Analyzing Chaotic Jerk System with Hyperbolic Sine: a new algorithm for Engineering Applications

this paper examines numerical solutions and applications of an innovative Chaotic Jerk System utilizing the Caputo-Fabrizio fractional derivative (CF-FD). Chaotic systems display complex and unpredictable behaviors, which are recognized as significant in secure communication, data encryption, and random number generation. Employing both the Caputo-Fabrizio fractional derivative and the theory of fractional calculus allows this study to present complex behaviors in compliance with memory effects and long-range interdependence. The computational approach has been introduced in the work and used to solve this chaotic system, which will eventually help us explore the dynamic behavior of such systems. At the same time, it scrutinizes a system's chaos, although closely recreating its action with. Initial conditions and parameters by executing numerical analysis, for example. Moreover, the paper gives an illustrative example for using this chaotic system in secure communication, data encryption, and random number generation. This article develops the phase form of a four-dimensional chaotic system using fractional calculus techniques and, hence, manipulates its chaos. The study provides a bridge between theory and application, an important function that allows for modern innovations in a wide spectrum of areas. We quantified the behavior of this system through numerical simulations and analytical exams, addressing methodologically different conditions thereon. These goals feed into a deeper knowledge of emergent complex chaotic dynamics as well as how the system might be implemented in applications such as secure communication and data encryption. Fractional calculus and chaotic systems have numerous and ever-expanding applications in engineering, helping to improve signal processing, modeling, communication, control, encryption, and our general knowledge of complicated dynamic systems.