Hybrid dynamics in delay-coupled swarms with mothership networks

Swarming behavior continues to be a subject of immense interest because of its centrality in many naturally occurring systems in physics and biology, as well as its importance in applications such as robotics. Here we examine the effects on swarm pattern formation from delayed communication and topological heterogeneity, and in particular, the inclusion of a relatively small number of highly connected nodes, or “motherships,” in a swarm's communication network. We find generalized forms of basic patterns for networks with general degree distributions, and a variety of dynamic behaviors including parameter regions with multistability and hybrid motions in bimodal networks. The latter is an interesting example of how heterogeneous networks can have dynamics that is a mix of different states in homogeneous networks, where high- and low-degree nodes have distinct behavior simultaneously.

Figure 1

Patterns for swarms with time-delayed interactions and an underlying network with a small fraction of highly connected nodes, or “motherships.” $CM$ and annealed network patterns are shown on the top and bottom rows, respectively, for bimodal networks with $p_0=0.95, k_0=5$ (blue), $K=50$ (red), and $N=500$. Arrows indicate the direction of motion. (a) Translating (b) ring (c) hybrid (d) rotating patterns.

Figure 2

Phase diagram for bimodal networks found by adiabatically changing $J$ and $\tau$ for each pattern until it loses stability [31]. (a) Heterogeneous network: (i) translating, ring, and hybrid states, (ii) ring state, (iii) ring and hybrid states, (iv) hybrid states, and (v) rotating states [32], where $p_0=0.9, K=60, k_0=3$, and $N=300$. The red circle marks a $\mathit{degenerate}-\mathit{Hopf}$ bifurcation. (b) Less heterogeneous network: (${\mathrm{i}}^{\prime }$) translating, ring, and hybrid states, (${\mathrm{ii}}^{\prime }$) ring and hybrid states, (${\mathrm{iii}}^{\prime }$) ring, hybrid, and rotating states, (${\mathrm{iv}}^{\prime }$) hybrid states, (${\mathrm{v}}^{\prime }$) hybrid and rotating states, and (${\mathrm{vi}}^{\prime }$) rotating states, where $p_0=0.95, K=25, k_0=5$, and $N=1000$.

Figure 3

Ring state rotation radii (a) and frequencies (b) for simulated $CM$ (green) and annealed (blue) bimodal networks compared to predictions [red, Eq. (5)]: $\tau =0.02, p_0=0.9, k_0=3, K=60$, and $N=300$. Parameters correspond to regions beneath the red line in Fig. 2.

Figure 4

Translating state centroid speeds (a) and displacements (b) for simulated $CM$ (green) and annealed (blue) bimodal networks compared to predictions [red, Eq. (7)] from region $\left(i\right)$ in Fig. 2: $J=0.333, p_0=0.9, k_0=3, K=60$, and $N=300$.

Figure 5

Rotating state phase differences for simulated bimodal $CM$ (green) and annealed network (blue) swarms compared to predictions [red, Eqs. (9) and (10)]: $J=0.8, p_0=0.95, k_0=5, K=25$, and $N=1000$. Parameters correspond to regions above the magenta line in Fig. 2.

Figure 6

Fourier $\left(\omega \right)$ power spectra of low-degree nodes for simulated $CM$ (green) and annealed (blue) bimodal networks. (a) Translating state spectrum for $\tau =0.05$ compared to predictions [red, Eq. (7)]. (b) Hybrid state spectrum for $\tau =0.65$ compared to predictions [red, Eq. (15)]. Large peaks in (a) and (b) correspond to the ring frequency. $J=0.333, p_0=0.9, k_0=3, K=60$, and $N=300$.

Figure 7

Mothership rotation radius (a) and frequency (b) in the hybrid state for simulated $CM$ (green) and annealed (blue) bimodal networks compared to predictions [red, Eqs. (13) and (14)]: $J=0.333, p_0=0.9, k_0=3, K=60$, and $N=300$. Parameters correspond to regions (iii) and (iv) in Fig. 2.

Figure 8

Spatial patterns for a power-law network in the ring (a) and rotating (b) states. (a) Ring state where clockwise and counterclockwise rotation are shown with red circles and squares, respectively, and with colors darkening with increasing $k$ [legend in (b)]. Predictions from Eq. (5) are shown in blue for degree classes in multiples of 5, i.e., $k=10,15,...,100$. Simulation parameters are $N=1000, J=0.15$, and $\tau =1.5$. (b) Rotating state with the same legend as (a). Prediction from Eqs. (9) and (10) are shown in blue for all $k$. Simulation parameters are $N=1000, J=0.15$, and $\tau =4.0$.