Topological Defects in Spiral Wave Chimera States

Abstract

Chimera states, where coherent and incoherent domains coexist, represent a key self-organization phenomenon in the study of synchronization in complex systems. We introduce a topological analysis method based on winding numbers to characterize the dynamics of spiral wave chimeras in a two-dimensional phase oscillator network. Our investigation reveals distinct scaling laws governing the system's evolution across the phase lag $α$. Perturbation analysis in the limit $α\to 0$ demonstrates that the incoherent core radius scales linearly with $α$. In contrast, within the stable chimera regime, the average total positive winding number $\overline{n_+}$ follows a clear exponential growth law $\overline{n_+} = ae^{bα}$. This divergence suggests a physical crossover from geometric core expansion to active topological excitation. Furthermore, we identify a statistical transition in the defect distribution from binomial-like to Poisson-like behavior at a critical threshold $α^*$, which we interpret as a shift from a constrained regime to an unconstrained state analogous to a BKT binding-unbinding transition. These results indicate that topological defects are not randomly distributed but possess quantifiable statistical order, proposing $\overline{n_+}$ as a robust physical quantity to analyze the topological structure and complexity of chimera states.

Figures (5)

• Figure 1: Order parameter profiles. Snapshots of (a) the phase $\Phi(\bm{r})$ and (b) amplitude $|z|(\bm{r})$ from the numerical simulation results for $N = 151$, $\alpha = 10^\circ$, and $t = 100$. (c-d) Comparison between the analytical expressions Eqs. (\ref{['eq:A']}),(\ref{['eq:f']}) and the diagonal cross-section of the numerical results [(a),(b)], where $r$ is the distance from the vortex.
• Figure 2: Spatial patterns of the phase $\phi(x,y)$ (first row), order parameter $|z|(x,y)$ (second row), and the diagonal cross-section the order parameter profile sim$|z|(x_{\rm diag}, x_{\rm diag})$ (third row) at $t = 110$ for different values of phase lag. (a) $0 < \alpha = 2^{\circ} < \alpha_0$: A pair of spiral waves without a random core. (b) $\alpha_0 < \alpha = 30^{\circ} < \alpha_1$: A pair of spiral waves with an incoherent core. (c) $\alpha_1 < \alpha = 45^{\circ} < \alpha_2$: A pair of spiral waves with a stable incoherent core. (d) $\alpha_2 < \alpha = 65^{\circ} < \pi/2$: Irregular patterns and turbulence.
• Figure 3: (a) The vector field $(\cos \phi, \sin \phi)$ and topological defects in a chimera state with $\alpha = 45^\circ$, observed in the region $x,y\in [45,99]$. (b)Time evolution of the total positive winding number $n_+$ under different values of phase lag. The y-axis is on a log scale. The dashed line shows the running time-average, $\langle n_+ \rangle (\tau) = \frac{1}{\tau} \sum_{t=1}^{\tau} n_+(t)$.
• Figure 4: Statistics of the total positive winding number $n_+$ under different phase lags. (a) $\alpha = 45^\circ$. (b) $\alpha = 55^\circ$. The left panels show the histogram of $n_+$ compared to a truncated Poisson distribution [Eq. \ref{['eq.Poi']}] and a binomial distribution [Eq. \ref{['eq.Binom']}]. The right panels show the corresponding time evolution of $n_+$ (blue line) and its running time-average (red dashed line), $\langle n_+ \rangle (\tau) = \frac{1}{\tau} \sum_{t=100}^{\tau} n_+(t)$ (Note that the $\alpha = 55^\circ$ case is not compared with the inomial distribution, because $\sigma^2 > \overline{n_+}$).
• Figure 5: (a) Dependence of the average total positive winding number $\overline{n_+}$ (left axis) and its standard deviation $\sigma$ (right axis) on the phase lag $\alpha$. The red markers show $\ln(\overline{n_+})$ for different interaction radii. The blue markers show $\ln(\sigma)$ for $R=6$. The lines represent linear fits to the corresponding data points. (b) Comparison of the measured entropy with the theoretical binomial and Poisson entropy.