A nonexistence criterion for closed orbits in planar flows and its application to fast limit cycle detection

Area of Science:

• Dynamical Systems Theory
• Nonlinear Dynamics
• Computational Mathematics

Background:

• Analytical criteria for ruling out limit cycles in 2D dynamical systems are scarce.
• Existing methods like the gradient and Dulac criteria have limitations.
• Limit cycle detection is computationally intensive.

Purpose of the Study:

• To present a new, generally applicable analytical criterion for limit cycle detection.
• To improve the efficiency of identifying limit cycles in dynamical systems.
• To provide a method for enhancing computational approaches to limit cycle analysis.

Main Methods:

• Reformulation of the time differential using the Moore-Penrose pseudoinverse of the velocity field.
• Derivation of a new analytical criterion based on this reformulation.
• Application of the criterion to a planar system with multiple limit cycles.

Main Results:

• A novel, generally applicable criterion for limit cycle detection is presented.
• The criterion provides one-dimensional manifolds that intersect limit cycles.
• Demonstrated a significant increase in efficiency for sampling initial conditions to identify limit cycles.

Conclusions:

• The new criterion offers a powerful tool for analyzing limit cycles in dynamical systems.
• It enhances computational efficiency, especially for systems with small attraction regions.
• The method provides a theoretical advancement for limit cycle detection and analysis.