Rotation sets for networks of circle maps

Area of Science:

• Dynamical Systems and Chaos Theory
• Network Science
• Topology

Background:

• Continuous maps of the torus are fundamental in dynamical systems.
• Coupled circle maps model complex systems with interacting components.
• Rotation sets characterize the long-term behavior of such maps.

Purpose of the Study:

• To investigate the impact of network architecture on rotation sets of coupled circle maps.
• To determine if network structure can constrain the dimensionality of rotation vectors.
• To analyze invertible maps on the torus and their associated network properties.

Main Methods:

• Analysis of continuous maps on the torus.
• Study of systems composed of coupled circle maps.
• Mathematical formulation relating network architecture to rotation set properties.

Main Results:

• Network architecture can significantly constrain the rotation set.
• For invertible torus maps, rotation vectors are forced into low-dimensional subspaces.
• All-to-all coupled systems of identical cells exhibit rotation sets confined to a line.

Conclusions:

• Network topology plays a crucial role in determining the dynamics of coupled systems.
• The dimensionality of the rotation set is not solely an intrinsic property of the map but is influenced by the coupling structure.
• Understanding these relationships offers insights into the collective behavior of complex systems.