Two-dimensional global manifolds of vector fields.

Area of Science:

• Dynamical Systems and Chaos Theory
• Computational Mathematics
• Geometric Dynamics

Background:

• Understanding the geometric structures of dynamical systems is crucial for analyzing their long-term behavior.
• Manifolds, specifically stable and unstable manifolds, provide key insights into the flow of trajectories around equilibrium points and attractors.
• Efficient computational methods are needed to explore these complex structures in three-dimensional systems.

Purpose of the Study:

• To develop and describe an efficient algorithm for computing two-dimensional stable and unstable manifolds of three-dimensional vector fields.
• To enable geometric study of manifolds and extract important features of dynamical behavior.
• To illustrate the algorithm's application with relevant examples.

Main Methods:

• An iterative algorithm that grows larger and larger pieces of a manifold.
• The process continues until a sufficiently long segment of the manifold is obtained.
• Application of the algorithm to specific dynamical systems.

Main Results:

• Successful computation of a two-dimensional stable manifold for a three-dimensional vector field.
• Demonstration of the stable manifold spiraling into the Lorenz attractor.
• Computation of an unstable manifold in a zeta(3)-model converging to an attracting limit cycle.

Conclusions:

• The developed algorithm is efficient for computing stable and unstable manifolds in 3D vector fields.
• The method facilitates geometric analysis and the identification of critical dynamical features.
• The computed examples showcase the algorithm's utility in understanding complex system dynamics.