[Paper Review] Attractor and synchronization for a complex network of reaction-diffusion systems of FitzHugh-Nagumo type

Benjamin Ambrosio, M. A. Aziz-Alaoui+1 more|arXiv (Cornell University)|Apr 29, 2015

Nonlinear Dynamics and Pattern Formation16 references3 citations

TL;DR

This paper establishes the existence of a global attractor and provides an $L^{∞}$-bound for a complex network of $n$ reaction-diffusion systems of FitzHugh-Nagumo type with partial diffusion. It proves a threshold coupling strength ensures identical synchronization across general network topologies and derives heuristic laws linking minimal coupling strength to network size and structure, validated numerically with spiral and homogeneous patterns.

ABSTRACT

We focus on the long time behavior of complex networks of reaction-diffusion (RD) systems. We prove the existence of the global attractor and a $L^{\infty}$-bound for a network of $n$ RD systems with $d$ variables each. This allows us to prove the identical synchronization for general class of networks and establish the existence of a coupling strength threshold value that ensures such a synchronization. Then, we apply these results to some particular networks with different structures (i.e. different topologies) and perform numerical simulations. We found out theoretical and numerical heuristic laws for the minimal strengh coupling needed for synchronization relatively to the number of nodes and the network topology, and discuss the link between spatial dimension and synchronization.

Motivation & Objective

• To analyze the long-time behavior of complex networks of reaction-diffusion systems of FitzHugh-Nagumo type.
• To establish the existence of a global attractor within which synchronization behavior is studied.
• To determine a theoretical threshold for coupling strength that guarantees identical synchronization across general network topologies.
• To derive heuristic laws for the minimal coupling strength required for synchronization based on network size and topology.
• To numerically investigate spatial effects on synchronization and pattern formation in such networks.

Proposed method

• Models the network as a graph where nodes are $d$-dimensional reaction-diffusion systems and edges represent coupling functions.
• Analyzes a system split into diffusive ($s$ variables) and non-diffusive ($d-s$ variables) subsystems, with diffusion and coupling only in the first $s$ components.
• Uses energy estimates and Gronwall-type inequalities to derive $L^2$ and $H^1$ bounds on solutions, ensuring precompactness.
• Applies a generalized Gronwall lemma (Theorem 4 and Corollary 3) to control growth of solution norms and establish uniform bounds.
• Employs numerical simulations with Neumann boundary conditions and various initial conditions to study synchronization and pattern formation.
• Fits numerical data to polynomial functions to derive heuristic laws for minimal coupling strength as a function of network size $n$.

Experimental results

Research questions

• RQ1 Does a global attractor exist for a network of $n$ coupled FitzHugh-Nagumo-type reaction-diffusion systems with partial diffusion?
• RQ2 What is the threshold value of coupling strength that ensures identical synchronization across arbitrary network topologies?
• RQ3 How does the minimal coupling strength required for synchronization scale with the number of nodes in the network?
• RQ4 What is the influence of network topology and spatial dimension on synchronization and pattern formation?
• RQ5 Can theoretical bounds on solution norms be used to prove synchronization and attractor existence in such systems?

Key findings

• The system admits a global attractor in $\mathscr{H} = (L^2(\Omega))^{nd}$, ensuring long-time convergence of all trajectories.
• An $L^\infty$-bound is established for the network, which is essential for proving synchronization and uniform boundedness.
• A theoretical threshold coupling strength exists that guarantees identical synchronization for general network topologies.
• Numerical simulations show that synchronization occurs when the coupling strength $g_3 \geq 0.02$ in a fully linearly connected 3-node network.
• A heuristic law $g_n = 0.0000167n^2 + 0.00062n + 0.02$ is derived, predicting the minimal coupling strength for synchronization in ring-connected networks as $n$ varies from 3 to 20.
• Asymptotic spatial patterns, including spiral waves, persist in synchronized networks regardless of initial conditions, indicating robust pattern formation.