Behavior of Limit Cycles in Nonlinear Systems

Abstract

In this paper, the behavior of limit cycles in second-order autonomous system will be analyzed based on the behavior of some appropriate equipotential curves which will be considered around the same limit cycles. 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 trajectories and another set plays a role as the lower band. It will be shown that the stability of the limit cycles in system can be assessed using the behavior of these two set of equipotential curves. It will be shown that asymptotic stability, semi-stability and instability of the limit cycles or oscillation behavior in the system need to analyze both the lower and upper bands set of the equipotential curves. The method can even detect a stable limit cycle appearing in the oscillation systems. The method is geometric and suitable for second-order nonlinear autonomous systems. Finally, some examples will be presented to verify the presented method.

How to Make Systems of Nonlinear Autonomous ODEs with Attractor-Behavior, by First Making the General Solutions: Part Two

“…In Part One [2], we made systems of ODEs where a distinct closed surface had attractor behavior, that is attracting the solution curves from both inside and outside the closed surface, and following the surface over and over in the same orbit as the parameter t goes to infinity [5]. The orbit is determined by the initial values.…”

“…A limit cycle is said asymptotic stable if all trajectories in the vicinity of the limit cycle converge to it as t → ∞ . Otherwise, the limit cycle is said semi-stable or unstable [5].…”