Time-delayed autosynchronous swarm control

In this paper a general Morse potential model of self-propelling particles is considered in the presence of a time-delayed term and a spring potential. It is shown that the emergent swarm behavior is dependent on the delay term and weights of the time-delayed function, which can be set to induce a stationary swarm, a rotating swarm with uniform translation, and a rotating swarm with a stationary center of mass. An analysis of the mean field equations shows that without a spring potential the motion of the center of mass is determined explicitly by a multivalued function. For a nonzero spring potential the swarm converges to a vortex formation about a stationary center of mass, except at discrete bifurcation points where the center of mass will periodically trace an ellipse. The analytical results defining the behavior of the center of mass are shown to correspond with the numerical swarm simulations.

Figure 1

The Morse potential as a function of agent separation.

Figure 2

The magnitude of the velocity history for 30 agents in the swarm given by numerical simulation with random initial conditions: (i) $b>a$, all velocities converge to zero (asymptotically stable), (ii) $a>b$, the velocities diverge rapidly (unstable), and (iii) $a=b$, the velocities are nonzero but bounded (marginally stable).

Figure 3

Real part of the rightmost eigenvalue is represented by the multicolored surface (grayscale surface) with varying $b$ and $\tau$ intersecting the plane defined by $\text{Re}\left[{\lambda }_0\right]=0$, represented by the single colored surface (black surface).

Figure 4

Evolution of the first four modes for different values of $b$ and $a=\tau =1$.

Figure 5

The potential surface as a function of agent separation and the distance $x$ from the origin.

Figure 6

The velocity magnitude history for 30 agents in the swarm: (i) $a=b=1$, all velocities are small and bounded, and (ii) $a=1.1,b=1$, velocities are bounded but their magnitudes become larger.

Figure 7

Characteristic roots of Eq. (21) for (i) $\tau =1$, where the rightmost eigenvalue has a negative real part, and (ii) $\tau =2\pi$, where the two rightmost eigenvalues lie on the imaginary axis.

Figure 8

The rightmost characteristic roots of the system (20) as a function of the delay term $\tau$.

Figure 9

Swarm of 30 agents forming a vortex independently of initial conditions: (i) random initial conditions, (ii) $t=10$, (iii) $t=20$, and (iv) $t=40$.

Figure 10

Trajectories of 30 agents with random initial conditions converging to a steady state for (i) $\tau =1$, where the center of mass stops and the swarm forms a vortex formation, and (ii) $\tau =2\pi$, where the center of mass oscillates about the origin and each agent winds around the origin.