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

In his famous book entitled $\mathit{\text{Theory}}\mathit{\text{of}}\mathit{\text{Oscillations}}$, Nicolas Minorsky wrote: “\textit{each time the system absorbs energy the curvature of its trajectory decreases} and $\mathit{\text{vice}}\mathit{\text{versa}}$”. 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 $\epsilon$ of its $\mathit{\text{slow}}\mathit{\text{invariant}}\mathit{\text{manifold}}$ equation. By using this method, we prove that, in the $\epsilon$-vicinity of the \textit{slow invariant manifold} of generalized Li'{e}nard systems, the \textit{curvature of trajectory curve} increases while the $\mathit{\text{energy}}$ of such systems decreases. Hence, we prove Minorsky’s statement for the generalized Li'{e}nard systems. Then, we establish a relationship between $\mathit{\text{curvature}}$ and $\mathit{\text{energy}}$ for such systems. These results are then exemplified with the classical Van der Pol and generalized Li'{e}nard \textit{singularly perturbed systems}.