Critical Switching in Globally Attractive Chimeras

We report on a new type of chimera state that attracts almost all initial conditions and exhibits power-law switching behavior in networks of coupled oscillators. Such switching chimeras consist of two symmetric configurations, which we refer to as subchimeras, in which one cluster is synchronized and the other is incoherent. Despite each subchimera being linearly stable, switching chimeras are extremely sensitive to noise: Arbitrarily small noise triggers and sustains persistent switching between the two symmetric subchimeras. The average switching frequency scales as a power law with the noise intensity, which is in contrast with the exponential scaling observed in typical stochastic transitions. Rigorous numerical analysis reveals that the power-law switching behavior originates from intermingled basins of attraction associated with the two subchimeras, which, in turn, are induced by chaos and symmetry in the system. The theoretical results are supported by experiments on coupled optoelectronic oscillators, which demonstrate the generality and robustness of switching chimeras.

Popular Summary

Imagine you and your friend sitting on the two ends of a seesaw. When disturbed, the seesaw can switch between its two stable states. As you go up, your friend goes down, and vice versa. When the disturbances are random, the seesaw can be modeled as a noisy bistable system. Theories on stochastic transitions predict that as the level of noise decreases, the waiting time between switching events increases exponentially—you could be stuck on the ground or in the air for a very long time when noise is small. Here, we introduce a class of bistable systems for which the switching rate decreases only algebraically with noise intensity. This unexpected behavior means that the system is extremely sensitive to noise: Even a light breeze would cause the seesaw to swing up and down erratically.

We demonstrate this anomalous switching behavior for the interesting case of chimera states, which are dynamical patterns in oscillator networks exhibiting the coexistence of coherence and incoherence. In our systems, noise induces power-law switching between the coherent and incoherent portions of the chimeras. In particular, switching persists for arbitrarily small noise even though each configuration is stable in the absence of noise.

Similar power-law switching dynamics are likely to occur in other systems, potentially even biological ones. Exploitation of this effect could also lead to new ways to detect small noise in experiments.

Figure 1

Globally attractive chimera state whose coherent and incoherent clusters switch under extremely small noise. (a) Network system, formed by two rings of logistic maps mutually coupled through weaker links [Eq. (1)]. (b) Parameter space color coded according to the linear stability of the possible states, namely, whether both rings can synchronize (cyan), only one ring can synchronize (purple), or neither ring can synchronize (red). The four dots mark the parameters used in Fig. 2. (c) Direct simulation of the system for $\sigma =1.7$ and $r=3.05$ [orange dot in (b)] for noise intensity $\xi ={10}^{-10}$, illustrating the dynamics of a switching chimera. The top and bottom show the oscillator states in each of the two rings (color coded by oscillator, where single-color segments indicate synchronization), while the center shows the synchronization error [defined in Eq. (2)] in each ring color coded accordingly.

Figure 2

Average switching period $\overline{T}$ as a function of the noise intensity $\xi$ for $\sigma =1.7$ and various values of $r$ [dots in Fig. 1]. The switching periods are extracted from long time series of switching chimeras obtained by simulating Eq. (1) for different values of $\xi$. The numbers indicate the scaling exponents and are obtained through least-square fit (slopes of the solid lines).

Figure 3

Modeling transitions in switching chimeras. (a) Illustration of a random walk model in the log-error space, where a switching event is triggered when the walker reaches the error ceiling. The time series is colored differently after each switching event. (b) Distribution of the local Lyapunov exponents associated with Eq. (1) for $\sigma =1.7$ and $r=3.05$, which is used to refine the random walk model for the switching chimeras. (c) Power-law scalings predicted by the random walk model and its refined versions (dashed lines). The scaling obtained from direct simulations of Eq. (1) is also shown for comparison (solid orange line).

Figure 4

Dominant transition pathway between the two symmetric subchimeras, which consists of the intermediary stages T1–T4. Only T2 requires activation from noise, which can be arbitrarily small but not strictly zero; all other transitions follow directly from the deterministic dynamics of Eq. (1). In particular, T3 and T4 follow the stable and unstable manifolds of the invariant saddle, respectively.

Figure 5

Projections of invariant sets and transition paths. (a) Symmetric subchimeras when projected onto the mean state of each ring. Each subchimera is indicated by a different color. (b) Invariant saddle in Fig. 4 (and its symmetric counterpart) projected onto the mean state of each ring. (c) Transition path with an action of ${10}^{-28}$ projected onto coordinates $e_1^{\prime }$ and $e_2^{\prime }$. Under this projection, the invariant saddle is projected onto the lower left corner. The stable and unstable manifolds of the invariant saddle are marked by $\mathrm{s}$ and $\mathrm{u}$, respectively. The path starts at the blue subchimera in the upper left corner and ends at the orange subchimera in the lower right corner. (d) Same transition path as in (c) projected onto $e_1$ and $e_2$. The perturbation that initiates the transition is marked by an arrow.

Figure 6

Action profile for transition paths. (a) Probability $p$ of finding small-action transition paths by introducing a short-wavelength perturbation of magnitude $\delta$ in a single iteration. The simulations are performed for $\xi =0$, and the other parameters are the same as in Fig. 2. Paths with arbitrarily small action exist, but small-action paths become increasingly more difficult to find as the available action is decreased, resulting in power-law relationships between the probability $p$ and the perturbation size $\delta$. Notice that the scaling exponents here match those in Fig. 2. (b) Minimum action ($\frac{1}{2}{\delta }^2$) needed to induce a transition by applying ${\mathbf{\Delta }}_{\mathrm{sw}}(\delta )$ at a given time $t$, for $\xi =0$, $\sigma =1.7$, and $r=2.95$. This highly structured profile can be regarded as a visualization of the transition-barrier landscape for switching chimeras.

Figure 7

Two-dimensional section of the state space showing intermingled basins of the two subchimeras. The two basins, shown in blue and orange, are fat fractals [85] intermingled with each other everywhere. Orange points are attracted to the subchimera where the first ring is synchronized, and the blue ones converge to the subchimera with the second ring synchronized. There is a symmetry between the two basins with respect to reflections across the diagonal, which originates from the reflection symmetry of the network. The areas marked for magnification are intentionally oversized to facilitate visualization. The choice of state-space section and system parameters are specified in the text.

Figure 8

Experimental realization of globally attractive switching chimeras. (a) Schematic diagram of the optoelectronic system, where the dashed box depicts our implementation of the coupling scheme. (b) Parameter space color coded according to direct simulations of Eq. (9). The regions shown include switching chimeras (purple), nonswitching chimeras (green), chimera death [18] (yellow), and incoherence (red). (c) Experimentally measured average switching period $\overline{T}$ as a function of the noise intensity $\xi$ for $\beta =1.3$ and two values of $\sigma$ [dots in (b)]. The scaling exponents annotated on the figure are obtained through linear least-square fitting applied to the relationship between $\log (\overline{T})$ and $\log (\xi )$. The exponents obtained from experiments are in good agreement with those predicted from simulations (shown in parentheses).

Figure 9

Statistics and dynamics of a switching chimera in the experiments. (a) Distribution of switching periods for $\beta =1.3$ and $\sigma =1.05$ [green dot in Fig. 8(b)]. (b) Portion of the experimentally measured time series used to generate (a). These measurements are performed at the base-noise level of the system, which is estimated to be 0.0019.

Figure 10

Average switching period $\overline{T}$ as a function of noise intensity $\xi$ for various $r$. The system is the network of logistic maps in Fig. 1(a) for $\sigma =1.7$, and the noise is Gaussian (but with the short-wavelength component filtered out). The flatness of the fitting lines below $\xi ={10}^{-9}$ confirms that short-wavelength bifurcation is the dominant route for chimera switching.

Figure 11

Transversal section of the intermingled basins that directly connects the two symmetric subchimeras. This plot corresponds to a different state-space section of the system considered in Fig. 7.

Figure 12

Effect of oscillator heterogeneity on the switching behavior determined from direct simulations. The solid line indicates the power-law scaling for $\xi \ge \mathrm{\Delta }$, which is precisely the scaling observed in the absence of oscillator heterogeneity. For each of the four levels of heterogeneity $\mathrm{\Delta }$ considered, when $\xi <\mathrm{\Delta }$, the effect of heterogeneity becomes dominant and the average switching period $\overline{T}$ becomes independent of $\xi$.