Infinitely Many Hidden Attractors in a Simple Six-Term Chaotic 2-torus Snap System with a Line of Equilibria

Chaotic systems with simple structures and controllable dynamics are highly desirable for practical applications, including secure communications, signal processing, and nonlinear control. However, most existing systems are too complex or lack flexibility in controlling signal amplitude and polarity. In this study, a simple chaotic snap system is considered that has a line of equilibria and is composed of six algebraic terms, with a single quadratic nonlinearity. In some regions of parameter space, a chaotic attractor coexists with a line equilibrium or with a limit cycle of period-2. The basin boundary of a limit cycle and chaotic attractor is smooth. Additionally, in specific regions, two limit cycles can coexist each other. A single quadratic term allows rescaling chaotic signal amplitude, while adding a constant to x controls polarity. The snap system can become self-reproducing via periodic functions, producing a 3D lattice of hidden attractors with identical Lyapunov exponents. Consequently, a chaotic signal of either polarity can be obtained by selecting different initial conditions without requiring an additional polarity converter. These properties allow the implementation of a digital circuit based on STM32 hardware to illustrate the physical feasibility of the proposed system. Finally, to validate the feasibility of the snap system for image encryption a random number generator (RNG) is implemented. The generated RNG successfully passes the National Institute of Standards and Technology (NIST) SP 800-22 statistical tests. Based on these random bits, an image encryption algorithm is developed. Security analysis confirms the system effectively encrypts and decrypts images with high performance.