A new numerical approach for tracing periodic solutions in nonlinear dynamical systems using CGM with PAC

In this research paper , we introduce a new numerical approach that combines the Conjugate Gradient Method (CGM) and Pseudo- Arclength Continuation (PAC) to trace periodic solutions in nonlinear dynamical systems. By minimizing Jacobian computations, the method reduces computational costs, making it efficient for large systems. We apply the method to the Lorenz and Duffing oscillator systems, utilizing Floquet theory for stability analysis. The results show that the proposed approach can accurately trace periodic solutions and provides a robust analysis of system stability. This method offers significant computational advantages while maintaining high accuracy in tracing periodic behaviors in complex systems.