Fixed points, periodic orbits, torus, chaos, hyperchaos, and busting firing activities in a novel 4D system with self-excited and hidden attractors: theory and digital circuit realization

The design of nonlinear dynamic systems with special characteristics remains a challenge. This paper introduces and analyzes the dynamics of a novel four-dimensional hyperchaotic system with nine terms, of which two are nonlinear, capable of simultaneously generating self-excited and hidden attractors simply by modifying one of its parameters. The system is designed by introducing linear feedback term into an existing three-dimensional chaotic system. The system is constructed so that it has two versions: one with unstable equilibrium points and the other without equilibrium points, corresponding, respectively, to the self-excited and hidden attractor models. The equilibrium points of the self-excited model are evaluated and analyzed. This analysis shows that the system exhibits a Hopf bifurcation. Both versions of the system are studied numerically with the help of numerical tools such as time traces, phase portraits, and bifurcation diagrams accompanied by Lyapunov exponents. The numerical results show that the different systems generate a very rich and varied range of complex self-excited and hidden dynamic behaviors, including fixed points, periodic attractors, 2D and 3D torus, chaos, hyperchaos, periodic and chaotic bursting, and offset boosting dynamics. The dynamic features of the proposed four-dimensional hyperchaotic system are compared with 20 other related systems published in recent literature. The results of the numerical simulations are supported by the implementation of the system under study using a microcontroller board. A good similarity is observed between the numerical results and those obtained from the digital circuit implementation.