Resumen de Dynamical Properties of Singularly Perturbed Renormalization Group Approach to Slow-Fast Systems
Ning Sun, Shaoyun Shi, Wenlei Li
Extensive research has been conducted on solving and analyzing the dynamical properties of slow-fast systems. Existing results mostly focus either on quantitative or qualitative analysis in isolation, creating a methodological dichotomy. This study presents a novel renormalization group (RG) framework that systematically bridges this methodological divide while elucidating the RG method’s inherent dynamical characteristics in slow-fast system investigations. Our results are threefold: First, we develop a rigorous RG formulation to construct arbitrary-order approximate solutions for initial value problems in slow-fast systems, complete with comprehensive error analysis. Next, we derive RG-approximated vector fields that preserve the essential dynamics of the original system, enabling detailed investigation of normally hyperbolic invariant manifolds through geometric singular perturbation theory. Furthermore, we establish an innovative RG-based methodology for constructing asymptotically accurate slow manifolds, complete with convergence proofs under asymptotic stability conditions. The analytical framework is complemented by numerical implementations demonstrating the efficacy of our approach through representative case studies. This unified RG methodology advances the field by providing both quantitative precision and qualitative insights within a single coherent framework.
