Transitions from chimeras to coherence: An analytical approach by means of the coherent stability function

Chimera states have been a vibrant subject of research in the recent past, but the analytical treatment of transitions from chimeras to coherent states remains a challenge. Here we analytically derive the necessary conditions for this transition by means of the coherent stability function approach, which is akin to the master stability function approach that is traditionally used to study the stability of synchronization in coupled oscillators. In chimera states, there is typically at least one group of oscillators that evolves in a drifting, random manner, while other groups of oscillators follow a smoother, more coherent profile. In the coherent state, there thus exists a smooth functional relationship between the oscillators. This lays the foundation for the coherent stability function approach, where we determine the stability of the coherent state. We subsequently test the analytical prediction numerically by calculating the strength of incoherence during the transition point. We use leech neurons, which exhibit a coexistence of chaotic and periodic tonic spiking depending on initial conditions, coupled via nonlocal electrical synapses, to demonstrate our approach. We systematically explore various dynamical states with the focus on the transitions between chimeras and coherence, fully confirming the validity of the coherent stability function. We also observe complete synchronization for higher values of the coupling strength, which we verify by the master stability function approach.

Figure 1

(a) Three-dimensional phase space of the bistable states: chaotic attractor in blue (black) and periodic orbit in red (gray). (b) Well-separated basins of attraction of the chaotic attractor [blue (black) region] and the periodic orbit [red (gray) region]. (c) The two-dimensional basin of attraction on the $h_{\text{Na}}=0.5$ plane. The dotted, dashed, and solid black lines, respectively, represent the V-shaped PIs, CIs, and MIs.

Figure 2

Snapshot (first column) of the state variable $V_i$, two-dimensional phase space (second column) in the $(m_{k{2}_i},V_i)$ plane, and local order parameter $L_i\left(t\right)$ (third column). V-shaped initial conditions are taken from the basin of attraction of the (a,b,c) periodic states PIs with $\epsilon =0.2$; (d,e,f) chaotic states CIs with $\epsilon =0.4$; and (g,h,i) mixed-type MIs with $\epsilon =0.3$. Other parameter: $P=20$.

Figure 3

(a) Snapshot, (b) 2D phase space, and (c) local order parameter of the coherent state for $\epsilon =10.0$ with V-shaped periodic initial conditions (PIs).

Figure 4

Variation of $V_{\text{SI}}$ (blue circle curve) and ${\mathrm{\Lambda }}_{\text{GS}}$ (red square curve) with respect to $\epsilon$ for (a) periodic initial conditions, (b) chaotic initial conditions, and (c) mixed initial conditions.

Figure 5

Two-parameter phase diagram in the $(P,\epsilon )$ plane with periodic initial conditions. Color bars represent the variation of (a) $V_{\text{SI}}$ and (b) ${\mathrm{\Lambda }}_{\text{GS}}$.

Figure 6

The variation of basin stability of incoherent [blue (black) bars], chimera [red (dark gray) bars], and coherent states [green (gray) bars] with respect to $\epsilon$ for $P=20$.

Figure 7

Periodic initial conditions near complete synchronization manifold (left panel): (a) Variation of $V_{\text{SI}}$ (blue circle line), ${\mathrm{\Lambda }}_{\text{GS}}$ (green square line), and ${\mathrm{\Lambda }}_{\text{CS}}$ (red diamond line) with respect to $\epsilon$. Snapshot of the state variable $V_i$ for (b) $\epsilon =15.0$ and (d) $\epsilon =27.0$, and corresponding spatiotemporal plots are in (c) and (e), respectively. PIs initial conditions (right panel): (f) Variation of $E_{\text{CS}}$ (left axis) and $B_{\text{CS}}$ (right axis) with respect to $\epsilon$. Snapshots at (g) $\epsilon =30.0$ and (i) $\epsilon =36.0$ and corresponding spatiotemporal behaviors are in (h) and (j), respectively. Inset figures show the corresponding trajectories in $(m_{\text{K}{2}_i},V_i)$ plane.

Figure 8

Two-parameter phase diagram in the $(P,\epsilon )$ plane with chaotic initial conditions. Color bars represent the variation of (a) $V_{\text{SI}}$ and (b) ${\mathrm{\Lambda }}_{\text{GS}}$.

Figure 9

Two-parameter phase diagram in the $(P,\epsilon )$ plane with mixed initial conditions. Color bars represent the variation of (a) $V_{\text{SI}}$ and (b) ${\mathrm{\Lambda }}_{\text{GS}}$.

Figure 10

Variation of the master stability function ${\mathrm{\Lambda }}_{\text{CS}}$ for the complete synchronization state in the $(P,\epsilon )$ parameter plane for periodic initial conditions.