Generalized Li\'enard systems, singularly perturbed systems, Flow Curvature Method
Abstract
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". According to the Flow Curvature Method, the location of the points where the curvature of trajectory curve, integral of such planar singularly dynamical systems, vanishes directly provides a first order approximation in of its slow invariant manifold equation. By using this method, we prove that, in the -vicinity of the slow invariant manifold of generalized Li\'enard systems, the curvature of trajectory curve increases while the energy of such systems decreases. Hence, we prove Minorsky's statement for the generalized Li\'enard systems. Then, we establish a relationship between curvature and energy for such systems. These results are then exemplified with the classical Van der Pol and generalized Li\'enard singularly perturbed systems.
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