\(N\)-tuple Compound and Compound Combination Synchronization in Different Chaotic Models and Their Circuits Implementation

In this paper, we introduced two definitions of \(N\)-tuple compound synchronization (NCS) and \(N\)-tuple compound combination synchronization (NCCS) in (\(2N+2\)) and (\(3N+2\)) different chaotic models, respectively. The analytical formulas for the control functions are derived from two established theorems in order to perform these types of synchronization. These new types of synchronization are considered a generalization of various types disccussed in the literature. Numerous applications in engineering and physics may benefit from these new types of synchronization, <span class="it">e.g.</span> , image encryption and electronic circuits. Using the active control technique for the choice \(N=5\), we study two examples of 5CS and 5CCS in 12 and 17 different chaotic models. The analytical forms of the control functions are used and good agreement is found. The Runge–Kutta method of order four is used in our numerical simulations. Other examples can be similarly studied. We designed an electronic circuit for the proposed \(N\)-tuple compound synchronization in 12 different chaotic models. Using <span class="sf">MATLAB/Simulink</span>, both numerical and simulation results show good agreement. For other drive models, similar circuit implementations can be created. Abstract Published by the Jagiellonian University 2025 authors