Flow curvature manifold and energy of generalized Liénard systems

In his famous book entitled Theory of Oscillations, Nicolas Minorsky wrote: “each time the system absorbs energy the curvature of its trajectory decreases and vice versa”. By using the Flow Curvature Method, we establish that, in the ε-vicinity of the slow invariant manifold of generalized Liénard systems, the curvature of trajectory curve increases while the energy of such systems decreases. Hence, we prove Minorsky's statement for the generalized Liénard systems. These results are then illustrated with the classical Van der Pol and generalized Liénard singularly perturbed systems.