A robust method for classification of chimera states

Abstract

Chimera states are one of the most intriguing phenomena in nonlinear dynamics, characterized by the coexistence of coherent and incoherent behavior in systems of coupled identical oscillators. Despite extensive studies and numerous observations in different settings, the development of reliable and systematic methods to classify chimera states and distinguish them from other dynamical patterns remains a challenging task. Existing approaches are often limited in scope and lack robustness. In this work, we propose a method based on Fourier analysis combined with statistical classification to characterize chimera behavior. The method is applied to a system of topological signals coupled via the Dirac operator, where it successfully captures the rich dynamical regimes exhibited by the model. We demonstrate that the proposed approach is robust with respect to variations in network topology and system parameters. Beyond the specific model considered, the framework provides a general and automated tool for distinguishing different dynamical regimes in complex systems.

Figures (20)

• Figure 1: Schematic illustration of orientation $1$, for the case $n=8$ and $P=3$. In the left panel, we report the original ring, where each one of the eight nodes is connected to $P = 3$ neighbors on either side. The middle panel refers to the case where $Q=1$ links incident to node $1$, have been reoriented (drawn in red). In the right panel, we show the case $Q=P=3$, where again we fixed node $1$ as reference (red links are the reoriented ones).
• Figure 2: Stability domain of the homogeneous solution $u_i=\sqrt{3}$, $v_j=-b/c$ for the FHN model defined on a ring of $n = 100$ nodes with a variable number of links controlled by the parameter $P$. The fraction $f = N'/N$ of initial conditions, starting within a ball of radius $R$ and remaining in the same ball after a sufficiently long time (i.e., staying "close" to the starting point), is shown as a function of $P$. A nonmonotonic behavior is observed: for small $P$, nearly $100\%$ of the orbits remain within a ball of radius $R \sim 0.6$. For larger $P$, the stability domain expands, with $f = 1$ up to $R \sim 1$ at $P = 20$. For even larger values of $P$, the stability domain slowly decreases.
• Figure 3: Typical dynamical behaviors. Numerical simulations of model \ref{['eq:eq1a']} and \ref{['eq:eq1b']} in orientation $1$, illustrating three distinct dynamical regimes: ordered behavior (panel (a), $Q=P=9$), a chimera state (panel (b), $Q=P=12$), and a disordered state (panel (c), $Q=P=21$). The remaining parameters are $n=50$, $a = 0.05$, $b = 0.5$, and $c = -0.01$. Simulations are performed using the Tsit5 solver over the time interval $[0,1000]$. For clarity, only the evolution of $u_i(t)$ over the final time window $t\in[950,1000]$ is shown.
• Figure 4: Fourier indicators. Average phases $\langle \theta\rangle$ (top row), average amplitudes $\langle a\rangle$ (middle row), and average frequencies $\langle \Omega\rangle$ (bottom row) obtained from the time series shown in Fig. \ref{['fig:9dyn']} for $P=9$, $P=12$, and $P=21$. Column (a) corresponds to ordered behavior, showing a smooth dependence of all three quantities on the node index. Column (b) displays a chimera state, characterized by the typical distribution of frequencies; here, the average amplitudes remain smooth, while the phases exhibit less regularity. Column (c) shows a disordered state, marked by small but irregular variations in both amplitude and frequency, and highly irregular phase behavior. The corresponding variation values $(V_\theta, V_a, V_\Omega)$ are $(0.0683, 0.082, 0.0022)$ for the ordered state in column (a), $(0.2301, 0.1110, 0.1087)$ for the chimera state in column (b), and $(0.4554, 0.04452, 0.0478)$ for the disordered state in column (c).
• Figure 5: Classification of dynamical behaviors by using the variations $V_\theta$, $V_a$ and $V_\Omega$, in the case of orientation $1$ and $Q=P$ links have been reoriented. In panel (a) we report the dendrogram obtained from the hierarchical clustering by using the values of $(V_\theta, V_a, V_\Omega)$. Branch lengths represent inter-cluster distances prior to merging, indicating two well-separated groups. The horizontal black line denotes the depth threshold used to determine the number of classes. In panel (b) we report the projection in the plane $(V_\theta,V_a)$ of the three obtained clusters by using the agglomerative clustering: the red cluster (class 1) corresponds to the ordered states, the blue cluster (class 2) correspond to chimeras, and the black cluster (class 3) corresponds to the disordered states. Panel (c) shows an alternative view of the classification as a function of $P=Q$ in the range $\{2,\dots, 24\}$; to each value of $P=Q$ we associate the dominant class, i.e., determined from the statistical modal cluster membership of $(V_\theta, V_a, V_\Omega)$ and we can obverse the presence of ordered states for $Q=P$ ranging from $2$ to $10$, chimera states for the $Q=P\in\{11,12,13\}$, and then disordered states for larger values of $Q=P$, but for the values $\{16,17,18\}$.