A simple method for constructing a class of discrete conservative chaotic maps

Abstract Conservative systems play a critical role in high-reliability chaotic encryption due to their phase-space conservation and stable dynamic behavior during long-term evolution. However, research on discrete chaotic mappings with conservative properties remains scarce. To address this gap, this paper proposes a class of 2D discrete-time conservative chaotic systems via nonlinear reconstruction of the generalized Gumowski-Mira mappings. Specifically, their conservative nature is rigorously verified through Liouville’s theorem, demonstrating phase-conservation properties and symmetric Lyapunov exponents. For deeper analysis, we select a representative model (2D-DTC) from this class, which transitions from a stable fixed point to multiple unstable states through parameter variations. The system’s rich dynamic behavior is illustrated using bifurcation diagrams, symmetric Lyapunov exponential spectra, and parameter-space chaos diagrams. Notably, heterogeneous coexisting chaotic orbits emerge under initial value perturbations, while complexity quantification using spectral entropy (SE) and permutation entropy (PE) validates their nonlinear characteristics. For practical implementation, an FPGA-based platform is designed for chaotic sequence generation and image encryption that integrates diffusion and permutation. Experimental results demonstrate superior encryption performance and effectiveness.