Generation of multi-folded multi-scroll hidden attractors with only stable equilibria

The multi-scroll hidden attractor (MSHA) systems are a growing research focus for their complex dynamic behaviors. This article proposes an approach to generate multi-folded multi-scroll hidden attractors (MFMSHAs) with only stable equilibria, thereby significantly increasing sequence complexity. This enhancement is particularly valuable for information security, where high sequence complexity is critical for robust encryption performance. Specifically, a three-step approach is proposed as followed. Initially, we develop an improved Sprott E system (ISES) with a single stable equilibrium. Subsequently, by applying the dimensional expansion to the ISES, we construct the dimensional expanded systems (DESs) generating one-dimensional (1D) and grid MSHAs. Then we perform the Julia fractal process on the DESs and realize the fractal dimensional expanded systems (FDESs) generating 1D and grid MFMSHAs. Using the proposed approach, a step-by-step enhancement of sequence complexity is achieved, as the Permutation Entropy (PE) index confirmed. More importantly, the proposed approach operates without altering the stability of the equilibria, thereby preserving the hidden characteristics of the seed system. Finally, the hardware circuit verification on an Field-Programmable Gate Array (FPGA) platform demonstrates the correctness and effectiveness of the theoretical investigation, paving the way for the future use in practical secure applications.