Generalized Li\'{e}nard systems, singularly perturbed systems, Flow Curvature Method

TL;DR

This paper applies the Flow Curvature Method to generalized Liénard systems, demonstrating how trajectory curvature relates to energy decrease near the slow invariant manifold, thus confirming Minorsky's energy-oscillation insight.

Contribution

It introduces a novel application of the Flow Curvature Method to generalized Liénard systems, linking curvature and energy dynamics near the slow invariant manifold.

Findings

Curvature increases as energy decreases near the slow invariant manifold.

Established a quantitative relationship between curvature and energy in these systems.

Validated results with classical Van der Pol and generalized Liénard systems.

Abstract

In his famous book entitled \textit{Theory of Oscillations}, Nicolas Minorsky wrote: "\textit{each time the system absorbs energy the curvature of its trajectory decreases} and \textit{vice versa}". According to the \textit{Flow Curvature Method}, the location of the points where the \textit{curvature of trajectory curve}, integral of such planar \textit{singularly dynamical systems}, vanishes directly provides a first order approximation in $\varepsilon$ of its \textit{slow invariant manifold} equation. By using this method, we prove that, in the $\varepsilon$-vicinity of the \textit{slow invariant manifold} of generalized Li\'{e}nard systems, the \textit{curvature of trajectory curve} increases while the \textit{energy} of such systems decreases. Hence, we prove Minorsky's statement for the generalized Li\'{e}nard systems. Then, we establish a relationship between \textit{curvature} and \textit{energy} for such systems. These results are then exemplified with the classical Van der Pol and generalized Li\'{e}nard \textit{singularly perturbed systems}.