Numerical analysis of complete period-adding structures in a novel chaotic system for message encoding

In this paper, a new three-dimensional chaotic system with signum and cubic nonlinearities is presented, and its dynamic evolution with variation of the system parameters is discussed. In all bifurcation diagrams, complete period-adding structures are also observed, which is a very interesting phenomenon. In these structures, all stable cycles of the periodic orbits, namely the sequence $$1-2-3-...$$ are embedded in chaotic regions. Moreover, the bifurcation analysis reveals other routes to chaos, including period-doubling and boundary crisis. The proposed system exhibits strong dynamical behavior, such as multiple attractors, quasi-periodic motion, Hopf bifurcation, and periodic orbits. In addition, the Sparse Identification of Nonlinear Dynamics (SINDy), a data-driven approach, is used to discover 1D Poincaré maps to determine periodic orbits with any cycle length. The influence of different combinations of parameters on the emergence of periodic orbits is also investigated. Finally, the concept of message encoding is introduced by defining an alphabet of n symbols based on periodic orbits, highlighting the significance of studying periodic orbits in this context.