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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,

2470-0045/2020/101(4)/042202(6) 042202-1 ©2020 American Physical Society

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 cde-

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−βr2

l˙

φ2

l+˙r2

lrl˙

φl−2m˙rl˙

φl

+a

N−1

j=l

rj(t−τ)sin[φj(t−τ)−φl],(2)

m¨rl=α−βr2

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 N1with 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]+2a

m[m˙

Bl+αBl]=a

N

jBτ

jsin 2π(j−l)

N−a

mτ+(Aτ

j−Al)mα

βacos 2π(j−l)

N−a

mτ,(5)

m¨

Bl−2mα

β

˙

Al=a

N

jBτ

jcos 2π(j−l)

N−a

mτ−Aτ

j−Almα

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

042202-2

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]

2ma

mλ−a

2ie−τ[λ+i√a

m]

+2a

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=m2−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−3βR22)+acos(τ )]Al+[2mλ−α+3βR22]Bl=ae−λτ

N

j

[Rcos(τ )Aj−sin(τ )Bj],(10)

[aR sin(τ )−2mRλ]Al+[mλ2−m2−λ(α−βR22)+a]Bl=ae−λτ

N

j

[Rsin(τ )Aj+cos(τ )Bj].(11)

042202-3

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

2mλ −asin(τ )

mλ2−λ(α−3βR22)+acos(τ )

−mλ2−m2−λ(α−βR22)+a

[α−3βR22−2mλ]=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 )=m2−a

(α−3βR22).(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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