphical as the fractional order β varies between 0<β⩽1. We study the proposed model in the both generalized case that is 0<β<1 and the case where β=1, which is the integer standard case. Due to the impact of the generalized case, the proposed model is able to maintain the coexistence of multi-scroll attractors.

Generation of 3-D Grid Multi-Scroll Chaotic Attractors Based on Sign Function and Sine Function

A three directional (3-D) multi-scroll chaotic attractors based on the Jerk system with nonlinearity of the sine function and sign function is introduced in this paper. The scrolls in the X-direction are generated by the sine function, which is a modified sine function (MSF). In addition, the scrolls in Y and Z directions are generated by the sign function series, which are the superposition of some sign functions with different time-shift values. In the X-direction, the scroll number is adjusted by changing the comparative voltages of the MSF, and the ones in Y and Z directions are regulated by the sign function. The basic dynamics of Lyapunov exponent spectrum, phase diagrams, bifurcation diagram and equilibrium points distribution were studied. Furthermore, the circuits of the chaotic system are designed by Multisim10, and the circuit simulation results indicate the feasibility of the proposed chaotic system for generating chaotic attractors. On the basis of the circuit simulations, the hardware circuits of the system are designed for experimental verification. The experimental results match with the circuit simulation results, this powerfully proves the correctness and feasibility of the proposed system for generating 3-D grid multi-scroll chaotic attractors.

DIGITALLY PROGRAMMABLE MULTI-SCROLL CHAOS GENERATOR ON FPGA

Multi-scroll chaotic attractors exhibit higher unpredictability than double-scroll attractors. However, the more number of scrolls cost the more usage of sources. To overcome this problem, the attractor design should be simplified. This paper presents a systematic approach that enables to realize digital piece wise linear (PWL) function in nonlinear dynamical system and to obtain whole behaviors in only one model. The proposed design requires only number of scroll as input and can realize chaotic PWL signal with a fewer number of FPGA resources. In the implementation stage of the study, the discrete mathematical equations of the chaotic attractor is modelled in Xilinx System Generator (XSG) platform and realized by using Xilinx Kintex-7 KC705 Evaluation Board.

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.

A Simple Method for Generating Mirror Symmetry Composite Multiscroll Chaotic Attractors

For further increasing the complexity of chaotic attractors, a new method for generating Mirror Symmetry Composite Multiscroll Chaotic Attractors (MSCMCA) is proposed. We take the Lorenz system as an example to explain the mechanism of the method. Firstly, by varying the signs and magnitudes of the nonlinear terms, the Lorenz system generates symmetrical attractors and different-magnitude attractors, respectively. Secondly, a modified Lorenz system is constructed by imposing several unified multilevel-logic pulse signals to the Lorenz system. The new system generates a novel chaotic attractor consisting of two pairs of different-magnitude symmetrical attractors. By adjusting the parameters of the pulse signals, the modified Lorenz system can also be controlled to generate novel grid multiscroll chaotic attractors, namely MSCMCA. Several dynamical behaviors of the new system are shown by equilibria analysis and Lyapunov exponent spectrum. Moreover, the method can be applied to other chaotic systems. Finally, a circuit of the modified Lorenz system is designed by Multisim software, and the simulation result proves the effectiveness of the method.

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.

Coexistence of multi-scroll chaotic attractors for fractional systems with exponential law and non-singular kernel

Abstract In this paper, we present mathematical analysis and numerical simulation of a three-dimensional autonomous fractional system with coexistence of multi-scroll chaotic attractors. We replaced the classical derivatives of such system with the Caputo-Fabrizio fractional derivative. This derivative combines both the exponential laws and non-singular kernels in its formulation which makes it special and useful. A two-step Adams-Bashforth scheme is derived for the approximation of the fractional derivative with exponential law and non-singular kernel. We then presented both numerical results and graphical results by considering many values of the fractional-order parameter β ∈ (0, ]. We demonstrate that the observed chaotic behavior conduct perseveres as the fractional-order parameter approaches 1.

Novel two-directional grid multi-scroll chaotic attractors based on the Jerk system

A new method is presented to generate two-directional (2D) grid multi-scroll chaotic attractors via a specific form of the sine function and sign function series, which are applied to increase saddle points of index 2. The scroll number in the x-direction is modified easily through changing the thresholds of the specific form of the sine function, while the scroll number in the y-direction is controlled by the sign function series. Some basic dynamical properties, such as equilibrium points, bifurcation diagram, phase portraits, and Lyapunov exponents spectrum are studied. Furthermore, the electronic circuit of the system is designed and its simulation results are given by Multisim 10.

Generation of Multi-Scroll Chaotic Attractors from a Jerk Circuit with a Special Form of a Sine Function

A novel chaotic system for generating multi-scroll attractors based on a Jerk circuit using a special form of a sine function (SFSF) is proposed in this paper, and the SFSF is the product of a sine function and a sign function. Although there are infinite equilibrium points in this system, the scroll number of the generated chaotic attractors is certain under appropriate system parameters. The dynamical properties of the proposed chaotic system are studied through Lyapunov exponents, phase portraits, and bifurcation diagrams. It is found that the scroll number of the chaotic system in the left and right part of the x-y plane can be determined arbitrarily by adjusting the values of the parameters in the SFSF, and the size of attractors is dominated by the frequency of the SFSF. Finally, an electronic circuit of the proposed chaotic system is implemented on Pspice, and the simulation results of electronic circuit are in agreement with the numerical ones.

Multiscroll chaotic attractors in Optical Injected Semiconductor Laser Driven by a Resonant Tunneling Diode Current

This paper shows a complex nonlinear dynamics of an optically injected laser diode (LD) directly modulated by electrical current from a resonant tunneling diode (RTD) circuit. The LD model evolves a new control parameter called effective gain coefficient (EGC) and can switch between a free-running laser and a perturbed laser. A bifurcation diagram analysis sustained Lyapunov exponents’ spectra reveals chaotic and quasi-periodic dynamics for restricted ranges of the frequency fin and by EGC variations, including the generation of the hyperchaotic regime and multiscroll chaotic attractors through a cascade of periodic oscillations to the chaos. Moreover, an implementation of this new coupled system also shows that when an AC signal Vacsin⁡2πfint is added to the DC voltage bias, the LD outputs show several optical signals such as periodic, coherence collapse and chaotic regimes with different strange complex attractors which are devoted to more improve secure optical telecommunications.

A novel amplitude control method for constructing nested hidden multi-butterfly and multiscroll chaotic attractors

Abstract A novel amplitude control method (ACM) is proposed to construct multiple self-excited or hidden attractors by scaling partial or total variables without changing their dynamic and topological properties. Various attractors including nested attractor, axisymmetric attractor, and centrosymmetric attractor can be obtained by multiplying signals with different amplitudes. An universal pulse control module is designed to realize the amplitude scale. Different number of scrolls can be adjusted by regulating the pulse signals without redesigning the nonlinear circuit. The classical Lorenz system and Jerk system are employed as examples to generate nested hidden multi-butterfly and multiscroll attractors. Some novel properties of ACM, such as nested morphology, amplitude modulation, and constant Lyapunov exponential spectrum, are analyzed theoretically and simulated numerically. The circuit design and PSpice simulation results are implemented to verify the availability and feasibility of the proposed approach.

Continuous Multi-Scroll Chaotic PWM and its Chaotic Signal Selection Method for EMI Suppression of Power Converters

Chaotic pulse width modulation (PWM) technique has been applied to suppress electromagnetic interference (EMI) in power converters. However, traditional discrete chaotic PWM has the shortcoming that the spreading spectrum distribution is triangular in shape and the practical spreading frequency bandwidth is much larger than the designed, which will further result in the limited EMI suppression. In this paper, a continuous chaotic PWM method based on the continuous multi-scroll chaotic attractors is put forward to suppress EMI more effectively, getting better spread spectrum distribution at switching frequency and its multiples for power converters. Moreover, to guide the selection of chaotic signals in real applications, the influence mechanism of multi-scroll chaotic signals on EMI spectrum distribution is firstly analyzed in this paper so as to provide a practical chaotic signal selection method. Finally, the experiments are presented followed by the simulations on a boost converter to verify the effectiveness of the proposed continuous multi-scroll chaotic PWM method and the correctness of the given chaotic signal selection method. This paper provides design guidance for the chaotic PWM in engineering applications.

Symmetries in Hidden Bifurcation Routes to Multiscroll Chaotic Attractors Generated by Saturated Function Series

In this paper, hidden bifurcation routes to multiscroll chaotic attractors generated by saturated function series are explored. The method to nd such hidden bifurcation routes (HBR) depending upon two parameters is similar to the method introduced by Menacer, et al. (2016) for Chua multiscroll attractors. These HBR are characterized by the maximal range extension (MARE) of their attractors and coding the appearance order of the scrolls under the control of the two parameters. Moreover, these HDR have interesting symmetries with respect to the two parameters. The novelty that this article introduces, is firstly the paradigm of MARE and the formula giving their approximate value depending upon parameters p and q, which is linked to the size of the scrolls; secondly the coding of the HBR which is dened for the first time including the basic cell; and thirdly unearthing the symmetries of these routes, allowing to obtain their coding without any numerical computation.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium provided the original work is properly cited.

Generating Multi-Scroll Chua’s Attractors via Simplified Piecewise-Linear Chua’s Diode

High implementation complexity of multi-scroll circuit is a bottleneck problem in real chaos-based communication. Especially, in multi-scroll Chua's circuit, the simplified implementation of piecewise-linear resistors with multiple segments is difficult due to their intricate irregular breakpoints and slopes. To solve the challenge, this paper presents a systematic scheme for synthesizing a Chua's diode with multi-segment piecewise-linearity, which is achieved by cascading even-numbered passive nonlinear resistors with odd-numbered ones via a negative impedance converter. The traditional voltage mode op-amps are used to implement nonlinear resistors. As no extra DC bias voltage is employed, the scheme can be implemented by much simpler circuits. The voltage-current characteristics of the obtained Chua's diode are analyzed theoretically and verified by numerical simulations. Using the Chua's diode and a second-order active Sallen-Key high-pass filter, a new inductor-free Chua's circuit is then constructed to generate multi-scroll chaotic attractors. Different number of scrolls can be generated by changing the number of passive nonlinear resistor cells or adjusting two coupling parameters. Besides, the system can be scaled by using different power supplies, satisfying the low-voltage low-power requirement of integrated circuit design. The circuit simulations and hardware experiments both confirmed the feasibility of the designed system.

Determining accurate Lyapunov exponents of a multiscroll chaotic attractor based on SNFS

In n-scroll attractors design using active devices, the behavior of the nonlinear part is, often, modeled by using a piecewise linear (PWL) approach. Although little effort is required to build a PWL model, the main performance parameters associated with the active device to be used in the physical design are not included. Moreover, the main metrics used for quantifying chaos are: Lyapunov exponent (LE), Kolmogorov–Sinai entropy, Kaplan–Yorke dimension, fractal power spectra and fractal dimension. Among them, the former not only is used as indicator of chaotic motion when a positive LE is found, but if its value positively increases, then the unpredictability grade of the chaotic system also increases. All reported works until today about the computed LEs on chaotic systems are derived using a PWL approach. This is a serious drawback since PWL models cannot model a dynamic behavior, and hence, introducing a level of error during the numerical analysis becomes evident when experimental and numerical results are examined at high frequency. Under this scheme, new results on the accurate computing of LEs for a chaotic oscillator based on a nonlinear function called saturated nonlinear function series (SNFS) are introduced. A mathematical model is used for characterizing the dynamical behavior of the SNFS that includes the main performance parameters of Op Amps such as slew rate, DC gain, gain bandwidth product and dynamic range, showing a much better accuracy than the traditional PWL approach. This behavioral modeling demonstrates that the positive LE increases in relation to the number of scrolls and even is also depending on the frequency of the system, overthrowing the results that have been handled in previous works, where a constant value of the LE was obtained when the number of scrolls increased.

A Class of Piecewise Linear Systems Without Equilibria With 3-D Grid Multiscroll Chaotic Attractors

In this brief a new class of piecewise linear dynamical system without equilibria which exhibits a 3-D grid multiscroll chaotic attractor is presented. The number of scrolls of the attractor generated can be easily changed by the number of linear parts of three piecewise constant functions. A particular system with a 3-D grid multiscroll attractor whose scrolls appear in an arrangement of ${3}{\times }{3}{\times }{3}$ is taken as a case study. Moreover, an electronic circuit realization is proposed for the particular system and simulation data as well as experimental data is provided.

Research and simulation of typical chaotic system model

A chaotic theory and its application now become a cutting edge issue in both science and engineering fields. Chua’s circuit possess multi-scroll chaotic attractor, abundant of dynamical behaviors and multivariate chaotic structure. It’s the emphasis

Chaos control for multi-scroll chaotic attractors generated by introducing a bipolar sigmoid function series

In this paper, a bipolar sigmoid function series is proposed to generate 1-D (one dimensional) and 2-D (two dimensional) multi-scroll chaotic attractors. The nonlinear dynamical behaviors of 1-D and 2-D multi-scroll chaotic systems are analyzed, including the equilibrium points, invariance, dissipation, Lyapunov exponents, fractional dimension and Poincare map. In addition, backstepping controllers are used to stabilize the 1-D and 2-D multi-scroll chaotic systems. The results show that the 1-D and 2-D multi-scroll chaotic attractors can be generated by introducing the bipolar sigmoid function series. The mechanism for generating multi-scroll chaotic attractors is as follows: The equilibrium points with index 2 (type two) are responsible for generating scrolls, and the equilibrium points with index 1 (type one) are responsible for connecting these scrolls. The chaotic behaviors in the 1-D and 2-D multi-scroll chaotic systems are stabilized to some equilibrium points using the backstepping controllers, without determining the desired targeting orbits beforehand.