Two-dimensional function-shaped hyperchaotic maps and their hardware implementation

Discrete chaotic maps hold significant application potential in information security and intelligent computing. However, existing chaotic maps generally suffer from several shortcomings, including discontinuous effective parameter ranges, susceptibility to dynamical degradation under finite precision, limited controllability of attractor morphology, and insufficient theoretical analysis. This paper proposes a general two-dimensional Function-Shaped Chaotic (2D-FSC) map framework, in which geometric constraints and a chaotic-kernel driving mechanism are introduced to realize both the topological inheritance of nonlinear-term geometries in phase space and tunable control of the resulting dynamics. Based on theoretical analysis, an analytical proof of the system’s chaotic behavior is provided, and sufficient conditions for hyperchaos are further derived for four typical nonlinear terms. Finite-precision period statistics and hardware experiments verify strong resistance to dynamical degradation. Numerical and dynamical analyses show that the resulting maps exhibit dense coverage within the function-shaped band and high algebraic complexity. In addition, the proposed maps are implemented on a digital hardware platform and applied to pseudorandom number generation (PRNG). Results show that 2D-FSC maps can stably generate desirable chaotic behavior and produce pseudorandom numbers with good statistical randomness.