Limit cycles in planar discontinuous piecewise linear Hamiltonian systems with three equal sectors

Area of Science:

• Differential Equations
• Dynamical Systems Theory
• Mathematical Analysis

Background:

• Discontinuous piecewise differential systems are increasingly important due to numerous applications.
• The presence and number of limit cycles are critical in analyzing these systems.
• Previous research has explored various configurations of such systems.

Purpose of the Study:

• To analyze a specific class of planar discontinuous piecewise differential systems.
• To determine the maximum number of crossing limit cycles possible in these systems.
• To address the extended 16th Hilbert problem for this system class.

Main Methods:

• The study focuses on systems divided into three sectors of angle 2π/3.
• Each sector is governed by a unique linear Hamiltonian system.
• The discontinuity set comprises three rays forming sector boundaries.

Main Results:

• It is proven that these differential systems can exhibit a maximum of three crossing limit cycles.
• An explicit example is constructed, demonstrating a system that achieves this upper limit.
• The findings contribute to solving the extended 16th Hilbert problem for this specific type of system.

Conclusions:

• The maximum number of crossing limit cycles for the studied systems is definitively established as three.
• The construction of an example reaching this bound validates the theoretical findings.
• This work provides a significant contribution to the understanding of limit cycle behavior in discontinuous piecewise differential systems.