Mokhtar Mahmoud Gamal

The development of novel models can lead to new findings, design new algorithms, and expand practical applications. In this article, we propose a new piecewise-continuous (PWC) system by introducing a threshold level for the states in both the Lorenz and Rabinovich systems. This threshold enables dynamical transitions between the Lorenz system and the Rabinovich system while preserving their fundamental characteristics. Further, this new system exhibits unique dynamic behaviors, including the simultaneous presence of a self-excited chaotic set and two-point attractors, the coexistence of self-excited and hidden attractors with distinct four wings, as well as the coexistence of two hidden scrolls and hidden chaotic attractors. Moreover, the electronic circuit using off-the-shelf components is designed on a schematic capture and SPICE simulator tool. A technique for achieving adaptive synchronization (AS) between switching dynamical systems with completely unknown parameters is proposed, employing sliding mode control. Numerical simulations are conducted to validate the reliability of the AS algorithm. Furthermore, by capitalizing on the inherent switching property of PWC systems, a technique for secure communications is proposed. The approach is based on the concept of dividing the message signal into segments and distributing them across a PWC chaotic system. Subsequently, these encrypted signals are transmitted through two channels. This methodology significantly enhances the overall security of the communication. Various statistical tests are conducted to evaluate the robustness of the method.

This work focuses on studying high-codimensional bifurcations in the glucose model with obesity’s effect. We examine the related dynamical behaviors, taking into account the risk and adverse health effects associated with obesity’s impact on the glucose model. Through the application of the normal form method, we demonstrate that the model exhibits codimension-1 and codimension-2 bifurcations such as saddle node bifurcation and cusp bifurcation. Additionally, we introduce and prove a theorem that establishes the presence of Hopf bifurcation in the model. The computation of the first Lyapunov coefficient is achieved using center manifold theory. To support our theoretical analysis and showcase the model’s complex dynamical behaviors, including periodic curve families, we provide numerical simulations. These simulations contribute to understanding clinical observations regarding the effects of obesity on glucose levels, oscillations and disorders. Ultimately, this research can aid in the early control of blood glucose levels.

This paper introduces the complex Rayleigh–van-der- Pol–Duffing oscillators (RVDOs), which are hyperchaotic and can be autonomous or nonautonomous. The fundamental dynamics of the autonomous and nonautonomous complex RVDOs, including dissipation, symmetry, fixed points, and stability, are studied. These oscillators are found in various necessary fields of physics and engineering. The paper proposes a scheme to achieve phase synchronization (PS) and antiphase synchronization (APS) for different dimensional models. These kinds of synchronization are considered a generalization of several other types of synchronization. We use the active control method based on Lyapunov’s stability theory for this scheme. By analytically determining the control functions, the scheme achieved PS and APS. Our scheme is applied to study the PS of hyperchaotic behaviors for two distinct hyperchaotic nonautonomous and autonomous complex RVDOs. Additionally, the scheme is employed to achieve the APS of a chaotic real nonautonomous RVDO and a hyperchaotic complex autonomous RVDO, including those with different dimensions. Our work presents numerical results that plot the amplitudes and phases of these hyperchaotic behaviors, demonstrating the achievement of the PS and APS. The encryption and decryption of grayscale images are researched based on APS. The experimental results of image encryption and decryption are computed with information entropy, visual analysis, and histograms.

This paper introduces the complex Rayleigh-van der Pol-Duffing oscillators (RVDOs), which are hyperchaotic and can be either autonomous or non-autonomous. The fundamental dynamics of the autonomous and nonautonomous complex RVDOs including dissipation, symmetry, fixed points and their stability are studied. These oscillators are found in various important fields of physics and engineering. The paper proposes a scheme to achieve phase synchronization (PS) and antiphase synchronization (APS) for different dimensional models. These kinds of synchronization are considered as generalization of several other types of synchronization. For this scheme we use the active control method based on Lyapunov stability theory. By analytically determining the control functions, the scheme achieved PS and APS. Our scheme is applied to study PS of hyperchaotic behaviours for two distinct hyperchaotic non-autonomous and autonomous complex RVDOs. Additionally, the scheme is employed to achieve APS of chaotic real non-autonomous RVDO and hyperchaotic complex autonomous RVDO, including those with different dimensions. Our work presents numerical results that plot the amplitudes and phases of these hyperchaotic behaviours, demonstrating the successful achievement of PS and APS.

For the first time, as we know, the generalization of combination synchronization (GCS) of chaotic dynamical models with integer, fractional and distributed-orders is studied in this paper. In the literature, this type of synchronization is considered as a generalization of numerous other kinds. We state the definition of GCS and it’s scheme using tracking control technique among two drive integer and fractional-order models and one response distributed-order model. A theorem is established and proven to give us the analytical formula for the control functions in order to achieve GCS. Numerical calculations are utilized to support these analytic results. We give an example to check the validity of the control functions to achieve GCS. Using the modified Predictor-Corrector method, we obtained numerical results for our models that are in good agreement with the analytical ones. In this work, also, we introduce both of the fractional-order hyperchaotic strongly coupled (FOHSC) Lorenz model and distributed-order hyperchaotic strongly coupled (DOHSC) Lorenz model. Since there are few articles on chaos desynchronization, we aim to study the chaos desynchronization of FOHSC and DOHSC Lorenz models. The encryption and decryption of color image are presented based on GCS between two drive integer and fractional-order models, respectively and one response distributed-order model. Information entropy, correlation analysis between adjacent pixels and histograms are determined together with the experimental results of color image encryption.

This article discusses, for the first time as we know, the basic dynamics of commensurate and noncommensurate fractional‐order simplified Lorenz models, such as invariance, dissipation, stationary points, and the coexistence of chaotic attractors. These models have chaotic attractors and circles of stationary points. Our suggested models are derived from the Lorenz model, which appears in physics, engineering, and medicine (e.g., secure communications). Based on the tracking control method, we investigate a scheme for studying dual compound combination synchronization (DCCS) among 10 fractional‐order chaotic models. This kind of synchronization is a generalization of many other types in the literature. In order to achieve this type of synchronization, a theorem is established and proven to give us the analytical formula for the control functions. These analytical results are supported by numerical calculations. The encryption and decryption of medical image are presented based on DCCS of 10 fractional‐order chaotic models. Information entropy, correlation analysis between adjacent pixels, and histograms are computed together with the experimental results of medical image encryption.