<p>Exact solutions to nonlinear systems are often obtained using elliptic functions. However, conventional elliptic functions have limitations when applied to specific problems. To address these challenges, this paper introduces new functions called Hyper-Elliptic and Hyper-Jacobi Elliptic functions. These newly developed functions aim to enhance and extend conventional elliptic functions, specifically designed to overcome their shortcomings. The significance of these new functions lies in their ability to solve high-order nonlinear systems and facilitate the evaluation of hyperelliptic integrals for polynomials of degree 5 or higher. In this paper, we apply these functions to derive solutions for various nonlinear systems and provide a comprehensive analysis of systems characterized by Quadratic – Cubic – Quartic – Quintic nonlinear oscillators. The results include analytical expressions for the exact solutions y(t) and the period T of the specific nonlinear systems considered. Additionally, we compare these findings with those of other authors and with solutions derived from conventional elliptic functions and numerical methods. The comparisons show that the proposed method is user-friendly, provides accurate results, and is broadly applicable to high-order nonlinear systems.</p>