Limit cycles and homoclinic networks in two-dimensional polynomial systems

Area of Science:

• Dynamical Systems and Chaos Theory
• Differential Equations
• Mathematical Analysis

Background:

• Planar polynomial dynamical systems are fundamental in analyzing complex behaviors.
• Understanding equilibrium properties, such as sources, sinks, and saddles, is crucial for characterizing system dynamics.
• Homoclinic networks play a significant role in the qualitative behavior of these systems.

Purpose of the Study:

• To investigate the properties of equilibriums in planar polynomial dynamical systems.
• To determine the number of sources, sinks, and saddles in self-univariate polynomial systems and their homoclinic networks.
• To analyze homoclinic networks of saddles and centers in crossing-univariate polynomial systems with limit cycles.

Main Methods:

• Development and application of theorems to determine the number of equilibrium points (sources, sinks, saddles, centers).
• Determination of first integral manifolds using polynomial functions.
• Illustrative examples of homoclinic networks to visualize geometric structures.

Main Results:

• A theorem is presented that determines the number of sources, sinks, and saddles in self-univariate polynomial systems, showing these networks lack centers.
• Another theorem establishes the number of saddles and centers in crossing-univariate systems with limit cycles, demonstrating the absence of sources and sinks.
• Maximum numbers of equilibrium points for systems of the same degree are discussed.

Conclusions:

• The study provides a novel approach to understanding and determining limit cycles within the context of the Hilbert 16th problem.
• The findings contribute to the classification and analysis of complex dynamics in planar polynomial systems.
• The presented theorems and illustrations offer a deeper insight into the structure of homoclinic networks.