Multi-scroll Chaotic Attractors Research Articles

Circuit Implementation of Multiscroll Chaotic Attractors Generated Using Step Function

A new chaotic system is proposed to generate multiscroll chaotic attractors. The major method used is for the step function to act as a nonlinear function. To prove that the proposed system can generate multiscroll chaotic attractors, the equilibrium point, the time domain waveform and the phase diagram of the proposed system are calculated. Finally, the design of the hardware circuit produces experimental results at a maximum of 8-scroll hardware. Theoretical analysis, simulation and hardware experimental results are fully matched, which further proves the existence of the proposed system and the physical realization. This provides the possibility for future applications in engineering.

Design of Grid Multiscroll Chaotic Attractors via Transformations

Three transformation approaches for generating grid multiscroll chaotic attractors are presented through theoretical analysis and numerical simulation. Three kinds of grid multiscroll chaotic attractors are generated based on one-dimensional multiscroll Chua system. The dynamics of the multiscroll chaotic attractors are analyzed by means of equilibrium points, eigenvalues, the largest Lyapunov exponent and complexity. As the experimental verification, we implemented the circular grid multiscroll attractor on DSP platform. The simulation and experimental results are consistent well with that of theoretical analysis, and it shows that the design approaches are effective.

Simulation and circuit implementation of 12-scroll chaotic system

Based on the typical Chua’s circuit and the latest research results of multi-scroll system, a model of the new system is constructed to produce multi-scroll chaotic attractors, replacing the typical Chua’s diode with the combination of sign function. The major method is to make equilibrium point located in the center of two adjacent breakpoints, and keep scrolls and bond orbits alternated with each other. The chaos generation mechanism is studied by analyzing the symmetry and invariance, the existence of the dissipation and attractor, the system equilibrium and stability. The fractal dimension, the K–S entropy, the time domain waveform and the initial value sensitivity are applied to verifying the chaotic behaviors. The numerical simulations show that the system generates n-double (n=1, 2, 3, 4, 5, 6) scroll chaotic attractors. Finally, the design of the hardware circuit produces at a maximum of 12-scroll hardware experimental results. Theoretical analysis, numerical simulation and hardware experimental results are full matched, which further proves the existence of the system and the physical realization.

A Systematic Design Method for 3-D Grid Multiscroll Chaotic Attractors and Its Circuit Implementation Employing CFOAs

A systematic design method for 3-D grid multiscroll chaotic attractors by using staircase nonlinear functions (SNFs) is proposed in this paper. Equations of parameters for 3-D grid multiscroll chaotic attractors are derived in the consideration of the dynamic range for the active device and the number of multiscroll chaotic attractors. 5 × 4 and 4 × 3 of 3-D grid multiscroll chaotic attractors are generated by this method. A chaotic circuit is designed only using current feedback operational amplifiers (CFOAs). The circuit consists of three integrators and two nonlinear circuits. The parameters of the circuit are calculated by the parameters of 3-D grid multiscroll chaotic attractors. The circuit can generate 3-D grid multiscroll chaotic attractors. The central frequency of the circuit is higher with fewer active devices and simpler circuit construction for the good frequency characteristic and port characteristic of the CFOA. Numerical simulation results and the circuit simulation results show that the method is feasible and the designed circuit is correct.

Bayesian Stable Mixture Model of State Densities of Generalized Chua's Circuit

In this paper, the probability density functions (PDFs) of the states of Generalized Chua's Circuit (GCC) have been modeled by Finite Mixture α-Stable (FMαS) distributions which is a Bayesian mixture model of α-stable distributions and it provides semiparametric characterization for the distributions of multiscroll chaotic attractors. Fully Bayesian approach has been applied to estimate the mixture parameters of multimodal distributions corresponding to the multiscroll chaotic attractors.

Switched systems based on unstable dissipative systems

In this work we present an unified theory of how to yield multi-scroll chaotic attractors based on step function, saturation, hysteresis and deterministic Brownian motion. This class of systems is constructed with a switching control law by changing the equilibrium point of an unstable dissipative system. The switching control law that governs the position of the equilibrium point changes according to the number of scrolls that is displayed in the attractor.

A novel multi-directional multi-scroll chaotic system and its CCII+ circuit implementation

This paper proposes a generalized formula of multi-directional multi-scroll chaotic system. Based on the generalized formula, by adding the nonlinear functions, and controlling the parameters of system, multi-directional multi-scroll chaotic attractors with different order system can be generated. According to this method, the correctness of generalized formula is verified. Then, basic dynamic characteristics of the system including the fractal, equilibrium points, Poincaré section, Lyapunov exponent spectrum and bifurcation diagram are analyzed. Finally, a nine-dimensional chaotic circuit based on CCII+ devices is designed, which verified the feasibility of the generalized formula and multi-directional multi-scroll chaotic circuit with CCII+ devices.

Generation and circuit implementation of multi-block multidirectional grid multi-scroll chaotic attractors

Due to the dynamic characteristics of the Chua's system, multi-scroll chaotic attractors are still confined in a single block and fail to break the limit. This paper proposes an approach for generating novel multi-block multidirectional grid multi-scroll chaotic attractors that can break the limit via novel nonlinear modulating functions. According to this method, the recursion rules used to generate multi-block multidirectional grid multi-scroll attractors are mathematically obtained. The new system is autonomous; the effectiveness of this method has been verified by theoretical analysis, numerical simulation, and circuit implementation.

High-Order Grid Multiscroll Chaotic Attractors Generated by the Second-Generation Current Conveyor Circuit

The multidirectional multiscroll chaotic attractors generated by second-generation current conveyor (CCII) are investigated. A fourth-order and a fifth-order grid multiscroll chaotic system are presented by designing staircase function and shifted voltage sign functions in the high-order Jerk system. Using the CCIIs, a chaotic circuit is designed and implemented, which can generate 4 × 2 × 2 × 2- and 4 × 2 × 2 × 2 × 2-scroll chaotic attractors, respectively, via switching control. Compared with the voltage operational-amplifier-based circuit, the proposed circuit has better performance in terms of circuit structure and bandwidth. Numerical simulations and experimental results are accordant.

Achieving the Multi-Scroll Chaotic Attractors’ Synchronization in the Four-Dimensional System by Driving-Response Synchronization Method

Multi-scroll chaotic attractors’ synchronization technology is one of the important research achievements in recent years.This paper uses driving-response synchronization method to achieve the two 6-scroll chaotic attractors’ synchronization in the four-dimensional syetem.The feasibility of this method is proved by theoretical analysis and numerical simulation.

Research on Multi-Scroll Chaotic Attractors Synchronization Technique

Multi-scroll chaotic attractors synchronization technique is a research focuses in recent years. This paper focuses on multi-scroll chaotic attractors in the four-dimensional system. It uses variable feedback synchronous method to achieve the two 4-scroll chaotic attractors’ synchronization. The feasibility of this method is proved by theoretical analysis and numerical simulation. Simultaneously, it gets a better synchronization results.

Grid multi-wing butterfly chaotic attractors generated from a new 3-D quadratic autonomous system

Due to the dynamic characteristics of the Lorenz system, multi-wing chaotic systems are still confined in the positive half-space and fail to break the threshold limit. In this paper, a new approach for generating complex grid multi-wing attractors that can break the threshold limit via a novel nonlinear modulating function is proposed from the firstly proposed double-wing chaotic system. The proposed method is different from that of classical multi-scroll chaotic attractors generated by odd-symmetric multi-segment linear functions from Chua system. The new system is autonomous and can generate various grid multi-wing butterfly chaotic attractors without requiring any external forcing, it also can produce grid multi-wing both on the xz-plane and yz-plane. Basic properties of the new system such as dissipation property, equilibrium, stability, the Lyapunov exponent spectrum and bifurcation diagram are introduced by numerical simulation, theoretical analysis and circuit experiment, which confirm that the multi-wing attractors chaotic system has more rich and complicated chaotic dynamics. Finally, a novel module-based unified circuit is designed which provides some principles and guidelines for future circuitry design and engineering application. The circuit experimental results are consistent with the numerical simulation results.

Generation and Circuit Implementation of Multi-Scroll Chaotic Attractors with Controllable Direction

An approach for generating multi-scroll chaotic attractors with controllable direction in one plane is proposed. Firstly, an appropriate nonlinear function is selected to control the number and direction of multi-scroll chaotic attractors in the three-order Jerk system. Then, we add new control function to Jerk system and observe Lyapunov exponent spectrum of relative coefficient and the change of equilibrium points. Different multi-scroll chaotic attractors with controllable direction are generated by adjusting the coefficient of the control function in a plane. The implementation of circuit verifies the feasibility of this method.

A Simple Grid Multiscroll Chaotic Electronic Oscillator Employing CFOAs

In this paper, a novel grid multiscroll chaotic electronic oscillator with simple circuit design is proposed. The proposed grid multiscroll chaotic circuit only needs two current feedback operational amplifiers (CFOAs) as current integrators, passive RC, and nonlinear elements in the form of current stair nonlinear function series (SNFS) realized with CFOAs. Compared with existing grid multiscroll circuits such as Chua's circuit and Jerk circuit, etc., its component count is reduced. Mathematical models of the proposed generators are derived. Theoretical analysis and numerical simulations of the derived models are included. Moreover, the experimental results are verified that the proposed simpler chaotic electronic oscillator can generate grid multiscroll chaotic attractors with higher frequency.

Frequency limitations in generating multi-scroll chaotic attractors using CFOAs

The current-feedback operational amplifier (CFOA) allows us to implement any kind of circuit useful in analogue signal processing applications. However, it has limited performance in implementing nonlinear circuits. That way, this investigation highlights the experimental results of implementing a multi-scroll chaotic oscillator by using the commercially available CFOA AD844. The chaotic oscillator is based on saturated nonlinear function (SNLF) series, and we show and discuss its frequency limitations to generate 3- to 10-scrolls from 1 kHz to 100 kHz. Finally, we conclude that the frequency limitations are due to the nonideal characteristics of the CFOA-based SNLF block, imposed by the AD844.

Design and application of multi-scroll chaotic attractors based on simplified Lorenz system

Two linear systems are obtained by employing linearization technique in a simplified Lorenz system, and a two-scroll chaotic attractor is generated via the control method. Multi-scroll chaotic attractors are generated by extending the saddle-focus equilibrium points with index 2. Dynamic characteristics of the multi-scroll chaotic system are analyzed by observing the phase diagrams, bifurcation diagrams, Poincaré sections and calculating the largest Lyapunov exponent. A circuit for the multi-scroll attractor is designed and simulated. The numerical simulation result and the circuit simulation result are consistent with each other. To apply the multi-scroll chaotic systems to image encryption, an improved hybrid encryption algorithm is designed based on the multi-scroll chaotic system and advanced encryption standard (AES), and its encryption performances are analyzed. The results show that the improved hybrid encryption has a higher security.

Grid multi-scroll chaotic circuit based on the second generation current conveyers

A three-dimensional grid multi-scroll chaotic system is presented by simplifying a generalized Jerk system. The analog circuit system is designed and realized by using positive-type second generation current conveyers. Compared with the chaotic circuit realized by traditional op amp, the chaotic circuit based on current conveyers can generate grid multi-scroll attractors at high frequency, and also has the advantages of simple circuit structure, and few components. Finally, the grid multi-scroll chaotic attractor are observed through using an oscilloscope, and the physical realizability of chaotic circuit is verified by circuit experiment. The results of circuit experiments and numerical simulations are accordant.

Behavioral Modeling of SNFS for Synthesizing Multi-Scroll Chaotic Attractors

Abstract Existing models of continuous nonlinear functions used for generating multi-scroll chaotic attractors are based on a piece-wise linear (PWL) approach. These models, although relatively easy to build, do not include any information related to the performance parameters of active devices, in the context of a possible physical implementation. This is a serious drawback, since the use of a PWL model introduces a level of inaccuracy into a numerical analysis which is more evident when numerical and experimental results are compared. This paper proposes a methodology to generate the behavioral model of continuous nonlinear functions, but unlike PWL approaches, real physical active device parameters are herein taken into account. In particular, we generate the behavioral model of a nonlinear function called saturated nonlinear function series (SNFS), but in general, the proposed approach can be used to generate the behavioral model of other continuous nonlinear functions. Our results indicate that the proposed approach yields a more realistic and accurate behavioral model than PWL models. As a consequence, not only the generation of chaotic attractors is more precise, but the metrics used to measure the complexity of a chaotic system can also be better predicted. Numerical and electrical simulation results at both domains, phase and time, illustrate the benefits of the new proposed model.

Multi-Scroll Chaotic System Based on Even-Symmetric Step-Wave Sequence Control

This paper presents a new approach for generating multi-scroll chaotic attractors. First, a new double scroll chaotic system with piecewise linearity and invariance under the transformationis introduced. Then, by using the even-symmetric step-wave sequence switching control method in this system to extend the number of saddle-focus points of index 2, the intended multi-scroll chaotic attractors can be obtained. A circuit for generating multi-scroll chaotic attractors is designed, and the experimental results are also given, confirming the consistency of the theory design and circuit implementation.

Generation and Circuit Implementation of Grid Multi-Scroll Chaotic Attractors

In this paper, grid multi-scroll chaotic attractors are firstly generated. The maximum Lyapunov exponent is further calculated to prove the existence of chaos in designed system. An improved module-based circuit is designed for realizing 5×3 grid scroll chaotic attractors, and the experimental results are also obtained, which is consistent with the numerical simulations.