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Persistence of chimera states and the challenge for synchronization in real-world networks

Riccardo Muolo, Joseph D. O'Brien, Timoteo Carletti, Malbor Asllani

TL;DR

This work links non-normal network structure to persistent coherent-incoherent patterns by presenting a symmetry-breaking, Laplacian-eigenmode framework that, near bifurcations, yields robust amplitude chimera states. It demonstrates amplitude chimeras and oscillon patterns in empirical networks and shows how leader/source nodes drive phase chimeras, challenging the feasibility of global synchronization in real systems. The results highlight the role of spectral properties and non-normality in shaping real-world synchronization phenomena and call for more sophisticated models beyond traditional MSF analysis.

Abstract

The emergence of order in nature manifests in different phenomena, with synchronization being one of the most representative examples. Understanding the role played by the interactions between the constituting parts of a complex system in synchronization has become a pivotal research question bridging network science and dynamical systems. Particular attention has been paid to the emergence of chimera states, where subsets of synchronized oscillations coexist with asynchronous ones. Such coexistence of coherence and incoherence is a perfect example where order and disorder can persist in a long-lasting regime. Although considerable progress has been made in recent years to understand such coherent and (coexisting) incoherent states, how they manifest in real-world networks remains to be addressed. Based on a symmetry-breaking mechanism, in this paper, we shed light on the role that non-normality, a ubiquitous structural property of real networks, has in the emergence of several diverse dynamical phenomena, e.g., amplitude chimeras or oscillon patterns. Specifically, we demonstrate that the prevalence of source or leader nodes in networks leads to the manifestation of phase chimera states. Throughout the paper, we emphasize that non-normality poses ongoing challenges to global synchronization and is instrumental in the emergence of chimera states.

Persistence of chimera states and the challenge for synchronization in real-world networks

TL;DR

This work links non-normal network structure to persistent coherent-incoherent patterns by presenting a symmetry-breaking, Laplacian-eigenmode framework that, near bifurcations, yields robust amplitude chimera states. It demonstrates amplitude chimeras and oscillon patterns in empirical networks and shows how leader/source nodes drive phase chimeras, challenging the feasibility of global synchronization in real systems. The results highlight the role of spectral properties and non-normality in shaping real-world synchronization phenomena and call for more sophisticated models beyond traditional MSF analysis.

Abstract

The emergence of order in nature manifests in different phenomena, with synchronization being one of the most representative examples. Understanding the role played by the interactions between the constituting parts of a complex system in synchronization has become a pivotal research question bridging network science and dynamical systems. Particular attention has been paid to the emergence of chimera states, where subsets of synchronized oscillations coexist with asynchronous ones. Such coexistence of coherence and incoherence is a perfect example where order and disorder can persist in a long-lasting regime. Although considerable progress has been made in recent years to understand such coherent and (coexisting) incoherent states, how they manifest in real-world networks remains to be addressed. Based on a symmetry-breaking mechanism, in this paper, we shed light on the role that non-normality, a ubiquitous structural property of real networks, has in the emergence of several diverse dynamical phenomena, e.g., amplitude chimeras or oscillon patterns. Specifically, we demonstrate that the prevalence of source or leader nodes in networks leads to the manifestation of phase chimera states. Throughout the paper, we emphasize that non-normality poses ongoing challenges to global synchronization and is instrumental in the emergence of chimera states.
Paper Structure (8 sections, 1 theorem, 8 equations, 6 figures)

This paper contains 8 sections, 1 theorem, 8 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Symmetry-breaking method
  3. Emergence of amplitude chimera states
  4. Hybrid chimera patterns: the oscillon states
  5. Are phase chimeras ubiquitous in the real world or is global synchronization elusive?
  6. Discussion and conclusions
  7. Effects of non-normality in the chimera pattern formation mechanism
  8. Supplementary Data

Key Result

Theorem 1

The right eigenvectors of a triangular matrix $\textbf{A}_{n\times n}$ form a triangular matrix $\textbf{P}_{n\times n}$ when considered as columns of $\textbf{P}$ and ordered according to the eigenvalues of $\textbf{A}$.

Figures (6)

  • Figure 1: Schematic representation of chimera pattern formation and leader nodes.a) The process follows a symmetry-breaking mechanism where the perturbation leave the unstable synchronized manifold (red empty circle) following the direction of the critical eigenvector (red arrow solid and shaded) and reaches the stable chimera state (blue filled circle) through a (quasi)linear orbit (black dashed line). b) Once the non-normality reaches a given threshold the terminal Strongly Connected Components (SCC) (orange curve) disappear to make space to the leader nodes (green curve).
  • Figure 2: Emergence of amplitude chimera state.$\textbf{a)}$ The almost triangular adjacency matrix of the macaques competition network ref41 ordered hierarchically. $\textbf{b)}$ The matrix where the columns are the Laplacian eigenvectors of the same network where the magenta rectangle shows the critical eigenvector. $\textbf{c)}$ The Master Stability Function (MSF) zoomed around the critical eigenvalue (the whole MSF is shown in the inset). $\textbf{d)}$ The comparison of the critical eigenvector (magenta stars) vs. the normalized amplitudes of the initial evolution (blue circles) and the final pattern (green diamonds). With the aid of the dashed lines, it is possible to notice that the shape of the final pattern is flipped (due to the choice of snapshot time) compared to the critical eigenvector, but otherwise is similar to it. $\textbf{e)}$ The time series for each oscillator (zoomed, lower part and the complete evolution, upper part). In the inset of the upper part the standard deviation for the first $4$ oscillators (red curve) and the last $12$ ones (blue curve) is shown. $\textbf{f)}$ The colormap representation of the oscillators dynamics evolution. The parameters for the Brusselator are $b=2.5$, $c=1$, $D_u=0.0168$, and $D_v=0.2112$ and the colorbars quantify either the magnitudes of the matrices entries or the oscillators amplitudes.
  • Figure 3: Emergence of oscillon patterns.$\textbf{a)}$ The Master Stability Function (dispersion relation) real (main, red stars) and imaginary (inset, blue circles) parts for the macaques competition network ref41. The critical eigenvalues correspond to the third and the fifth igenvectors shown in Fig. 1 $b)$. $\textbf{b)}$ Comparison between the critical eigenvectors 3 (magenta stars) and 5 (red crosses), respectively, and the normalized amplitudes of the initial evolution (blue circles) and the final pattern (green diamonds). $\textbf{c)}$ The time series for each nodes where it can be noticed that for the nodes from $7$ to $16$, (almost) no oscillations are present. $\textbf{d)}$ The colormap representation of the evolution of the oscillon patterns. The colorbar quantifies the oscillators amplitudes. The parameters for the Zhabotinsky model are $c_1=c_3=28.5$, $c_2=c_4=15.5$, $c_5=c_6=1$, $c_7=25.65$, $c_8=3.1$, $D_x=D_y=0$ and $D_z=0.1$.
  • Figure 4: Emergence of phase chimera states.$\textbf{a)}$ The graphic representation of the ants dominance network ref44 ordered hierarchically. $\textbf{b)}$ The Master Stability Function shows no (local) instability, however, there are $9$ zero overlaying eigenvalues in the MSF curve. $\textbf{c)}$ The time series for each oscillator zoomed (lower part) and the complete evolution (upper part). $\textbf{d)}$ The colormap representation of the oscillators dynamics evolution with the synchronized clusters emphasized with the magenta rectangle. The colorbar quantifies the oscillators amplitudes. Notice that due to the random initial perturbation of the system, some of the sources belong to the synchronized cluster just by chance. The parameters for the Brusselator model are $b=2.5$, $c=1$, $D_u=0.0175$, and $D_v=0.075$.
  • Figure 5: Non-normality driven chimera patterns.$\textbf{a)}$ The Master Stability Function (red stars) for the dominance among macaques network ref31. The most expressed eigenvalue correspond to the eigth eigenvector which is clearly farer from the threshold of instability than other modes. $\textbf{b)}$ Comparison between the most expressed eigenvector (magenta stars) and the normalized amplitudes of the initial evolution (blue circles) and the final pattern (green diamonds). Notice that although quantitatively very similar the comparison is less neat than previously. $\textbf{c)}$ The time series for each nodes where due to many nodes involved it is almost impossible to establish the shape of the pattern obtained. $\textbf{d)}$ The colormap representation of the evolution of the amplitude chimera state where the colorbar quantifies the magnitude of the patterns. The parameters for the Brusselator model are $b=2.5$, $c=1$, $D_u=0.007$, and $D_v=0.083$.
  • ...and 1 more figures

Theorems & Definitions (1)

  • Theorem 1