Synchronization and desynchronization of chaotic models with integer, fractional and distributed-orders and a color image encryption application

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.