In this article, we introduced fractional and distributed order hyperchaotic Lü, Chen and Lorenz systems with both line and parabola of equilibrium points (EPs). Their dynamics which include invariance, dissipation, EPs and their stability, chaotic and hyperchaotic solutions are studied. Numerically we calculated the values of the systems parameters and the fractional order at which these systems have chaotic, hyperchaotic attractors and solutions that approach EPs. Those systems with no line and parabola of EPs have no hyperchaotic attractors of order 2 and 3. We discussed the difference between fractional order hyperchaotic (FOH) systems with line and parabola of EPs and distributed order hyperchaotic (DOH) systems have line and parabola of EPs. Finally we presented a scheme to investigate the generalization of combination-combination synchronization (GCCS) between two FOH systems and two DOH systems. A theorem is stated and proved to provide us with the analytical formula of the control functions to achieve this kind of synchronization. These analytical results are confirmed via numerical computations.

The combination synchronization (CS) and combination-combination synchronization (CCS) for chaotic and hyperchaotic dynamical systems with the same dimensions are introduced and studied in the literature. In this paper, we introduce the definition of CS and CCS for those systems with different dimensions. We state two schemes to achieve these kinds of synchronization based on the active control technique. Two theorems are stated and proved to provide us with analytical expressions for the control functions. We state four hyperchaotic dynamical systems with different dimensions which are used as examples to achieve CS and CCS. These examples are hyperchaotic detuned laser, Lorenz, van der Pol and dynamos systems. These systems appeared in many important applications in applied science, e.g., a ring laser system of two-level atoms, vacuum tube circuit and two coupled dynamos system. The validity of the analytical control functions are tested numerically and good agreement is found between them. The numerical solutions of ODE systems are calculated by using the method of Runge-Kutta of order 4. Other systems can be similarly studied.

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.