Susceptibility of transient chimera states

Chaotic dynamics of a dynamical system is not necessarily persistent. If there is (without any active intervention from outside) a transition towards a (possibly nonchaotic) attractor, this phenomenon is called transient chaos, which can be observed in a variety of systems, e.g., in chemical reactions, population dynamics, neuronal activity, or cardiac dynamics. Also, chimera states, which show coherent and incoherent dynamics in spatially distinct regions of the system, are often chaotic transients. In many practical cases, the control of the chaotic dynamics (either the termination or the preservation of the chaotic dynamics) is desired. Although the self-termination typically occurs quite abruptly and can so far in general not be properly predicted, previous studies showed that in many systems a ‘terminal transient phase” (TTP) prior to the self-termination existed, where the system was less susceptible against small but finite perturbations in different directions in state space. In this study, we show that, in the specific case of chimera states, these susceptible directions can be related to the structure of the chimera, which we divide into the coherent part, the incoherent part and the boundary in between. That means, in practice, if self-termination is close we can identify the direction of perturbation which is likely to maintain the chaotic dynamics (the chimera state). This finding improves the general understanding of the state space structure during the TTP, and could contribute also to practical applications like future control strategies of epileptic seizures which have been recently related to the collapse of chimera states.

Figure 1

Exemplary snapshots of chimera states (a) in a one-dimensional ring of oscillators [Eq. (1)] with $N=40$, and the (b) two-dimensional system [Eq. (3)] for system size $A=3\times 3=9\mathrm{arb}.\mathrm{units}$ (discretized on a square grid of $30\times 30$ oscillators). In both cases spatially separated regions of coherent and incoherent dynamics of the phase oscillators exist simultaneously.

Figure 2

The logarithms of the average transient lifetime ${\langle T\rangle }_{\mathrm{IC}}$ for varying system sizes in (a) the one-dimensional system and (b) the two-dimensional system, respectively. In both cases, the average lifetimes for single domain sizes were determined via a nonlinear least-squared fit, shown in Fig. 6. The resulting average lifetimes are shown here, where the error bars are too small to be visible. They are fitted regarding Eq. (6) (with a nonlinear least-squared regression). The fit parameters are given in Table 1.

Figure 3

The averaged phase velocity during the final episode of an exemplary chimera state (1D) is shown in subplot (a) [self-termination occurs at $t=10470\mathrm{arb}.\mathrm{units}$, vertical (gray) line], with the associated order parameter $Z_{1\mathrm{D}}\left(t\right)$ (dashed black) and the local Lyapunov exponent ${\lambda }_1\left(t\right)$ (solid blue) in subplot (b). The length of the TTP is in (a) and (b) depicted as the (gray) shaded region. In (c) and (e) ${\mathcal{T}}_{\mathrm{sus}}\left(t\right)$ is plotted, where the average was taken over all perturbations (c) and over perturbations associated to coherent ($\times$), incoherent ($+$) and boundary regions ($•$), in (e), respectively. The error bars in (c) are standard deviations due to averaging over different initial conditions [Eq. (7), for the sake of clarity not shown in (e)]. Subplots (d) and (f) both depict the state of the chimera (black dots) at time ${\mathrm{t}}_1$ during the TTP (also marked in the other subplots), together with the distribution of $T_{\mathrm{pert}}\left({\mathrm{t}}_1\right)$ [solid red, subplot (d)] and the Lyapunov vector ${\gamma }_1$ which corresponds to the largest Lyapunov exponent. The boundary regions of the chimera state in (d) and (f) are marked as the (gray) shaded areas. Note, that while subplots (a), (b), (d), and (f) show results regarding a single exemplary trajectory, subplots (c) and (e) show quantities which are averaged over 50 initial conditions.

Figure 4

For an exemplary 2D transient chimera state three snapshots of the phase $\varphi$ at times ${\mathrm{t}}_1$, ${\mathrm{t}}_2$, and ${\mathrm{t}}_3$ before self-termination are shown in subplot (a). These points in time are also marked in subplots (b), where the order parameter $Z_{2\mathrm{D}}\left(t\right)$ (dashed black) and the (largest) local Lyapunov exponent (solid blue) is shown for the same initial condition shown in (a). The TTP is depicted in (b) as the gray shaded area. In (c) and (d) ${\mathcal{T}}_{\mathrm{sus}}$ is plotted, where the average was taken over all perturbations (c) and over perturbations associated to coherent ($\times$), and incoherent ($+$) regions, in (d), respectively. The error bars in (c) are standard deviations due to averaging over different initial conditions [Eq. (7), for the sake of clarity not shown in (d)]. This distinction between coherent and incoherent parts of the chimera state is discussed in Fig. 5. Note that, while subplots (a) and (b) show results regarding a single exemplary trajectory, subplots (c) and (d) show quantities which are averaged over 20 initial conditions.

Figure 5

The first row shows the local order parameter $Z_{2\mathrm{D}}^{\mathrm{loc}}\left(t\right)$ [Eq. (5)] of the system for the exemplary trajectory discussed in Fig. 4 for three points in time $t_1$, $t_2$, and $t_3$, corresponding to the snapshots of $\varphi$ shown in Fig. 4. The dashed orange line indicates the isolines of $Z_{2\mathrm{D}}^{\mathrm{loc}}\left(t\right)=0.7$ which is used to distinguish coherent $[Z_{2\mathrm{D}}^{\mathrm{loc}}\left(t\right)>0.7]$ from incoherent regions $[Z_{2\mathrm{D}}^{\mathrm{loc}}\left(t\right)<0.7]$. In the second and third rows, the patterns of $T_{\mathrm{pert}}$ [subplot (b)] and the Lyapunov vector ${\gamma }_1$ corresponding to the largest Lyapunov exponent [subplot (c)] are depicted, respectively, for the same points in time as in (a). In (a) and (b), the dashed lines separate coherent regions and incoherent regions [additionally marked by a gray shaded overlay in (b)] obtained from the isolines in (a) to compare the patterns of $T_{\mathrm{pert}}$ and the Lyapunov vector with the coherent or incoherent parts of the chimera state.

Figure 6

In solid gray, the number of initial conditions $N_{\mathrm{ch}}\left(t\right),$ which show chaotic dynamics at time $t$ is shown, which typically decreases exponentially. The average lifetime ${\langle T\rangle }_{\mathrm{IC}}$ was determined by fitting $N_{\mathrm{ch}}\left(t\right)$ (nonlinear least-squared). The fit is depicted as the dashed red line. An amount of time $T_1$ was discarded for the fit to separate out those initial conditions which converge to the resting state immediately [initial drop of $N_{\mathrm{ch}}\left(t\right)$]. Also, the fit was performed till $N_{\mathrm{ch}}\left(t\right)$ dropped below 50 initial conditions (marked by $T_2$), due to low statistics for $t>T_2$.