Fractional-order Chaotic Oscillators Research Articles

Solution of fractional order chaotic oscillator system

Solution of fractional order chaotic oscillator system

Synchronization of time-varying fractional-order chaotic systems in the presence of unknown disturbance using a fast disturbance identification

This paper studies the synchronization of fractional-order chaotic systems in the presence of unknown external disturbances. A new disturbance identification method using fractional-order algebraic identification is proposed to handle unknown disturbances. In order to guarantee system stability, the nonlinear state-feedback controller is designed based on the Lyapunov theory. All system parameters and orders, as well as the external disturbances, are considered time-varying in this paper. The simulation results show that the estimation of unknown disturbance is executed more quickly and accurately using the proposed method, which also leads to fast and precise synchronization of the chaotic systems. A circuit implementation is also performed to identify the unknown disturbances in a fractional-order electronic chaotic oscillator system to evaluate the practicability of the proposed approach.

Chaos in Analog Electronic Circuits: Comprehensive Review, Solved Problems, Open Topics and Small Example

This paper strives to achieve a comprehensive review of chaos in analog circuits and lumped electronic networks. Readers will be guided from the beginning of the investigations of simple electronic circuits to the current trends in the research into chaos. The author tries to provide the key references related to this issue, including papers describing modern numerical algorithms capable of localizing chaotic and hyperchaotic motion in complex mathematical models, interesting full on-chip implementations of chaotic systems, possible practical applications of entropic signals, fractional-order chaotic systems and chaotic oscillators with mem-elements.

On the Sizing of CMOS Operational Amplifiers by Applying Many-Objective Optimization Algorithms

In CMOS integrated circuit (IC) design, operational amplifiers are one of the most useful active devices to enhance applications in analog signal processing, signal conditioning and so on. However, due to the CMOS technology downscaling, along the very large number of design variables and their trade-offs, it results difficult to reach target specifications without the application of optimization methods. For this reason, this work shows the advantages of performing many-objective optimization and this algorithm is compared to the well-known mono- and multi-objective metaheuristics, which have demonstrated their usefulness in sizing CMOS ICs. Three CMOS operational transconductance amplifiers are the case study in this work; they were sized by applying mono-, multi- and many-objective algorithms. The well-known non-dominated sorting genetic algorithm version 3 (NSGA-III) and the many-objective metaheuristic-based on the R2 indicator (MOMBI-II) were applied to size CMOS amplifiers and their sized solutions were compared to mono- and multi-objective algorithms. The CMOS amplifiers were optimized considering five targets, associated to a figure of merit (FoM), differential gain, power consumption, common-mode rejection ratio and total silicon area. The designs were performed using UMC 180 nm CMOS technology. To show the advantage of applying many-objective optimization algorithms to size CMOS amplifiers, the amplifier with the best performance was used to design a fractional-order integrator based on OTA-C filters. A variation analysis considering the process, the voltage and temperature (PVT) and a Monte Carlo analysis were performed to verify design robustness. Finally, the OTA-based fractional-order integrator was used to design a fractional-order chaotic oscillator, showing good agreement between numerical and SPICE simulations.

CMOS OTA-Based Filters for Designing Fractional-Order Chaotic Oscillators

Fractional-order chaotic oscillators (FOCOs) have shown more complexity than integer-order chaotic ones. However, the majority of electronic implementations were performed using embedded systems; compared to analog implementations, they require huge hardware resources to approximate the solution of the fractional-order derivatives. In this manner, we propose the design of FOCOs using fractional-order integrators based on operational transconductance amplifiers (OTAs). The case study shows the implementation of FOCOs by cascading first-order OTA-based filters designed with complementary metal-oxide-semiconductor (CMOS) technology. The OTAs have programmable transconductance, and the robustness of the fractional-order integrator is verified by performing process, voltage and temperature variations as well as Monte Carlo analyses for a CMOS technology of 180 nm from the United Microelectronics Corporation. Finally, it is highlighted that post-layout simulations are in good agreement with the simulations of the mathematical model of the FOCO.

Experimental verification of the multi-scroll chaotic attractors synchronization in PWL arbitrary-order systems using direct coupling and passivity-based control

Because one of the main applications of chaos synchronization is oriented to data encryption and random number generators, fractional-order chaotic systems showing multi-scroll attractors have pointed out as feasible solutions to improve the performance of secure communications schemes. Nowadays, recent works have reported the analysis and control of fractional-order synchronization schemes for multi-scroll chaotic systems. However, the synchronization schemes from those works present mostly theoretical numerical simulations and give circuit simulations employing a simulator software rather than physical implementations. That way, our first contribution is to describe herein the experimental realization of a synchronization scheme for a fractional-order multi-scroll chaotic oscillator with a 4-scroll attractor and fractional-order q=0.9. First, we implemented the multi-scroll systems using an opamp-based analog circuit and an RC-based fractance device to emulate a fractional-order integrator. As a second contribution, we analytically demonstrate the complete synchronization between two identical multi-scroll systems employing two fractional-order controllers and the fractional-order derivative in the Caputo sense. By utilizing the Laplace transform theory, we analyze the one-way coupling method where the response (slave) system’s behavior depends on an identical drive (master) system. As a third contribution, we propose a passivity-based synchronization controller. This synchronization strategy consists of designing a virtual state feedback input and then based on the fractional-order Lyapunov theory, a control law obtains by designing an appropriate storage function to stabilize the system around its stable zero equilibrium. The experimental measures of the implemented synchronization scheme present suitable results compared to analytical and numerical findings.

On the variable-order fractional memristor oscillator: Data security applications and synchronization using a type-2 fuzzy disturbance observer-based robust control

Abstract In the present paper, for the first time, we propose a variable-order hyperchaotic system for information security. Firstly, we study the dynamical behaviors of a memristor oscillator through well-known numerical and analytical tools, such as the Lyapunov exponents, stability of equilibria, and bifurcation diagram. Then as an engineering application, a variable-order fractional version of the system is proposed for sound encryption. In comparison with integer and conventional constant fractional-order chaotic memristor oscillator, the proposed variable-order fractional system shows more complex characteristics and more degrees of freedom due to the existence of time-varying fractional derivatives. Thus, the proposed system is an appropriate choice for data transmission and information security. To illustrate the proper performance of the suggested system for encryption purposes, sound encryption is successfully performed, and its excellent results are demonstrated. The predictor-corrector method is utilized for numerical simulation. Then, a new type-2 fuzzy disturbance observer-based robust control is offered for synchronization of the variable-order hyperchaotic system. The stability and convergence of the disturbance estimator and closed-loop system are proven. Lastly, the synchronization results, which confirm the appropriate performance of the proposed method in the presence of the external disturbances, are demonstrated.

Optimization of the Kaplan-Yorke dimension in fractional-order chaotic oscillators by metaheuristics

The optimization of the Kaplan-Yorke dimension (DKY) of fractional-order chaotic oscillators (FOCOs) is shown herein by applying two metaheuristics, namely: differential evolution and particle swarm optimization algorithms. The optimization process is performed in two stages, and the cases of study are five commensurate (all derivatives have the same fractional-order) and incommensurate (all derivatives have different fractional-order) FOCOs. The first optimization stage evaluates the equilibrium points and eigenvalues of the individuals or particles to verify if they accomplish the minimum fractional-order value to guarantee chaotic behavior. The individuals or particles accomplishing this requirement are passed to the second optimization stage evaluating their DKY. This saves computing time because the individuals or particles that do not accomplish the minimum fractional-order value, are not simulated in the time domain. We show that this optimization approach generates feasible values of the parameters of the FOCOs, providing better DKY values compared to the ones already reported in the literature.

Fractional-Order Hidden Attractor Based on the Extended Liu System

In this paper, a new commensurate fractional-order chaotic oscillator is presented. The mathematical model with a weak feedback term, which is named hypogenetic flow, is proposed based on the Liu system. And with changing the parameters of the system, the hidden attractor can have no equilibrium points or line equilibrium. What is more interesting is that under the occasion that no equilibrium point can be obtained, the phase trajectory can converge to a minimal field under the lead of some initial conditions, similar to the fixed point. We call it the virtual equilibrium point. On the other hand, when the value of parameters can produce an infinite number of equilibrium points, the line equilibrium points are nonhyperbolic. Moreover than that, there are coexistence attractors, which can present hyperchaos, chaos, period, and virtual equilibrium point. The dynamic characteristics of the system are analyzed, and the parameter estimation is also studied. Then, an electronic circuit implementation of the system is built, which shows the feasibility of the system. At last, for the fractional system with hidden attractors, the finite-time synchronization control of the system is carried out based on the finite-time stability theory of the fractional system. And the effectiveness of the controller is verified by numerical simulation.

FPAA-based implementation of fractional-order chaotic oscillators using first-order active filter blocks

Fractional-order chaotic oscillators (FOCOs) have been widely studied during the last decade, and some of them have been implemented on embedded hardware like field-programmable gate arrays, which is a good option for fast prototyping and verification of the desired behavior. However, the hardware resources are dependent on the length of the digital word that is used, and this can degrade the desired response due to the finite number of bits to perform computer arithmetic. In this manner, this paper shows the implementation of FOCOs using analog electronics to generate continuous-time chaotic behavior. Charef’s method is applied to approximate the fractional-order derivatives as a ratio of two polynomials in the Laplace domain. For instance, two commensurate FOCOs are the cases of study herein, for which we show their dynamical analysis by evaluating their equilibrium points and eigenvalues that are used to estimate the minimum fractional-order that guarantees their chaotic behavior. We propose the use of first-order all-pass and low-pass filters to design the ratio of the polynomials that approximate the fractional-order. The filters are implemented using amplifiers and synthesized on a field-programmable analog array (FPAA) device. Experimental results are in good agreement with simulation results thus demonstrating the usefulness of FPAAs to generate continuous-time chaotic behavior, and to allow reprogramming of the parameters of the FOCOs.

An experimental synthesis methodology of fractional-order chaotic attractors

A novel synthesis methodology of fractional-order chaotic systems, from the level of nonlinear systems until their experimental verification using microcontrollers, is presented. Firstly, the integer-order behavioral model of the Lorenz’s, Rossler’s, Chen’s, Liu’s, Saturated Nonlinear Function Series at one- and two-direction systems is briefly reviewed. Secondly, a first-order transfer function that approximates the behavior of a fractional-order integrator, based on continued fraction expansion method, is substituting the integer-order integrator inside the revised previously chaotic systems. Thirdly, the minimum phase Al-Alaoui’s transformation method is used for synthesizing all fractional-order chaotic systems in the discrete domain and are programmed in MATLAB and in the Arduino DUE development board. For a fair comparison, the minimum phase Al-Alaoui’s algorithm not only is used for solving the integer-order chaotic systems and it is also programmed in MATLAB and Arduino DUE board, but same initial conditions are also used for both interger- and fractional-order chaotic systems. Finally, experimental results of the integer- and fractional-order chaotic oscillators are shown. The results obtained not only allow a simple synthesis methodology of fractional-order chaotic attractors and their experimental evidence on reconfigurable hardware, but also demonstrate the viability of fractional attractors to be used in various applications such as secure communications, robot control, cryptography and so on.

Synchronization of fractional-order Lü chaotic oscillators for voice encryption

In this paper, the encryption improvement via modulation of the fractional order chaotic oscillators state variables, is presented. A network of N-coupled fractional order Lü chaotic oscillators, is synchronized. A voice message is encrypted via additive encryption, by using a state variable of the synchronized network. The selected state variable is modulated and used to encrypt the message again. The results are compared.

FPGA-based implementation of different families of fractional-order chaotic oscillators applying Grünwald–Letnikov method

Fractional-order chaotic oscillators are a hot topic of research and nowadays many new mathematical models have been introduced, which can be suitable for the development of novel applications in all related fields of science and engineering. However, the challenge is their implementation that can be performed using electronic devices. In this manner, we highlight the implementation of different families of fractional-order chaotic oscillators using field-programmable gate arrays (FPGAs). We detail the hardware implementation when solving the mathematical models applying the Grünwald–Letnikov method, and highlight the short-memory principle, which memory length is designed using specialized random-access-memory and read-only-memory blocks. In addition, we show how to reduce hardware resources by reusing blocks that are controlled by an especial architecture that is introduced herein, in order to perform an efficient processing of the data. Finally, using Cyclone IV GX FPGA DE2i-150 from Altera, DAS1612 digital-to-analog converter and fixed-point arithmetic of 32 bits, we provide experimental results that were observed in a Lecroy’s oscilloscope showing working frequencies of fractional-order chaotic attractors between 77.59 and 84.9 MHz.

Synchronization and FPGA realization of complex networks with fractional–order Liu chaotic oscillators

In this paper synchronization of fractional–order Liu chaotic oscillators is addressed. Two networks of fractional–order Liu chaotic oscillators are synchronized by appealing to results from complex systems theory for integer–order systems. We use and implement complex dynamical networks composed by nine Liu chaotic oscillators. Two scenarios are considered: (i) complex network with regular topology, and (ii) complex network with irregular topology. Synchronization of both complex networks is achieved by coupling the fractional–order chaotic oscillators through their second state. We use numerical simulations to verify the results, we also show an FPGA realization of the complex networks previously synchronized.

Chaotic synchronization of regular and irregular complex networks with fractional order oscillators

Synchronization of N-coupled fractional-order (FO) chaotic oscillators arranged in regular and irregular topologies is numerically studied. Synchronization is achieved based on the coupling matrix from the complex systems theory. In particular, we consider complex dynamical networks composed by Lorenz, Volta, Duffing and Financial FO chaotic oscillators, where the interaction of the nodes is defined by coupling only one state of each FO oscillator.

Wavelet Phase Synchronization of Fractional-Order Chaotic Systems

Nowadays, fractional-order systems are attracting more and more attention. There are several ways available for analyzing fractional-order systems, among which wavelet transform is an efficient method for analyzing system dynamics in both time and frequency domains. We investigate the wavelet phase synchronization employing wavelet transform to explore the phase synchronization behaviors of fractional-order chaotic oscillators. We analyze in detail the synchronization behaviors with changes to the coupling strength, the central frequency Ω0, and the time scale of the wavelet.

Adaptive feedback control and synchronization of non-identical chaotic fractional order systems

This paper addresses the reliable synchronization problem between two non-identical chaotic fractional order systems. In this work, we present an adaptive feedback control scheme for the synchronization of two coupled chaotic fractional order systems with different fractional orders. Based on the stability results of linear fractional order systems and Laplace transform theory, using the master-slave synchronization scheme, sufficient conditions for chaos synchronization are derived. The designed controller ensures that fractional order chaotic oscillators that have non-identical fractional orders can be synchronized with suitable feedback controller applied to the response system. Numerical simulations are performed to assess the performance of the proposed adaptive controller in synchronizing chaotic systems.

PHASE AND LAG SYNCHRONIZATION IN COUPLED FRACTIONAL ORDER CHAOTIC OSCILLATORS

Chaotic dynamics of fractional (non-integer) order systems have begun to attract much attention in recent years. In this paper, we study the phase and lag synchronization in coupled two fractional order chaotic oscillators. It is shown that with the increase of coupling strength, the system first undergoes a transition to phase synchronization, and when the coupling strength is increased further, another transition to lag synchronization occurs. It is further shown that the system with two chaotic oscillators of different orders can also exhibit phase synchronization phenomena, although the identical synchronization is impossible in this case.

Projective Synchronization in Fractional Order Chaotic Systems and Its Control

The chaotic dynamics of fractional (non-integer) order systems have begun to attract much attention in recent years. In this paper, we study the projective synchronization in two coupled fractional order chaotic oscillators. It is shown that projective synchronization can also exist in coupled fractional order chaotic systems. A simple feedback control method for controlling the scaling factor onto a desired value is also presented.

Master–slave synchronization of coupled fractional-order chaotic oscillators

The dynamics of fractional-order systems have attracted increasing attentions in recent years. In this paper, we study the synchronization technique based on the master?slave synchronization scheme and apply it to the synchronization of two coupled nonlinear fractional-order electronic chaotic oscillators. Simulations show that two coupled fractional-order chaotic oscillators can be brought to an exact synchronization with appropriate coupling strength. It is interesting that the synchronization rate of the fractional-order chaotic oscillators is slower than its integer-order counterpart; however, with the increase of system order, the curves of synchronization error can be smoothened, which indicates that the master?slave synchronization of two coupled fractional order oscillators can be smoothened and stabilized.