Arbitrarily large heteroclinic networks in fixed low-dimensional state space.

Area of Science:

• Dynamical Systems and Mathematical Biology
• Topology and Geometric Methods in Dynamics

Background:

• Heteroclinic networks are crucial for understanding complex dynamics, but their realization in low-dimensional spaces is challenging.
• Existing methods often require high-dimensional spaces, limiting practical applications.

Purpose of the Study:

• To develop a robust and efficient method for realizing heteroclinic networks with a specific connection structure.
• To investigate the minimum required spatial dimension for such realizations, irrespective of network size.

Main Methods:

• Utilizing a construction in a one-dimensional space with connections in coordinate planes.
• Employing polynomial vector fields for network realization.
• Analyzing the stability properties of the constructed heteroclinic objects.

Main Results:

• Successfully realized heteroclinic networks in R6 for any number of nodes (n).
• Demonstrated that the required spatial dimension is bounded, regardless of the network's size (n → ∞).
• Established a novel phenomenon of dimension-bounded realization for these specific network structures.

Conclusions:

• The proposed method offers an efficient approach to constructing heteroclinic networks.
• The dimension-bounded realization is a significant advancement for theoretical and applied dynamical systems.
• Further stability analysis of these heteroclinic objects warrants additional investigation.