A universal discrete multi-attractor chaotic system framework: Construction, analysis and application

This paper proposes a universal construction framework for three-dimensional discrete chaotic systems featuring programmable multi-attractor topologies. Its core innovation is a parametric piecewise-linear controller that dynamically generates unstable equilibrium points, enabling precise digital control over the number of scroll-like attractors via integer parameters m and k. The framework is systematic and map-agnostic (independent of the base map): any bounded one-dimensional chaotic map can be normalized, hyperchaotically coupled, and augmented with this controller to yield a new 3D system with scalable attractor counts. Theoretical analyses elucidate the attractor layering mechanism, confirm the universal instability of all equilibria, and demonstrate robust hyperchaotic dynamics invariant to initial offsets. The feasibility and randomness of the generated systems are validated through numerical simulations, DSP-based hardware implementation, and successful passage of NIST SP800–22 tests. As a demonstrative application, a chaos-based timed one-time password scheme is designed, highlighting the framework's potential for cryptographic engineering. This work establishes a foundational methodology for the systematic design of discrete multi-attractor chaotic systems with tailored dynamical complexity.