Computation Of Limit Cycle With Renormalization Group Analysis

Periodic orbits and limit cycles play a crucial role in understanding nonlinear dynamics and provide methods for describing and controlling system behavior.While classical asymptotic analysis offers approaches to tackle these problems,it becomes challenging when dealing with highdimensional systems.Challenges include determining boundary layer thickness or generating asymptotic sequences in complex dynamic systems.To address these issues,we propose exploring the renormalization group(RG)method as a more generalized approach.The RG method,originally used in theoretical physics,eliminates singularities in perturbation theory by appropriately transforming parameters.It has also been extended to solve large-scale asymptotic solutions of perturbed nonlinear differential equations.Compared to traditional progressive methods,the RG method is simpler and involves a more concise calculation process.In this paper,we present an extension of the RG method to higher-dimensional nonlinear systems.To validate the effectiveness of our approach,we apply the new method to three specific examples that highlight different aspects of the RG method.In the first example,we employ the standard RG method and compare it with the traditional multi-scale method for handling singular perturbations.In the second example,we investigate the impact of increasing the order of approximation and perturbation coefficient on the RG method.The third example demonstrates how the current RG method can be easily extended to approximate analytical solutions of limit cycles in three-dimensional models.We emphasize the significance of this method in dealing with various limit cycle models generated by Hopf bifurcations.By leveraging the RG method and its extension,we aim to provide a more comprehensive and effective analysis tool for understanding and controlling periodic behavior in complex systems.The results obtained from applying this method to the examples mentioned above showcase the potential and value of the RG method in the study of nonlinear dynamics.This research contributes to advancing our understanding of periodic orbits and limit cycles,paving the way for further applications in diverse fields...