Nonlinear distributed-order models: Adaptive synchronization, image encryption and circuit implementation

The main aim of present work is to investigate the dynamics of the chaotic nonlinear distributed order Lu model (DOLM). The distributed order (DO) derivative is used for describing the viscoelasticity of various technical models and materials. The modified spectral numerical method is used to evaluate the numerical solutions for DOLM. Using nonlinear feedback control and the Lyapunov direct approach, the adaptive synchronization of two chaotic distributed order models (DOMs) is presented. We state a theorem to drive analytical controllers which are used to achieve our synchronization. The DOLM is introduced as an example of DOMs to verify the validity of our analytical results. Numerical computations are displayed to show the agreement between both analytical and numerical results. The DOMs appear in many applications in engineering and physics, e.g., image encryption and electronic circuits (ECs). Based on our proposed synchronization, the encryption and decryption of color images are studied. Information entropy, visual analysis and histograms are calculated, together with the experimental results of image encryption and decryption. We design the EC of the DOLM using the Multisim circuit simulator for the first time to our knowledge. Using electronic circuit simulation, we achieved the same results for the numerical treatment of our synchronization. Other ECs can be similarly designed for other DOMs.