On the Dynamics of a Hyperjerk System and Its Reduction to the Generalized Michelson System

This paper presents a novel, simple hyperjerk system that includes six terms: one nonlinear term and a line of equilibria. The study has two main objectives: to analyze the proposed 4D system and to investigate its reduction to a 3D system. For the 4D system, a zero Hopf bifurcation is displayed in the neighborhood of the origin, with a periodic solution obtained through averaging theory that bifurcates at the origin. It is proven that no [Formula: see text] first integral exists in the neighborhood of this periodic solution. Moreover, employing a polynomial first integral allows the hyperjerk system to be reduced to a Jerk system, known as a generalized Michelson system. A sufficient condition for the reduced Jerk system is derived using the existence and uniqueness theorem and the stability of the equilibrium points is examined. Two local bifurcations, namely saddle-node and Hopf bifurcations, are analyzed, with a periodic solution bifurcating from the Hopf bifurcation proven to correspond to a stable limit cycle. Under appropriate conditions on the parameters, the Hopf bifurcation is shown to be supercritical near the nontrivial equilibrium point. The global dynamics are investigated using Poincaré compactification and the chaotic behavior of the generalized Michelson system is analyzed through bifurcation diagrams and Lyapunov exponents, revealing a self-excited attractor. In some parameter spaces, the generalized Michelson system exhibits multistability, where a periodic attractor coexists with a chaotic attractor. The dynamical regions and basins of attraction are presented for both fixed and varying parameters. The basin boundaries of periodic and chaotic attractors exhibit a smooth fractal structure. In addition, the amplitude and polarity of the chaotic signals can be controlled through the introduction of new control constants, making the system well-suited for various chaos-based applications. Furthermore, a digital circuit based on STM32 hardware is implemented to demonstrate the physical feasibility of the proposed system.