This paper introduces a generalized piecewise linear (PWL) function capable of simultaneously approximating both single-variable and multi-variable coupled nonlinearities, overcoming the dimensional limitations of conventional PWL methods. This mathematical enhancement provides an efficient framework for modeling complex nonlinear dynamics. Crucially, we embed this generalized PWL function to construct and investigate novel discrete-time fractional-order chaotic systems. A rigorous error analysis of the resulting chaotic sequences is conducted. Through comprehensive numerical investigations—utilizing discrete phase diagrams, Lyapunov exponent spectra, and bifurcation diagrams—the system's underlying exponential dissipative properties and intricate dynamical behaviors are systematically revealed. Furthermore, we propose a novel variable-order offset enhancement method tailored for fractional-order systems. By dynamically coupling the fractional derivative order with the system's offset quantity, this mechanism drastically expands the multidimensional parameter space, enabling the system to exhibit unprecedented topological complexity and significantly richer chaotic characteristics. Based on the afore-mentioned system, a chaotic sequence odd-even index jumping method is proposed to construct a dynamic chaotic substitution box (CS-box). The performance metrics of the CS-box, including nonlinearity, bit independence criterion, differential approximation probability, and strict avalanche criterion, are evaluated. The results fully demonstrate the feasibility of deploying the proposed discrete-time system in practical applications.
Yan et al. (Tue,) studied this question.