Exploring Multi-Rhythmicity in the light of Hilbert's 16th Problem : A Li\'enard--Levenson-Smith Perspective and the Potency of Perturbative Methods

This study explores the dynamics of nonlinear systems, particularly focusing on a class of oscillatory kinetic equations in two variables and their connection to the second part of Hilbert?s 16th problem, which concerns the behavior of limit cycles in polynomial vector fields. We examine the reduction of kinetic equations to second-order differential equations in the generalized Li\'enard--Levenson--Smith (LLS) form, offering a systematic classification of nonlinear oscillators into the the Li\'enard, extended Li\'enard, and Rayleigh families. These classifications provide insight into universal rhythmic patterns, especially in biological systems such as heartbeats, circadian rhythms, and neural oscillations, when analyzed using the LLS condition. The review also delves into various perturbative techniques, including the Renormalization Group (RG) and Krylov--Bogoliubov (KB) methods, highlighting their role in understanding amplitude-phase dynamics. We discuss how these methods can be used to estimate upper bounds on the number of limit cycles in generalized LLS systems, offering theoretical insights to address Hilbert's 16th problem. Furthermore, the review critically examines the limitation of the LLS framework, particularly the assumption of a unique limit cycle, by presenting counterexamples of the generalized systems which admit multiple limit cycles. By synthesizing mathematical theory, perturbative analysis, and real-world applications, the review provides a comprehensive framework for understanding nonlinear oscillations, with broad implications for several interdisciplinary fields ranging from physics to biology and engineering.