Matrix projective combination synchronization of time-delayed chaotic system and its application in image encryption

External disturbance, uncertainty and time delay are basic ideas of nonlinear systems that are highly related to the control and synchronization of nonlinear systems. This paper introduces a novel approach for achieving synchronization in non-identical chaotic systems in the presence of model uncertainty, external disturbances and time delay. The proposed method utilizes a matrix projective combination synchronization strategy, demonstrating its efficacy in scenarios both with and without time delay. Notably, the application of this strategy has proven to be valuable and efficient, particularly in the domain of encrypting images. This work proceeds by combining two drive systems and one response system using Lyapunov stability theory and a suitable active control technique. Numerical examples are taken to verify the effectiveness of the proposed method and numerical simulations have been performed to demonstrate that the theoretical results are completely consistent with the graphical results. A significant idea for encryption and decryption algorithms is that the secure transmission of images using affine encryption. In affine encryption algorithm, the key is based on the solution of synchronized chaotic delayed disturbed systems and the birth date and birth time of the sender and receiver. The proposed encryption and decryption algorithms have been applied to plain images. Their effectiveness has been validated through key security metrics. Additionally, a comparative analysis with previously published methods demonstrates the robustness and efficiency of the approach.