Symmetric patterns in linear arrays of coupled cells

Area of Science:

• Mathematical modeling
• Chemical kinetics
• Pattern formation

Background:

• Coupled cell systems are fundamental in various scientific disciplines.
• Understanding pattern formation is key to explaining complex phenomena.
• Turing structures in continuous media are well-studied, but discrete analogs are less explored.

Purpose of the Study:

• To develop a method for finding patterned solutions in linear arrays of coupled cells.
• To establish a discrete analog of Turing structures.
• To demonstrate the existence of patterned solutions in coupled Brusselator systems.

Main Methods:

• Embedding a linear array into a circular array with double the number of cells.
• Utilizing the symmetry properties of the circular array and bifurcation theory.
• Applying abstract results to a specific system of coupled Brusselators.

Main Results:

• A novel method for identifying patterned solutions in linear coupled cell systems was established.
• The patterned solutions were shown to be discrete analogs of Turing structures.
• The existence of patterned solutions in coupled Brusselator systems was proven.

Conclusions:

• The embedding technique provides a robust framework for analyzing pattern formation in discrete systems.
• The identified patterned solutions are analogous to Turing structures, bridging continuous and discrete models.
• Patterned solutions in Brusselator systems can be found numerically, indicating potential asymptotic stability.