Chimera states in population dynamics: Networks with fragmented and hierarchical connectivities

We study numerically the development of chimera states in networks of nonlocally coupled oscillators whose limit cycles emerge from a Hopf bifurcation. This dynamical system is inspired from population dynamics and consists of three interacting species in cyclic reactions. The complexity of the dynamics arises from the presence of a limit cycle and four fixed points. When the bifurcation parameter increases away from the Hopf bifurcation the trajectory approaches the heteroclinic invariant manifolds of the fixed points producing spikes, followed by long resting periods. We observe chimera states in this spiking regime as a coexistence of coherence (synchronization) and incoherence (desynchronization) in a one-dimensional ring with nonlocal coupling and demonstrate that their multiplicity depends on both the system and the coupling parameters. We also show that hierarchical (fractal) coupling topologies induce traveling multichimera states. The speed of motion of the coherent and incoherent parts along the ring is computed through the Fourier spectra of the corresponding dynamics.

Figure 1

(a) Limit cycle for various values of $p_1$ above the Hopf bifurcation. Fixed points $Q_1,Q_2$ (not shown), $Q_3$, and $Q_4$ (for $p_1=10)$ are marked with crosses. (b) Peak-to-peak amplitude of the $x$ variable and (c) period of the limit cycle as a function of $p_1$. Panels (d) and (e) show the time series of $x$ and $y$, respectively, corresponding to the limit cycles in (a). Other parameters: $p_2=0.5$ and $p_3=0.8$.

Figure 2

(a) Snapshot of the variable $x_k$ of Eqs. (3) and (b) corresponding mean phase velocity profile. (c) Limit cycle of the uncoupled system (black solid line) and snapshot in the $(x,y)$ plane (red dots). (d) Space-time plot of variable $x_k$. Parameters: $p_1=300,p_2=0.5,p_3=0.8,N=1000,R=350$, and $\sigma =0.015$.

Figure 3

Snapshots of the $x$ variable (red dots) and corresponding ${\omega }_k$ profiles (black dots) for various values of the coupling range: (a) $R=100$; (b) $R=120$; (c) $R=190$; (d) $R=230$; (e) $R=280$; (f) $R=410$. Other parameters are as in Fig. 2.

Figure 4

Snapshots of the $x$ variable (red dots) and corresponding ${\omega }_k$ profiles (black dots) at time $t=3000$ for a fixed coupling range $R=350$ and various values of $p_1$: (a) $p_1=20$; (b) $p_1=40$; (c) $p_1=90$; (d) $p_1=170$; (e) $p_1=210$; (f) $p_1=370$. Other parameters are as in Fig. 2.

Figure 5

Space-time plots of the $x$ variable (top) and order parameter (bottom) for fixed $R=350$ and increasing $p_1$: (a) $p_1=90$; (b) $p_1=170$; (c) $p_1=210$; and (d) $p_1=370$. Other parameters are as in Fig. 2.

Figure 6

Schematic representation of the hierarchical connectivity using the triadic Cantor set. (a) “Bottom-up” construction of the triadic Cantor set, (b) linear connectivity arrangement, and (c) ring connectivity arrangement for one reference node.

Figure 7

Space-time plots of two traveling chimeras for (a) $d_f=ln4/ln6=0.774$ and initiation string “001111” and (b) $d_f=ln5/ln6=0.898$ and initiation string “110111”. All other parameters are as in Fig. 2.

Figure 8

(a) Position of the oscillator displaying maximum $x$ value with time for $\sigma =0.015$ and (b) Fourier transforms for $\sigma =0.015$ (red spectrum) and $\sigma =0.020$ (black spectrum). The inset in (b) shows in detail the lowest part of the two spectra where the frequencies associated with the traveling motion around the ring are discerned. All other parameters are as in Fig. 7.

Figure 9

Traveling speed $v_{\text{tr}}$ of (in) coherent regions around the ring. All other parameters are as in Fig. 7.

Figure 10

Chimera states (left) and corresponding mean phase velocity profiles (right) for gap size variation. The gap sizes are indicated in the right panels, while $R_1=100$ and $R_2=300$. All simulations start from the same initial conditions. Other parameters are as in Fig. 2.

Figure 11

The four measures of coherence as a function of the gap size $G$: (a) $\langle {\omega }_{\text{coh}}\rangle$; (b) $\mathrm{\Delta }\omega ={\omega }_{\text{max}}-{\omega }_{\text{min}}$; (c) $N_{\text{incoh}}$; (d) $M_{\text{incoh}}$. Averages are taken over ten different sets of initial conditions. Parameter values are as in Fig. 10.

Figure 12

Chimera states and corresponding mean phase velocity with variations on the position of a gap of constant size $G_R=100$. (a) $R_1=50,R_2=350$; (b) $R_1=200,R_2=200$; (c) $R_1=300,R_2=100$; (d) $R_1=400,R_2=0$. All runs start from the same initial conditions. Other parameters are as in Fig. 2.

Figure 13

The four measures of coherence as a function of the inner connectivity radius $R_1$, while $R_1+R_2=400$: (a) $\langle {\omega }_{\text{coh}}\rangle$; (b) $\mathrm{\Delta }\omega ={\omega }_{\text{max}}-{\omega }_{\text{min}}$; (c) $N_{\text{incoh}}$; (d) $M_{\text{incoh}}$. The gap size is kept to $G_R=100$. Other parameter values are as in Fig. 10. Averages are calculated over ten different initial conditions.