Symmetry Analysis of Coupled Scalar Systems under Time Delay

TL;DR

This work addresses how symmetry constrains bifurcations in networks of $n$ identical scalar units with time delay. It combines linearization around the zero solution with spectral analysis of the coupling matrix $C$ and applies equivariant degree theory (via the EDML package) to classify stationary and Hopf bifurcations by symmetry, including submaximal isotropy through secondary dominating orbit types. The authors derive that only extreme eigenvalues $ \xi_{ ext{min}}, \xi_{ ext{max}}$ govern the initial destabilization and provide a complete symmetry-based framework for dihedral ring networks, with explicit invariants $ \omega_0$ and $ \nomega_1$ and detailed results for the $D_{12}$ case, complemented by simulations of near-neighbor coupling. The methodology yields a scalable, symmetry-aware approach to predicting spatio-temporal patterns in time-delayed networks and remains valid under symmetry-preserving perturbations, with implications for chimera-like phenomena and other structured dynamical states.

Abstract

We study systems of coupled units in a general network configuration with a coupling delay. We show that the destabilizing bifurcations from an equilibrium are governed by the extreme eigenvalues of the coupling matrix of the network. Based on the equivariant degree method and its computational packages, we perform a symmetry classification of destabilizing bifurcations in bidirectional rings of coupled units. Both stationary and oscillatory bifurcations are discussed. We also introduce the concept of secondary dominating orbit types to capture bifurcating solutions of submaximal nature.

Figures (7)

• Figure 1: Bifurcation diagram of the characteristic equation (\ref{['eq:char']}). The curves indicate the parameter values for which the characteristic equation has a root on the imaginary axis. The curves separate the $\alpha$--$\beta$ parameter plane into regions in which the number of characteristic roots with positive real part is a constant, the value of which is indicated in the figure. Hence "0" indicates the region where the origin is stable, which is bounded from above by the straight line L1 and from below by the curve C2.
• Figure 2: The largest eigenvalue of the coupling matrix $C$ for a circular arrangement of 12 cells where each cell is connected to four others, with coupling strength $d_1$ to its two immediate neighbors on each side and with coupling strength $d_2$ to the second-nearest neighbors.
• Figure 3: The smallest eigenvalue of the coupling matrix $C$ for a circular ring of size 12, in terms of the coupling strengths. (See the caption of Figure \ref{['F:max_eig_G12']} for explanation.)
• Figure 4: System \ref{['neural']} with excitatory coupling ($\kappa=1$) approaching a uniform steady-state solution from random initial conditions (left). The negative of the final state is also a stable equilibrium which can be observed for a different choice of initial conditions. Adding self-coupling yields the diffusively coupled system \ref{['eq-b']}, which exhibits a stable oscillatory pattern of two clusters (right): Cells $\{2, 3, 6, 7, 10, 11\}$ form a synchronized cluster (blue curve) that oscillate in anti-phase with cells $\{1, 4, 5, 8, 9, 12 \}$ (red curve).
• Figure 5: System \ref{['neural']} with inhibitory coupling ($\kappa=-1.2$). When $\tau=1$, the network approaches a two-cluster steady-state solution from random initial conditions (left). The clusters are the same as in the oscillatory pattern of Figure \ref{['F:excitatory']}. Increasing the delay to $\tau=3$ yields spatially uniform synchronized oscillations shown on the right.

Theorems & Definitions (23)

• Example 2.1
• Example 2.2
• Example 2.3
• Example 2.4
• Definition 2.5
• Remark 2.6
• Example 2.7
• Lemma 4.1
• proof
• Remark 4.2