<p>In this paper, a new type of controllable and switchable multi-scroll chaotic system family is proposed based on sign function and system switching. The mathematical model of the multi-scroll chaotic system includes switching function terms. By selecting different switching functions, the system can present honeycomb-shaped or twist-shaped multi-scroll chaotic attractors in phase plane. The number of generated scrolls can be controlled by adjusting the upper bound of the summation function. When different combinations of function expressions are selected, the system will generate different odd or even number of scrolls in both horizontal and vertical directions. The 4 × 3 switchable multi-scroll chaotic system is chosen as an example to analyze the chaotic dynamic behavior. The corresponding analog circuit of the 4 × 3 multi-scroll chaotic system is designed to verify the correctness and feasibility of the proposed system. Finally, the designed multi-scroll chaotic system is combined with the improved Hilbert curve and dynamic RNA diffusion for multi-image encryption. Simulation experiments and security analysis demonstrate that the proposed encryption algorithm has excellent encryption performance.</p>