A Geometric Method for Analyzing of Second-Order Autonomous Systems and Its Applications

In this paper, the behavior of a second-order dynamical system around its equilibrium point will be analyzed based on the behavior of some appropriate equipotential curves which will be considered around the same equilibrium point. In fact two sets of equipotential curves are considered so that a set of the equipotential curves has a role as the upper band of the system trajectory and another set plays a role as the lower band. It will be shown that stability of the system around its equilibrium point can be assessed using the behavior of these two set of equipotential curves. As will be shown, asymptotically stability and instability analysis of the system only need the analysis of the upper band set of the equipotential curves, but oscillation behavior analysis of the system need to analyze both the lower band set of the equipotential curves and the upper band set. The method can even detect a stable limit cycle appearing in the oscillation systems. The proposed method is geometric and has some applications such as designing of oscillators. Finally, some examples, practical designing of oscillators and simulation results will be presented to verify and validate the presented method. Key-Words: Equilibrium point, stability, instability, autonomous system.