Synchronization in Dynamically Coupled Fractional-Order Chaotic Systems: Studying the Effects of Fractional Derivatives

This study presents the effectiveness of dynamic coupling as a synchronization strategy for fractional chaotic systems. Using an auxiliary system as a link between the oscillators, we investigate the onset of synchronization in the coupled systems and we analytically determine the regions where both systems achieve complete synchronization. In the analysis, the integration order is considered as a key parameter affecting the onset of full synchronization, considering the stability conditions for fractional systems. The local stability of the synchronous solution is studied using the linearized error dynamics. Moreover, some statistical metrics such as the average synchronization error and Pearson’s correlation are used to numerically identify the synchronous behavior. Two particular examples are considered, namely, the fractional-order Rössler and Chua systems. By using bifurcation diagrams, it is also shown that the integration order has a strong influence not only on the onset of full synchronization but also on the individual dynamic behavior of the uncoupled systems.

A Simple Integration Order Test: An Alternative to Unit Root Testing

The paper introduces order of integration test (OIT) which serves as a simple alternative to unit root test built generally using auxiliary autoregressive AAR(3) model. The parametric boundary conditions necessary and sufficient for testing the null hypothesis that the non-stationary variable under test is integrated order zero I(0) were estimated via generalized least squares (GLS). The decision on the hypothesis is evaluated using t-statistic. The test procedure was applied to a simulated non-stationary series (y1) of sample size n = 2000 and a known non-stationary time series data (y2) with two unit roots. The results showed that y1 is integrated order one (I(1)) and y2 is I(2). These results were confirmed by Augmented Dickey Fuller (ADF); Phillips-Perron (PP); Kwiatkowski, Phillips, Schmidt, and Shin (KPSS); Elliot, Rothenberg, and Stock Point Optimal (ERS) and Ng and Perron (NP) unit root tests. For logarithm transformed variable, the divergent opinions of other unit root tests in clear-cut solution of the integrated order of such variable makes the new test procedure a better alternative. Nevertheless, the simplicity and aptness of the integration order test give it leverage over conventional methods of unit root test.

THE IMPACT OF THE AUTOMOTIVE INDUSTRY ON THE MOROCCAN ECONOMY

This study explains the relationship between the Moroccan automotive industry, employment and gross domestic product GDP, for this reason, a literature review on the subject was conducted, which allowed us to understand the relationship between all variables. After the analysis, the results show that the variables integration order allowed us to apply the error correction model (ECM), in fact, this model confirmed the relationships between the variables, and showed that the automotive industry in Morocco has a significant positive impact on employment and GDP. JEL: L60; L62 <p> </p><p><strong> Article visualizations:</strong></p><p><img src="/-counters-/edu_01/0795/a.php" alt="Hit counter" /></p>

Multistability Emergence through Fractional-Order-Derivatives in a PWL Multi-Scroll System

In this paper, the emergence of multistable behavior through the use of fractional-order-derivatives in a Piece-Wise Linear (PWL) multi-scroll generator is presented. Using the integration-order as a bifurcation parameter, the stability in the system is modified in such a form that produces a basin of attraction segmentation, creating many stable states as scrolls are generated in the integer-order system. The results here presented reproduce the same phenomenon reported in systems with integer-order derivatives, where the multistable regimen is obtained through a parameter variation. The multistable behavior reported is also validated through electronic simulation. The presented results are not only applicable in engineering fields, but they also enrich the analysis and the understanding of the implications of using fractional integration orders, boosting the development of further and better studies.

On the theoretical basis of memory-free approaches for fractional differential equations

PurposeA nonclassical method, usually called memory-free approach, has shown promising potential to release arithmetic complexity and meets high memory-storage requirements in solving fractional differential equations. Though many successful applications indicate the validity and effectiveness of memory-free methods, it has been much less understood in the rigorous theoretical basis. This study aims to focus on the theoretical basis of the memory-free Yuan–Agrawal (YA) method [Journal of Vibration and Acoustics 124 (2002), pp. 321-324].Design/methodology/approachMathematically, the YA method is based on the validity of two fundamental procedures. The first is to reverse the integration order of an improper quadrature deduced from the Caputo-type fractional derivative. And, the second concerns the passage to the limit under the integral sign of the improper quadrature.FindingsThough it suffices to verify the integration order reversibility, the uniform convergence of the improper integral is proved to be false. Alternatively, this paper proves that the integration order can still be reversed, as the target solution can be expanded as Taylor series on [0, ∞). Once the integration order is reversed, the paper presents a sufficient condition for the passage to the limit under the integral sign such that the target solution is continuous on [0, ∞). Both positive and counter examples are presented to illustrate and validate the theoretical analysis results.Originality/valueThis study presents some useful results for the real performance for the YA and some similar memory-free approaches. In addition, it opens a theoretical question on sufficient and necessary conditions, if any, for the validity of memory-free approaches.

Effect of Strapdown Integration Order and Sampling Rate on IMU-Based Attitude Estimation Accuracy

This paper deals with the strapdown integration of attitude estimation Kalman filter (KF) based on inertial measurement unit (IMU) signals. In many low-cost wearable IMU applications, a first-order is selected for strapdown integration, which may degrade attitude estimation performance in high-speed angular motions. The purpose of this research is to provide insights into the effect of the strapdown integration order and sampling rate on the attitude estimation accuracy for low-cost IMU applications. Experimental results showed that the effect of integration order was small when the angular velocity was low and the sampling rate was large. However, as the angular velocity increased and the sampling rate decreased, the effect of integration order increased, i.e., obviously, the third-order KF resulted in better estimations than the first-order KF. When comparing the case where both transient matrix and process noise covariance matrix are applied to the corresponding order and the case where only the transient matrix is applied to the corresponding order but the process noise covariance matrix for the first-order is still used, both cases had almost equivalent estimation accuracy. However, in terms of the calculation cost, the latter case was more economical than the former, particularly for the third-order KF (i.e., the ratio of the former to the latter is 1.22 to 1).

Proactive model based testing and evaluation for component-based systems

Embedded software systems are getting more and more complex. The demand for new features and functions led to an increasing complexity in the design and development of these systems. There are frequent reports in the media about software systems crashing and damages occurring due to software errors. One reason for this is that there are many software testing methods and techniques but they are often non-practical and difficult to use. The aim of the study was to improve existing testing methods and their practicality especially from the integrator viewpoint. Component-based system development, components of different granularities must be tested. Furthermore, an optimization approach based on simulated annealing is presented which is used to derive an integration order with respect to the proposed parameters in a powerful and reliable manner. The paper discusses explicit properties and the requirements that are to be verified, imposed upon software-intensive systems by their environment and by their users.