The identification of fractional order systems by multiscale multivariate analysis

The article aims to study the reduced-order anti-synchronization between projections of fractional order hyperchaotic and chaotic systems using active control method. The technique is successfully applied for the pair of systems viz., fractional order hyperchaotic Lorenz system and fractional order chaotic Genesio-Tesi system. The sufficient conditions for achieving anti-synchronization between these two systems are derived via the Laplace transformation theory. The fractional derivative is described in Caputo sense. Applying the fractional calculus theory and computer simulation technique, it is found that hyperchaos and chaos exists in the fractional order Lorenz system and fractional order Genesio-Tesi system with order less than 4 and 3 respectively. The lowest fractional orders of hyperchaotic Lorenz system and chaotic Genesio-Tesi system are 3.92 and 2.79 respectively. Numerical simulation results which are carried out using Adams-Bashforth-Moulton method, shows that the method is reliable and effective for reduced order anti-synchronization.

Compared with fractional-order chaotic systems with a large number of dimensions, three-dimensional or integer-order chaotic systems exhibit low complexity. In this paper, two novel four-dimensional, continuous, fractional-order, autonomous, and dissipative chaotic system models with higher complexity are revised. Numerical simulation of the two systems was used to verify that the two new fractional-order chaotic systems exhibit very rich dynamic behavior. Moreover, the synchronization method for fractional-order chaotic systems is also an issue that demands attention. In order to apply the Lyapunov stability theory, it is often necessary to design complicated functions to achieve the synchronization of fractional-order systems. Based on the fractional Mittag–Leffler stability theory, an adaptive, large-scale, and asymptotic synchronization control method is studied in this paper. The proposed scheme realizes the synchronization of two different fractional-order chaotic systems under the conditions of determined parameters and uncertain parameters. The synchronization theory and its proof are given in this paper. Finally, the model simulation results prove that the designed adaptive controller has good reliability, which contributes to the theoretical research into, and practical engineering applications of, chaos.

Fractional calculus is a 300 years topic, which has been introduced to real physics systems modeling and engineering applications. In the last few decades, fractional-order nonlinear chaotic systems have been widely investigated. Firstly, the most used methods to solve fractional-order chaotic systems are reviewed. Characteristics and memory effect in those method are summarized. Then we discuss the memory effect in the fractional-order chaotic systems through the fractional-order calculus and numerical solution algorithms. It shows that the integer-order derivative has full memory effect, while the fractional-order derivative has nonideal memory effect due to the kernel function. Memory loss and short memory are discussed. Finally, applications of the fractional-order chaotic systems regarding the memory effects are investigated. The work summarized in this manuscript provides reference value for the applied scientists and engineering community of fractional-order nonlinear chaotic systems.

Exponential synchronization of fractional-order complex chaotic systems and its application

In order to figure out the dynamical behaviour of a fractional-order chaotic system and its relation to an integer-order chaotic system, in this paper we investigate the synchronization between a class of fractional-order chaotic systems and integer-order chaotic systems via sliding mode control method. Stability analysis is performed for the proposed method based on stability theorems in the fractional calculus. Moreover, three typical examples are carried out to show that the synchronization between fractional-order chaotic systems and integer-orders chaotic systems can be achieved. Our theoretical findings are supported by numerical simulation results. Finally, results from numerical computations and theoretical analysis are demonstrated to be a perfect bridge between fractional-order chaotic systems and integer-order chaotic systems.

In this paper, a generalized scheme for synchronizing a fractional order chaotic system with another one or with a linear combination of two other fractional order chaotic systems is presented. Static (time-independent) or dynamic (time-dependent) synchronization that could generate multiple scaled versions of the response is discussed for some fractional order continuous chaotic systems based on differential equations. Non-Standard finite difference method suitable for fractional order chaotic systems is used to solve each system and get the responses. Analysis in the generalized fractional order case is presented, as well as results of simulations that match the corresponding analysis for different values of fractional order parameters.

This contribution deals with identification of fractional-order dynamical systems. System identification, which refers to estimation of process parameters, is a necessity in control theory. Real processes are usually of fractional order as opposed to the ideal integral order models. A simple and elegant scheme of estimating the parameters for such a fractional order process is proposed. This method employs fractional calculus theory to find equations relating the parameters that are to be estimated, and then estimates the process parameters after solving the simultaneous equations. The said simultaneous equations are generated and updated using particle swarm optimization (PSO) technique, the fitness function being the sum of squared deviations from the actual set of observations. The data used for the calculations are intentionally corrupted to simulate real-life conditions. Results show that the proposed scheme offers a very high degree of accuracy even for erroneous data.

Parameter estimation for fractional-order chaotic systems by improved bird swarm optimization algorithm

Time delays are frequently appearing in many real-life phenomena and the presence of time delays in chaotic systems enriches its complexities. The analysis of fractional-order chaotic real nonlinear systems with time delays has a plenty of interesting results but the research on fractional-order chaotic complex nonlinear systems with time delays is in the primary stage. This paper studies the problem of hybrid projective synchronization (HPS) of fractional-order chaotic complex nonlinear systems with time delays. HPS is one of the extensions of projective synchronization, in which different state vectors can be synchronized up to different scaling factors. Based on Laplace transformation and the stability theory of linear fractional-order systems, a suitable nonlinear controller is designed to achieve synchronization between the master and slave fractional-order chaotic complex nonlinear systems with time delays in the sense of HPS with different scaling factors. Finally, the HPS between fractional-order delayed complex Lorenz system and fractional-order delayed complex Chen system and that of fractional-order delayed complex Lorenz system and fractional-order delayed complex Lu system are taken into account to demonstrate the effectiveness and feasibility of the proposed HPS techniques in the numerical example section.

This paper focuses on the finite-time generalized synchronization problem of non-identical fractional order chaotic (or hyper-chaotic) systems by a designing adaptive sliding mode controller and its application to secure communication. The effects of both disturbances and model uncertainties are taken into account. A novel fractional order integral sliding mode surface is designed and its stability to the origin is proved in a given finite time. By the aid of the fractional Lyapunov stability theory, a robust controller with adaptive update laws is proposed and its finite-time stability for generalized synchronization between two non-identical fractional-order chaotic systems in the presence of model uncertainties and external disturbances is derived. Numerical simulations are provided to demonstrate the effectiveness and robustness of the presented approach. All simulation results obtained are in good agreement with the theoretical analysis. According to the proposed generalized finite-time synchronization criterion, a novel speech cryptosystem is proposed to send or share voice messages privately via secure channel. Security and performance analyses are given to show the practical effect of the proposed theories.

Fractional-order memristor-based chaotic jerk system with no equilibrium point and its fractional-order backstepping control

Compared to integer-order chaotic systems, fractional-order chaotic systems have more complex dynamical features due to the introduction of order. The application of fractional-order chaotic systems to chaotic cryptosystems makes the cryptosystems with higher security properties. In this paper, we developed a new 3D fractional-order chaotic system from a 3D integer-order chaotic system, and investigate the dynamical behaviors of this fractional-order system with different parameters and orders. Moreover, self-excited attractors appeared at lower orders through circuit simulations. Furthermore, the synchronization of the new fractional-order chaotic system in the presence of systematic uncertainties and perturbations was achieved using the sliding mode control technique, which sets the stage for the implementation of communication. Finally, offset boosting control was used to investigate the utility of the new chaotic system in engineering applications.

The problem of analyzing the parameters of a chaotic system of fractional order is considered. As an example, a chaotic system based on a Chua circuit of an integer order was chosen. To obtain a Chua circuit of fractional order, it is proposed to replace the capacitive elements in the original circuit with elements with fractal impedance. The purpose of the work is to create a mathematical model of a fractional order Chua circuit with real values of the circuit elements, on the basis of which to evaluate existing software products in the Matlab environment suitable for modeling chaotic fractional order systems, as well as to search for software products suitable for calculating the spectrum of Lyapunov exponents. An analysis of ready-made solutions in the Matlab environment, from the literature known to us, showed that the most suitable option for modeling a system of differential equations of a fractional order is the «fde_pi12_pc» program. The selected program allows you to set all the system parameters of interest and has a relatively high accuracy due to the use of the predictor-corrector algorithm. Based on this program, a domain for determining fractional indicators of the Chua system was constructed in which the presence of a characteristic attractor is visually observed. Existing software products in the Matlab environment are analyzed, allowing the calculation of Lyapunov exponents for chaotic systems described by fractional order differential equations. Of the solutions considered, the best, in our opinion, is the «FO_NC_Lyapunov» program. Using the selected program, a spatial graph of the spectrum of Lyapunov exponents was constructed when changing the fractional exponents of the Chua system. Based on the spatial graph of the spectrum of Lyapunov exponents and the criteria for the chaotic behavior of the system based on the values of Lyapunov exponents, a definition area was constructed with visualization of zones of chaotic behavior of the Chua system. The domains of definition obtained by visual analysis of the presence of an attractor and calculating the spectrum of Lyapunov exponents for the same mathematical model did not coincide; therefore, there are currently no software products for reliably assessing the chaos of fractional order systems by calculating the spectrum of Lyapunov exponents.

Our aim is to study the Hopf bifurcation and synchronisation of a fractional-order butterfly-fish chaotic system. First, we derived the existence of a chaotic attractor in the fractional-order system and also synchronisation problem between two identical fractional-order chaotic systems is studied. Also, control design for the synchronisation with a suitable linear controller is tested in the response system. Finally, numerical simulation results are provided to confirm the theoretical analysis.

This paper describes multi-switching synchronization schemes between three similar chaotic fractional-order systems. The Rossler fractional system is considered to apply the technique. With multi-switch combination synchronization, two drive system state variables are synchronized with different response system state variables at the same time. The multi-switching combination synchronization of three fractional-order equivalent systems was investigated based on the stability of fractional-order chaotic systems. Appropriate controllers have been configured to synchronize three similar fractional-order chaotic systems. Numerical calculations were carried out in Matlab to support the theoretical argument.