The Krylov-Bogoliuvob-Mitropolsky averaging method for polynomial dynamical systems
Frank Ernesto Alvarez, Mariano Rodriguez Ricard
TL;DR
This work tackles the existence and characterization of limit cycles in polynomial planar dynamical systems, in the spirit of Hilbert's sixteenth problem. It introduces a constructive polynomial change of variables $\mathcal{H}(X)$ that reduces the system to a second-order ODE, enabling Krylov-Bogoliubov-Mitropolsky (KBM) averaging to obtain an existence result and an explicit asymptotic description of the limit cycle. A key contribution is a general proof that for any polynomial nonlinearity of degree $n$, there exists a polynomial $\mathcal{H}$ of degree $m\le\left\lceil\frac{2n-5+\sqrt{8n^2-16n+25}}{2}\right\rceil$ with a locally invertible power-series inverse, together with an explicit expansion $\mathcal{H}^{-1}(Y)=\Gamma^{-1}Y+\Xi_2\Lambda_2(Y)+\Xi_3\Lambda_3(Y)+\cdots$. The KBM analysis then yields an averaged limit cycle with amplitude $\bar{\varsigma}(t)=r_0\sin(\omega_0 t)$, where $r_0=\sqrt{\delta_{\alpha}/(2|p_3(0)|)}$ and a frequency shift $\omega_0=1-\frac{\tau_{\alpha}}{2}\frac{p_3(0)}{q_3(0)}$, and maps this back to the original variables to obtain an explicit $\bar{X}(t)$; stability follows from Hopf-bifurcation-type conditions. This framework extends KBM analysis beyond linear changes of variables to general polynomial transformations, enabling systematic asymptotic study of periodic orbits in polynomial systems.
Abstract
We describe the transformation of a polynomial planar dynamical system into a second order differential equation by means of a polynomial change of variables. We then, by means of the Krylov-Bogoliubov-Mitropolsky averaging method, identify sufficient conditions involving said change of variables so that a limit cycle exists.