Unstable modes and bistability in delay-coupled swarms

Abstract

It is known that introducing time delays into the communication network of mobile-agent swarms produces coherent rotational patterns, from both theory and experiments. Often such spatiotemporal rotations can be bistable with other swarming patterns, such as milling and flocking. Yet, most known bifurcation results related to delay-coupled swarms rely on inaccurate mean-field techniques. As a consequence, the utility of applying macroscopic theory as a guide for predicting and controlling swarms of mobile robots has been limited. To overcome this limitation, we perform an exact stability analysis of two primary swarming patterns in a general model with time-delayed interactions. By correctly identifying the relevant spatiotemporal modes, we are able to accurately predict unstable oscillations beyond the mean-field dynamics and bistability in large swarms-- laying the groundwork for comparisons to robotics experiments.

Full text

PHYSICAL REVIEW E 101, 042202 (2020)
Unstable modes and bistability in delay-coupled swarms
Jason Hindes,1Victoria Edwards,2Sayomi Kamimoto ,3Ioana Triandaf,1and Ira B. Schwartz 1
1U.S. Naval Research Laboratory, Code 6792, Plasma Physics Division, Washington, DC 20375, USA
2U.S. Naval Research Laboratory, Code 5514, Navy Center for Applied Research in Artificial Intelligence, Washington, DC 20375, USA
3Department of Mathematics, George Mason University, Fairfax, Virginia 22030, USA
(Received 7 February 2020; accepted 12 March 2020; published 6 April 2020)
It is known that introducing time delays into the communication network of mobile-agent swarms produces
coherent rotational patterns, from both theory and experiments. Often such spatiotemporal rotations can be
bistable with other swarming patterns, such as milling and flocking. Yet, most known bifurcation results related
to delay-coupled swarms rely on inaccurate mean-field techniques. As a consequence, the utility of applying
macroscopic theory as a guide for predicting and controlling swarms of mobile robots has been limited. To
overcome this limitation, we perform an exact stability analysis of two primary swarming patterns in a general
model with time-delayed interactions. By correctly identifying the relevant spatiotemporal modes, we are able to
accurately predict unstable oscillations beyond the mean-field dynamics and bistability in large swarms—laying
the groundwork for comparisons to robotics experiments.
DOI: 10.1103/PhysRevE.101.042202
I. INTRODUCTION
In nature, swarms consist of individual agents with limited
dynamics and simple rules, which interact, sense, collaborate,
and actuate to produce emergent spatiotemporal patterns. Ex-
amples include schools of fish [1–3], flocks of starlings [4,5]
and jackdaws [6], colonies of bees [7], ants [8], locusts [9],
and bacteria [10], as well as crowds of people [11]. Given the
many examples across a wide range of space and timescales,
significant progress has been made in understanding swarm-
ing by studying simple dynamical models with general prop-
erties [12–14].
Deriving inspiration from nature, embodied artificial
swarm systems have been created to mimic emergent pattern
formation—with the ultimate goal of designing robotic
swarms that can perform complex tasks autonomously
[15–18]. Recently robotic swarms have been used
experimentally for applications such as mapping [19],
leader following [20,21], and density control [22]. To
achieve swarming behavior, often robots are controlled
based on models where swarm properties can be predicted
exactly [23–27]. Such approaches rely on strict assumptions
to guarantee behavior. Any uncharacterized dynamics can
cause patterns to be lost or changed. This is particularly the
case for robotic swarms that move in uncertain environments
and must satisfy realistic communication constraints.
In particular, in both robotic and biological swarms, there
is often a delay between when the time information is per-
ceived and the reaction time of agents. Such delays have
been measured in swarms of bats [28], birds [29], fish [30],
and crowds of people [31]. Delays naturally occur in robotic
swarms communicating over wireless networks, due to low
bandwidth [32] and multihop communication [33]. In general,
time delays in swarms result in the multistability of rota-
tional patterns in space and the possibility of switching be-
tween patterns [34–42]. Though observed in simulations and
experiments, swarm bistability due to time delay has lacked
an accurate quantitative description, which we provide in this
work.
Consider a system of mobile agents, or swarmers, moving
under the influence of three forces: self-propulsion, friction,
and mutual attraction. In the absence of attraction, each
swarmer tends to a fixed speed which balances propulsion and
friction but has no preferred direction. The agents are assumed
to communicate through a network with time delays. Namely,
each agent is attracted to where its neighbors were at some
moment in the past. A simple model which captures the basic
physics is
m¨
rl=[α−β|˙
rl|2]˙
rl+a
N−1
j=l
[rj(t−τ)−rl]+ξl(t),
(1)
where mis the mass of each agent, αis a self-propulsion
constant, βis a friction constant, ais a coupling constant,
τis a characteristic time delay, Nis the number of agents,
rlis the position vector for the lth agent in two spatial
dimensions, and ξl(t) is a small noise source [34,43–46].
Equation (1) has been implemented in experiments with sev-
eral robotics platforms, including autonomous cars, boats, and
quadrotors [35,36]. Note: In this work we consider the simple
case of spring interaction forces and global communication
topology for illustration and ease of analysis; however, these
assumptions can be relaxed with predictable effects on the
dynamics [35,47–49].
II. SWARMING PATTERNS AND STABILITY
From generic initial conditions a swarm described by
Eq. (1) tends to one of two spatiotemporal patterns: a ring
(milling) state or a rotating state, depending on initial con-
ditions and parameters [46]. The two patterns can be seen
in Fig. 1(b). Note that the snapshots in time are drawn
from simulations of Eq. (1) with Gaussian white noise,
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JASON HINDES et al. PHYSICAL REVIEW E 101, 042202 (2020)
a
0.5 1 1.5 2 2.5 3 3.5 4
0
2
4
6
8
10 (a) (b)
FIG. 1. Stability diagram for delay-coupled swarms. (a) The blue
(upper) curve denotes a Hopf bifurcation for the ring state (b, upper).
The red (lower) curve denotes a combined line for saddle-node and
double-Hopf bifurcations for the rotating state (b, lower). Points
denote simulation-determined stability changes for N=600: a ring
state with all agents rotating in the same direction (blue circles),
a ring state with half the agents rotating in the opposite direction
(blue squares), and a rotating state (red diamonds). (b) Snapshots for
both states in the x-yplane (a=1,τ=2.6, N=100). Positions are
drawn with red circles and velocities with blue arrows. In all panels,
m=α=β=1.
ξ(c)
l(t)ξ(c)
j(t)=0.02 ×δ(t−t)δljδcc, where cand cde-
note the Cartesian components, xor y. The emergence and
stability of the ring and rotating patterns are often qualita-
tively described using mean-field approximations in which
the motions of agents relative to the swarm’s center of mass
are neglected [34,50]. Though useful, such descriptions do
not capture bistability and noise-induced switching, let alone
the more complex motions observed in experiments [35,36].
Additionally, higher-order approximation techniques predict
bistability qualitatively but suffer from quantitative inaccu-
racy and are difficult to analyze [51]. Hence, an analyzable
and accurate description of stability is needed, especially for
robotics experiments which use Eq. (1) (and its generaliza-
tions) as a basic autonomy controller. In support of such
experiments, we analyze the linear stability of the ring and
rotating states exactly for large Nin the noiseless limit and
compare our predictions to simulations.
A. Ring state
First, since the ring and rotating states are effectively
two types of phase-locked solutions with different phase
distributions and frequencies, it is useful to transform Eq. (1)
into polar coordinates where each can be naturally represented
as fixed-point solutions in appropriately chosen rotating refer-
ence frames. Introducing the coordinate transformations rl≡
rlcos(φl),rlsin(φl), substituting into Eq. (1) and neglect-
ing noise, we obtain
mrl¨
φl=α−βr2
l˙
φ2
l+˙r2
lrl˙
φl−2m˙rl˙
φl
+a
N−1
j=l
rj(t−τ)sin[φj(t−τ)−φl],(2)
m¨rl=α−βr2
l˙
φ2
l+˙r2
l˙rl+mrl˙
φ2
l
+a
N−1
j=l{rj(t−τ) cos[φj(t−τ)−φl]−rl}.(3)
For large Nwe can approximate the restricted sums in
Eqs. (2) and (3) over all but one of the agents, with sums
over all of the agents. In this case, ring-state formations are
solutions of Eqs. (2) and (3), where radii and frequencies are
constant [34], and phases are splayed uniformly:
rj(t)=mα
βa,φ
j(t)=2π(j−1)
N+a
mt.(4)
This is easy to check by direct substitution. In general, many
related ring states also exist, i.e., where some number of
agents have the opposite frequency, −√a/m, and are dis-
tributed uniformly around a concentric ring. In our stability
analysis below, we focus on the case where all agents rotate
in the same direction for three reasons: this case persists
when small repulsive forces are added (as in robotics experi-
ments [35,36]), the stability of any given ring pattern has only
a weak dependence on the number of nodes rotating in each
direction (as demonstrated with simulations), and analytical
tractability.
To determine the local stability of the ring state, we need
to understand how small perturbations to Eq. (4)grow(or
decay) in time. Our first step is to substitute a general pertur-
bation, rj(t)=√mα/βa+Bj(t) and φj(t)=2π(j−1)/N+
√a/mt+Aj(t), into Eqs. (2) and (3) and collect terms to
first order in Aj(t) and Bj(t) (assuming |Aj|,|Bj|1∀j).
The result is the following linear system of delay-differential
equations for N1with constant coefficients; the latter
property is a consequence of our transformation into the
proper coordinate system and is what allows for an analytical
treatment:
mα
βa[m¨
Al+2α˙
Al]+2a
m[m˙
Bl+αBl]=a
N
jBτ
jsin 2π(j−l)
N−a
mτ+(Aτ
j−Al)mα
βacos 2π(j−l)
N−a
mτ,(5)
m¨
Bl−2mα
β
˙
Al=a
N
jBτ
jcos 2π(j−l)
N−a
mτ−Aτ
j−Almα
βasin 2π(j−l)
N−a
mτ,(6)
where Aτ
j≡Aj(t−τ) and Bτ
j≡Bj(t−τ).
Given the periodicity implied by the equally spaced phase
variables in Eq. (4), it is natural to look for eigensolutions of
Eqs. (5) and (6) in terms of the discrete Fourier transforms of
Aj(t) and Bj(t). In fact, by inspection we can see that only
the first harmonic survives the summations on the right-hand
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UNSTABLE MODES AND BISTABILITY IN … PHYSICAL REVIEW E 101, 042202 (2020)
sides of Eqs. (5) and (6), because of the sine and cosine
terms, and hence we look for particular solutions: Aj(t)=
Aexp{λt−2πi(j−1)/N}and Bj(t)=Bexp{λt−2πi(j−
1)/N}. Substitution and a fair bit of algebra gives the follow-
ing transcendental equation for the stability exponent, λ,of
the ring state:
mλ2+2αλ −a
2e−τ[λ+i√a
m]
2ma
mλ−a
2ie−τ[λ+i√a
m]
+2a
m[mλ+α]−a
2ie−τ[λ+i√a
m]
mλ2−a
2e−τ[λ+i√a
m]=0.(7)
In general, the ring state will be linearly stable if there
are no solutions to Eq. (7) with Re[λ]>0. In fact, varying
aand τwhile fixing the other parameters, we discover a Hopf
bifurcation, generically, at which λ=±iωc[52]. An example
Hopf line is shown in Fig. 1(a) in blue for m=α=β=1.
Based on our analysis, we expect the ring state to be locally
stable below the blue line and unstable above it. For compari-
son, the blue circles in Fig. 1(a) denote simulation-determined
transition points: the largest τ(a) for which a swarm of 600
agents, initially prepared in a ring state with a small random
perturbation (i.e., independent and uniformly distributed Aj
and Bjover [−10−5,10−5]), returns to a ring configuration
after an integration time of t=20 000. Numerical predictions
from Eq. (7) show excellent agreement with these simulation
results. Similarly determined transition points for a ring for-
mation in which half the agents rotate in one direction and half
rotate in the opposite direction are shown with blue squares.
We can see that the ring’s Hopf transition line still gives a
good approximation for this more general case, especially for
larger values of a.
In addition to the transition points, we can check the
frequency of oscillations around the ring state, implied by the
existence of an unstable mode for τ(a) slightly above the Hopf
bifurcation. First we perform a simulation initially prepared
in the ring state with a small perturbation (as described in
the preceding paragraph) and compute the peak frequency,
ω∗, in the Fourier spectrum of the swarm’s center of mass,
R(t)≡jrj/N. An example is shown in the inlet panel of
Fig. 2(a) for (a=3.243, τ=1.565); the symbol P denotes
the absolute value of the Fourier transform. Second, we plot
ωc=ω∗−√a/mand compare to predictions from solutions
of Eq. (7) with λ=±iωc= 0 for a range of time delays. The
comparison is shown in Fig. 2(a) with excellent agreement.
B. Rotating state
Next, we perform a similar stability analysis for the ro-
tating state, which has a different bifurcation structure and
unstable modes. Unlike the ring state, the rotating state entails
=
!c
(a)
(b)
12345 6 78 910
0.1
0.2
0.3
0.4
0.5
123
1
3
5
7
P
5.30.35.20.25.15.00.1 4.0
0.1
0.3
0.5
0.7
P
1
3
5
7
1245
1
c
FIG. 2. Frequency of unstable modes near bifurcation. (a) Un-
stable frequency for the ring state at the Hopf bifurcation (black
line) determined from the power spectrum of the swarm’s center of
mass (red circles). (b) Unstable frequency for the rotating state at
the double-Hopf bifurcation (black line) determined from the power
spectrum for a single agent. Inset panels show example spectra for
both states: (a) when (a=3.243, τ=1.565) and (b) when (a=3.5,
τ=1.059). In all panels, m=α=β=1.
a collapse of the swarm onto the center of mass with complete
phase and amplitude synchronization (in the noiseless limit).
In polar coordinates, the agents satisfy rj(t)=Rand φj(t)=
t[34], where
0=m2−a[1 −cos τ ],(8)
R=1
||α−asin(τ )/
β.(9)
In order to determine the local stability of the rotating
state, we substitute rj(t)=R+Bjexp{λt}and φj(t)=t+
Ajexp{λt}into Eqs. (2) and (3) and, again, collect terms to
first order in Ajand Bj(assuming |Aj|,|Bj|1∀j). The
result is another linear system of equations with constant
coefficients. After some algebra and replacing the restricted
sums in Eqs. (2) and (3) by sums over all particles, we obtain
R[mλ2−λ(α−3βR22)+acos(τ )]Al+[2mλ−α+3βR22]Bl=ae−λτ
N
j
[Rcos(τ )Aj−sin(τ )Bj],(10)
[aR sin(τ )−2mRλ]Al+[mλ2−m2−λ(α−βR22)+a]Bl=ae−λτ
N
j
[Rsin(τ )Aj+cos(τ )Bj].(11)
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JASON HINDES et al. PHYSICAL REVIEW E 101, 042202 (2020)
There are two primary categories of solutions to Eqs. (10)
and (11). The first is Al=Aand Bl=B, which we call
the homogeneous modes. Because all agents move together
(equal to the center-of-mass motion), the stability entailed
by the homogeneous modes should match the mean-field
approximation mentioned above and analyzed in [34]. Be-
cause the mean field is known to be quantitatively inaccurate
for capturing stability [36,51], we focus on the second set
of solutions: jAj/N=0 and jBj/N=0. The stability
exponents, λ, for these modes satisfy
2mλ −asin(τ )
mλ2−λ(α−3βR22)+acos(τ )
−mλ2−m2−λ(α−βR22)+a
[α−3βR22−2mλ]=0.(12)
Equation (12) has four complex solutions.
In general, the rotating state will be linearly stable if there
are no solutions to Eq. (12) with Re[λ]>0. In practice, we
find that changing aand τwhile keeping all other parameters
fixed produces saddle-node, Hopf, and double-Hopf bifurca-
tions [52–54]. In the former case, a single real eigenvalue
approaches zero when
tan(τ )=m2−a
(α−3βR22).(13)
Equation (13) gives the stability line for the rotating state
with small aand large τ.Forlargeaand small τ,
the stability changes through a double-Hopf bifurcation
where two frequencies become unstable simultaneously, λ=
±iω1,±iω2= 0. Figure 1(a) shows the predicted composite
stability curve for the rotating state, combining both bifurca-
tions. Plotted is the maximum τfor fixed a, where Re[λ]>0.
Above the red line the rotating state is expected to be locally
stable, and below it, unstable (see Sec. IV for an enlarged view
of the bifurcation curves for the rotating state).
As with the ring state, we compare our stability predictions
to simulations and determine the smallest value of τ(a)for
which a swarm of N=600 agents, initially prepared in the
rotating state with a small, random perturbation, returns to
a rotating state after a time of t=20 000. These points are
shown with red diamonds in Fig. 1(a) for several values of
coupling. Again, we find excellent agreement with predic-
tions. Another consequence of our analysis is the clear quan-
titative prediction of swarm bistability (between the red and
blue curves in Fig. 1) and noise-induced switching between
ring and rotating patterns, which can now be precisely tested
in experiments [35,36,40].
Lastly, just as with the ring state, we can compare the
frequency of oscillations around the rotating state for τ(a)
slightly below the double-Hopf bifurcation values, where we
expect weak instability of modes orthogonal to the center-of-
mass motion. First we perform a simulation initially prepared
in the rotating state with a small perturbation and compute
the peak frequency, ω∗, in the Fourier spectrum of rj−R,
where jis a randomly selected agent. An example is shown
in the inlet panel of Fig. 2(b) for (a=3.5, τ=1.059). This
peak frequency is compared to predictions from numerical
solutions of Eq. (12) with λ=±iω1,±iω2= 0 for a range
of coupling strengths. In Fig. 2(b) the smaller of the two
frequencies, ω1, is plotted along with ω∗, showing excellent
agreement. Note that in this comparison, we do not subtract
the rotating state’s frequency , since rjdoes not oscillate in
the rotating state but is equal to R.
III. CONCLUSION
In this work we studied the stability of ring and ro-
tational patterns in a general swarming model with time-
delayed interactions. We found that ring states change stability
through Hopf bifurcations, where spatially periodic modes
sustain oscillations in time. On the other hand, rotating states
undergo saddle-node, Hopf, and double-Hopf bifurcations,
where modes with orthogonal dynamics to the center-of-
mass motion change stability. For both states, the unstable
oscillations correspond to dynamics not captured by standard
mean-field approximations. Our results were verified in detail
with large-agent simulations. Future work will extend our
analysis to include the effects of repulsive forces, noise,
and incomplete (and dynamic) communication topology—all
of which are necessary for parametrically controlling real
swarms of mobile robots.
ACKNOWLEDGMENTS
J.H., I.T., and I.B.S. were supported by funding from the
U.S. Naval Research Laboratory (No. N0001419WX00055)
and the Office of Naval Research (No. N0001419WX01166
and No. N0001419WX01322). T.E. was supported through a
U.S Naval Research Laboratory Karles Fellowship. S.K. was
supported through a GMU Provost PhD award as part of the
Industrial Immersion Program.
APPENDIX
Close inspection of Fig. 1(a) shows that there is a small,
apparent discontinuity in the stability line for the rotating
state. This apparent discontinuity is a consequence of several
bifurcation curves intersecting in a small region in the (a,τ)
plane for m=α=β=1. For completeness, we show an
enlarged version of the bifurcation curves for the rotating state
0.83 0.84 0.85 0.86 0.87 0.88
1.94
1.98
2.02
2.06
2.08
a
FIG. 3. Bifurcation curves for the rotating state near the inter-
section of saddle-node (red line), Hopf (green dotted line), and
double-Hopf (blue dashed line) bifurcations. Other parameters are
m=α=β=1.
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UNSTABLE MODES AND BISTABILITY IN … PHYSICAL REVIEW E 101, 042202 (2020)
in Fig. 3. The rotating state is linearly stable in the region
to the right of the red line (saddle-node) and blue dashed
line (double-Hopf) bifurcations. The combined stability line
drawn in Fig. 1for the rotating state plots whichever value of
τis larger between the red and blue-dashed curves in Fig. 3,
for each value of a. Note that the blue dashed curve in Fig. 3
is a bifurcation of the rotating state and is different from the
blue curve in Fig. 1(a).
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