Torus bifurcations of large-scale swarms having range dependent communication delay
Ira B Schwartz, Victoria Edwards, Sayomi Kamimoto, Klimka Kasraie, Ioana Triandaf, M. Ani Hsieh, Jason Hindes
TTorus bifurcations of large-scale swarms having range dependentcommunication delay
Ira B. Schwartz, a) Victoria Edwards, Sayomi Kamimoto, Klimka Kasraie, M. Ani Hsieh, Ioana Triandaf ,and Jason Hindes U.S. Naval Research Laboratory, Code 6792, Plasma Physics Division, Washington, DC 20375,USA U.S. Naval Research Laboratory, Code 5514, Navy Center for Applied Research in Artificial Intelligence, Washington, DC 20375,USA Department of Mathematics, George Mason University, Fairfax Virginia, 22030, USA Aerospace, Transportation and Advanced Systems Laboratory of the Georgia Tech Research Institute, Atlanta,GA 30332 Mechanical Engineering and Applied Mechanics University of Pennsylvania, Philadelphia,PA 19104 USA (Dated: 27 April 2020)
Dynamical emergent patterns of swarms are now fairly well established in nature, and include flocking and rotationalstates. Recently, there has been great interest in engineering and physics to create artificial self-propelled agents thatcommunicate over a network and operate with simple rules, with the goal of creating emergent self-organizing swarmpatterns. In this paper, we show that when communicating networks have range dependent delays, rotational stateswhich are typically periodic, undergo a bifurcation and create swarm dynamics on a torus. The observed bifurcationyields additional frequencies into the dynamics, which may lead to quasi-periodic behavior of the swarm.
Swarming behavior occurs when a large number ofself-propelled agents interact using simple rules. Natu-ral swarms of biological systems have been observed ata range of length scales forming complex emergent pat-terns. Engineers have drawn inspiration from these nat-ural systems, resulting in the translation of swarm theoryto communicating robotic systems. Example applicationsof artificial swarms include: exploration and mapping,search and rescue, and distributed sensing and estimation.Through continued development, an additional parameterof delay in communication between artificial agents hasbecome important to consider. Specifically, it was pre-viously discovered, that communication delay will createnew rotational patterns which are not observed withoutdelay, both theoretically and experimentally. Here we ex-tend the understanding of communication delays to revealthe effects of range dependent delay, where the commu-nication between agents depends on the distance betweenagents. The results of the research show that by includ-ing range dependent delay, new rotational states are in-troduced. We show how these new states emerge, discusstheir stability, and discuss how they may be realized inlarge scale robotic systems. In improving our theoreti-cal understanding of predicted swarm behavior modeledin simulation we can better anticipate what will happenexperimentally. Additionally, it is possible to leverage thepredicted autonomous behaviors to try and force differentswarm behavior. a) Electronic mail: [email protected]
I. INTRODUCTION
Swarming behavior, which we define as the emergence ofspatio-temporal group behaviors from simple local interac-tions between pairs of agents, is widespread and observed overa range of application domains. Examples can be found inbiological systems over a range of length scales, from aggre-gates of bacterial cells and dynamics of skin cells in woundhealing to dynamic patterns of fish, birds, bats, and evenhumans . These systems are particularly interesting becausethey allow simple individual agents to achieve complex tasksin ways that are scalable, extensible, and robust to failures ofindividual agents. In addition, these swarming behaviors areable to form and persist in spite of complicating factors suchas delayed actuation, latent communication, localized numberof neighbors each agent is able to interact with, heterogeneityin agent dynamics, and environmental noise. These factorshave been the focus of previous theoretical research in de-scribing the bifurcating spatial-temporal patterns in swarms,as seen for example in Refs. . Likewise, the application ofswarms have been experimentally realized in areas, such asmapping , leader-following , and density control . Toguarantee swarming behavior experimentally, control is typ-ically employed to prove convergence to a given stateby relying on strict assumptions to guarantee the desired be-havior. However, by relaxing certain assumptions, a numberof studies show that even with simple interaction protocols,swarms of agents are able to converge to organized, coher-ent behaviors in a self-emergent manner; i.e. autonomouslywithout control. Different mathematical approaches yielded awide selection of both agent-based and continuum mod-els that predict swarming dynamics. . In almost all mod-els, since the agents have just a few simple rules, there ex-ists only a relatively small number of controllable parameters.The parameter set usually consists of a self-propulsion force, a r X i v : . [ n li n . AO ] A p r a potential function governing attracting and repelling forcesbetween agents, and a communicating radius governing thelocal neighborhood at which the agents can sense and interactwith each other.In both robotic and biological swarms, an additional pa-rameter appears as a delay between the time information isperceived and the actuation (reaction) time of an agent. Suchdelays have now been measured in swarms of bats, birds, fish,and crowds of people . The measured delays are longerthan the typical relaxation times of the agents, and may bespace and time dependent. Robotic swarms experience com-munication delays which provide similar effects to the delayexperienced in natural swarms. Incorporating stationary de-lays along with a minimal set of parameters in swarm modelsresults in multi-stability of rotational patterns in space . Inparticular, for delays that equal and fixed, one observes threebasic swarming states or modes: Flocking, which is a trans-lating center of mass, Ring state, where the agents are splayedout on a ring in in phase about a stationary center of mass, anda Rotating state, where the center of mass itself rotates.Synthetic robotic swarms have communication delays thatnaturally occur over wireless networks, as a result of lowbandwidth resulting in delayed communication and multi-hop communication . In cases where the delays are fixed andequal, and the communication occurs on a homogeneous net-work, it is known that delays create new rotational patterns ,as has been verified both theoretically and experimentally .However, in situations with robots, even simple communica-tion models are based on the distance between agents .Following from these models, if one assumes that the delaysare range dependent, the problem becomes one of studyingstate dependent delays where delays depend implicitly on therelative positions between agents.When placing swarms in realistic complex environments,delays are not necessarily a continuous function of range,but rather it is the increasing probability of delays increasingstochastically when agents move further away from one an-other beyond a certain radius . That is, the rate of commu-nication becomes spatially dependent, whereby near agentssee a signal with a fast rate of communication, but due to shad-ing and fading of signals, communication rates are slowed andcomplex outside a given radius. Underwater communicationis an excellent swarm example where delays outside a signif-icant radius impart rates of communication of one to two or-ders of magnitude greater than local communication rates .The swarm model that follows takes a globally coupledswarm, and explicitly relaxes the fixed delay assumption, byincluding range dependent delay based on a fixed communi-cation radius. We show that when range dependent delays areincluded, new frequencies are introduced and generate bifur-cations to a torus. The result is a milling type of swarm thatdepends on just a few parameters. The results here are impor-tant for robotic swarming where one of the goals is to producedesired patterns autonomously, without external controls. Thepattern formations predicted here show how delayed infor-mation, whether coming from communication, actuations, orboth, impacts the stability of swarm states, such as ring and/orrotating states. By revealing those parameter regions where patterns are destabilized, we provide a comprehensive charac-terization of the autonomously accessible swarm states in thepresence of range-dependent delay. II. THE SWARM MODEL
Consider a swarm of delay-coupled agents in R . Eachagent is indexed by i ∈ { , . . . , N } . We use a simple butgeneral model for swarming motion. Each agent has a self-propulsion force that strives to maintain motion at a preferredspeed and a coupling force that governs its interaction withother agents in the swarm. The interaction force is defined asthe negative gradient of a pairwise interaction potential U ( · , · ) .All agents follow the same rules of motion; however, mechan-ical differences between agents may lead to heterogeneous dy-namics; this effect is captured by assigning different acceler-ation factors (denoted κ i ) to the agents. In this paper, we as-sume κ i = i . For the effect of heterogeneity on theswarm bifurcations, see .Agent-to-agent interactions occur along a graph G = { V , E } , where V is the set of vertexes v i in the graph and E is the set of edges e i j . The vertices correspond to individualswarm agents, and edges represent communication links; thatis, agents i and j communicate with each other if and onlyif e i j ∈ E . All communications links are assumed to be bi-directional, and all communications occur with a time delay τ . That is, range dependence is not included. Let r i ∈ R de-note the position of the agent i and let N i = { v j ∈ V : e i j ∈ E } denote its set of neighbors of agent i . The motion of agent i isgoverned by the following equation:¨ r i = κ i ( −(cid:107) ˙ r i (cid:107) ) ˙ r i − κ i ∑ j ∈ N i ∇ x U ( r i ( t ) , r τ j ( t )) , (1)where superscript τ is used to denote time delay, so that r τ j ( t ) = r j ( t − τ ) , (cid:107)·(cid:107) denotes the Euclidean norm, and ∇ x de-notes the gradient with respect to the first argument of U . Thefirst term in Eq. 1 governs self-propulsion, where the speedhas been normalized to unity. That is, without coupling theagents always asymptote to unit speed.To analyze the dynamics of a large scale swarm, we use aharmonic interaction potential with short-range repulsion. U ( r i , r τ j ) = c r e − (cid:107) r i − r j (cid:107) lr + a N (cid:13)(cid:13)(cid:13) r i − r τ j (cid:13)(cid:13)(cid:13) . (2)In Eq. 1, it is assumed that the communication delay, τ ,is independent of the distance, or range, between any pair ofagents. (Notice that the exponent of the repulsion term is inde-pendent of the delay since the repulsion force is local.) Withthe addition of delays in the network, it was shown in ho-mogeneous communication networks that in addition to theusual dynamical translating and milling (or ring) states, forsufficiently large τ , new rotational states emerge . In par-ticular, for a a given attractive coupling strength, there is adelay that destabilizes the periodic ring state into a rotatingstate, in which the agents coalesce to a small group and movearound a fixed center of rotation; this behavior is quite dif-ferent from the ring state where agents are spread out in asplay state phase. The rotating state is only observed with de-lay introduced in the communication network, and it appearsthrough a Hopf bifurcation.However, in real-world robotic swarms, communication de-lays are not uniform between all pairs of agents; delays maybe stochastic or even state-dependent. For example, if agentsare communicating over a multi-hop network, the delay willincrease with the number of hops required to send a messagefrom one agent to the other, and in general will scale withthe separation between them. In order to handle range depen-dent delays, we will make an approximation that depends ona communication range radius. A. Approximating range dependent delayed coupling
For the coupling term, we are interested in introducing anapproximation to range based coupling delay. Since all com-municating agents send signals with some delay, we computerelative distances defined as D τ i , j ≡ || r i − r τ j || . (3)We define a Heaviside function, H ( x ) , that is zero when x ≤ ε ≥ ε , then sensingbetween two agents is almost immediate. In practice, the timeneeded for sensing depends on several factors, such as actu-ation times, and so distances in practice are computed withdelay. Therefore, we model the coupling term for the i th agentas C i ( r i , r j , r τ j , ε ) = − aN ( ∇ x U ( r i ( t ) , r τ j ( t ))) H ( D τ i , j − ε ) − aN ( ∇ x U ( r i ( t ) , r j ( t )))( − H ( D τ i , j − ε )) , (4)where the first coupling term has delay turned on since thedistance is outside a ball of radius ε , while the second termhas no delay since the distance is within the ε ball. Theresulting swarm model with range dependence from Eq. 4 isnow ¨ r i = κ i ( −(cid:107) ˙ r i (cid:107) ) ˙ r i − κ i ∑ j ∈ N i C i ( r i , r j , r τ j , ε ) . (5)If the delayed distance is within an ε ball, then we eval-uate the coupling without delay. Otherwise the coupling isdelayed. Thus the coupling function takes into account whendelay is active or not between pairs of communicating agents,and depends on the range radius, ε .The Heaviside function of the right hand side of Eq. 9 ren-ders the differential delay equation derivatives discontinuous,and as such poses a numerical integration problem. To mol-lify the lack of smoothness, we approximate H ( x ) by letting H ( x ) ≈ π arctan ( kx )+ , where k (cid:29) k → ∞ .Using only the delayed distance to compute a rangedependent coupling assumes that any measurement is notinstantaneous. If one were to be able to compute the idealsituation where delay would not be a sensing factor, thencertain issues would need to be resolved, which we do notconsider here. B. Numerical simulations of full swarms
Examples of simulations using the swarm model with therange dependent coupling are shown below. Here the numberof agents, N = a = .
0. Forthe remainder of the analysis, we set c r =
0, and note that theattractors persist when the repulsive amplitude is sufficientlysmall . (See supplementary material for a video of the dy-namics with small repulsion.) FIG. 1. Three snapshots of swarm state in space for ε = . , a = . , τ = .
75. Sample times t , t = t + , t = t +
40 .
Note that even when ε is very small, as shown in Fig. 1,we observe a mix of clustered states which are a combina-tion of pure ring and rotation states. The agents tend to clus-ter into local groups, and the clusters move in clockwise andcounter-clockwise directions as in the ring state. Here, how-ever, the phase differences between agents are non-uniform.When examining a single random agent, as shown in Fig 2,it is periodic with a sharp frequency of rotation, and the rela-tive positions of all individual agents are phase locked. Whenconsidering the center of mass of the positions over all agents, R ≡ N ∑ i r i , the center of mass does small amplitude oscilla-tions about a fixed point (not shown).As the radius ε increases, instability of the periodic mixedstate occurs, giving rise to more complicated behavior, as seenin Fig. 3. New frequencies are introduced, causing the ringstate to appear as a quasi-periodic attractor. Moreover, the FIG. 2. Swarm ring state for ε = . , a = . , τ = .
75. (a)Timeseries of the x-component of a single agent. (b)The power spectrumshowing a sharp frequency. (c)A phase portrait of the orbit of a singleagent. The red point denotes the center of mass. dynamics of the center of mass has its own non-trivial dy-namics which includes the effects of new frequencies. Byexamining the Poincare map of the attractors, the instabilitygives rise to dynamics which we conjecture is motion on atorus. Letting ( M x , M y ) denote the time averaged center ofmass over all agents, we compute the sequence x ( t i ) , i = .. M when y ( t i ) = x ( t i ) > M x . The result is shown in thetwo panels in Fig. 4. Panel (a) shows a complicated toroidalmotion after transients are removed of the center of mass inFig. 3c. For a single frequency, the dynamics of the centerof mass would be a single fixed point. The addition of newfrequencies is revealed in the Poincare map as complicatedmotion on a torus. For larger values of ε , the motion on thetorus converges to a periodic attractor in panel (b). III. MEAN-FIELD EQUATION OF RANGE DEPENDENTDELAY COUPLED SWARM
In order to shed some light on the origin of the bifurcationto dynamics on a torus, we examine the full swarm modelfrom a mean-field perspective. The mean field is much lowerdimensional, and a full bifurcation analysis may be done. Weconsider the case of all-to-all communication. Let
FIG. 3. Swam instability ε = . , a = . , τ = .
75. (a)Time seriesof the x-component of a single agent. (b)The Power spectrum show-ing a slight broadening and birth of a new frequency. (c)A phaseportrait of the orbit of a single agent.
RRR = N N ∑ i = r i and r i = RRR + δ r i , where δ r i is a fluctuation term with the identity, and N ∑ i = δ r i = . (6)Then we can write Eq. 5 as¨ RRR + δ ¨ r i = ( − | ˙ RRR + δ ˙ r i | )( ˙ RRR + δ ˙ r i ) − aN N ∑ j = , j (cid:54) = i (( RRR + δ r i ) − ( RRR τ + δ r τ j )) C , i − aN N ∑ j = , j (cid:54) = i (( RRR + δ r i ) − ( RRR + δ r j )) C , i , (7)where C , i = H ( (cid:107) r i − r τ j (cid:107) − ε )= H ( (cid:107) ( RRR + δ r i ) − ( RRR τ + δ r τ j ) (cid:107) − ε )= H ( (cid:107) RRR − RRR τ + δ r i − δ r τ j (cid:107) − ε ) and C , i = − C , i . We use the following to reduce the equations of motion to themean field: From Eq. 6, we note N ∑ i = δ r τ i = N ∑ j = , j (cid:54) = i δ r τ j + δ r τ i = ⇐⇒− N ∑ j = , j (cid:54) = i δ r τ j = δ r τ i . (8)We further assume that all perturbations from the mean, δ r i , are all negligible. (This is always true if the coupling am-plitude is sufficiently large.) In addition, we use the fact that a ( N − ) N limits to a , as N → ∞ . Therefore, we obtain meanfield approximation for the center of mass of range dependentcoupled delay case:¨ RRR = ( − | ˙ RRR | ) · ˙ RRR − a ( RRR − RRR τ ) · H ( (cid:107) RRR − RRR τ (cid:107) − ε ) (9) IV. NUMERICAL ANALYSIS OF THE MEAN FIELDEQUATIONA. Examples of rotational attractors
As in the case for the full multi-agent system, we see theexistence of periodic behavior for τ sufficiently below an in-stability threshold, as shown in the time series of Fig. 5. As we FIG. 4. Poincare map of Eqs. 1-4 for (a) ε = .
25, (b) ε = .
5. Otherparameters are fixed: a = . , τ = .
75. See text for details. FIG. 5. Periodic motion of the mean field Eq. 9 for ε = . , a = . , τ = .
6. (a) Time series of the x-component of the mean field.(b) Power spectra of the time series. increase τ , we expect the periodic orbit to lose stability, result-ing in a new attractor. In particular, one notices the emergenceof a new frequency in addition to the existing dominant one,as shown in Fig. 6 The additional frequency usually impliesa bifurcation to dynamics on a torus, or a higher dimensionaltorus.We now investigate this transition by tracking the stabilityvia monitoring the Floquet exponents corresponding to the pe-riodic orbit. For a general differential delay equation given by˙ xxx ( t ) = FFF ( xxx ( t ) , xxx ( t − τ )) , if φφφ ( t ) = φφφ ( t + T ) for all t ≥
0, thenstability is determined by examining the linearized equationalong φφφ ( t ) :˙ XXX ( t ) = ∂ FFF ∂ xxx ( t ) ( φφφ ( t ) , φφφ ( t − τ )) XXX ( t )+ ∂ FFF ∂ xxx ( t − τ ) ( φφφ ( t ) , φφφ ( t − τ )) XXX ( t − τ ) . (10)The stability of the periodic solution is determined by thespectrum of the time integration operator U ( T , ) which inte-grates Eq. 10 around φ ( t ) from time t = 0 to t = T. This opera-tor is called the monodromy operator and its (infinite numberof) eigenvalues, which are independent of the initial state, arecalled the Floquet multipliers . For autonomous systems, itis necessary and sufficient there exists a trivial Floquet multi-plier at 1, corresponding to a perturbation along the periodicsolution . The periodic solution is stable provided all mul-tipliers (except the trivial one) have modulus smaller than 1; itis unstable if there exists a multiplier with modulus larger than1. Bifurcations occur whenever Floquet multipliers move intoor out of the unit circle. Generically three types of bifurcationsoccur in a one parameter continuation of periodic solutions: aturning point, a period doubling, and a torus bifurcation wherea branch of quasi-periodic solutions originates and where acomplex pair of multipliers crosses the unit circle .We have tracked a set of stable periodic orbits for variousradii of ε , and located the change in stability by computing the FIG. 6. Quasi-periodic motion of the mean field Eq. 9. (a) Timeseries of the x-component of the mean field. Solid (red) line denotesperiod length of dominant spectral peak. Dashed line denotes periodlength of secondary peak. (b) Power spectra of the time series.FIG. 7. Bifurcation plot showing the norm of the periodic orbits asa function of delay τ . Parameter a=0.68. Red (blue) markers denoteunstable (stable) orbits.Cyan symbols denote the change in stabilitywhere a pair of complex eigenvalues cross the imaginary axis. Floquet multipliers. The results plotted in Fig. 7 show that fora range of radii ε , there exists a bifurcation to a torus at somedelay. Notice that as ε increases, there results an increase inthe size of the orbits, which qualitatively agrees with our fullagent based simulations.Since there exists a range of delays which destabilize peri-odic swarm dynamics for each ε , we summarize the onset oftorus bifurcations by plotting the locus of points at which sta-bility changes as a a function of coupling amplitude and delay.The results are plotted in Fig. 8.Figure 8 is revealing in that it shows a functional relation-ship of the bifurcation onset that is similar over a range of ε .For larger values of ε , it is clear that lower values of delayand coupling are required to generate bifurcations. This holdstrue over two orders of ε . For a fixed value of ε , we also seemonotonic relationship between delay and coupling strength,so that it is easier for smaller delays to destabilize periodicmotion for larger coupling strengths. FIG. 8. Plotted is the locus of points at which torus bifurcationsemerge as a function of coupling amplitude a , delay τ for variousrange radii ε for the mean field Eq. 9. V. CONCLUSIONS
We considered a new model of a swarm with delay coupledcommunication network, where the delay is considered to berange dependent. That is, given a range radius, delay is onif two agents are outside the radius, and zero otherwise. Theimplication is that small delays do not matter if the agents areclose to each other.The additional range dependence creates a new set of bi-furcations not previously seen. For general swarms withoutdelay, the usual states consist of flocking (translation) or ring/ rotational state (milling), with agents spread in phase. Withthe addition of a fixed delay, a rotational state bifurcates thathas all agents in phase and rotate together . Range depen-dence introduces a new rotational bifurcating state that ex-hibits behavior observed as a new mixed state combining dy-namics of both ring and rotating states.The radius parameter ε , was used to quantify the bifurca-tion of the rotational mixed state. For small ε , we see dy-namics for the full swarm shows clustered counter-rotationalbehavior that is periodic. This agrees for small radius valuesin the mean field description as well. As the radius increases,the mixed periodic state generates new frequencies in the fullmodel, which are manifested as torus bifurcations in the meanfield. Mean field analysis was done by tracking Floquet mul-tipliers that cross the imaginary axis as complex pairs. Fre-quency analysis explicitly shows the additional frequencies inthe mean field.Finally, we tracked the locus of coupling amplitudes anddelay for various values of ε locating the parameters at whichtorus bifurcation occur. The results reveal that as ε increases,torus bifurcations onset at lower values of coupling amplitudeand delay. The implications are that more complicated be-havior than periodic motion has a greater probability of beingobserved in both theory and experiment if range dependenceof delay is included. VI. SUPPLEMENTARY MATERIAL
The videos show the attractor of a swarm consisting ofN=300 agents. Fixed parameters for the three videos are a = . , τ = .
75 The parameters for zero radius (delay is onall the time) are ε = . , c r = .
05, and l r = .
05 for a base-line, are shown in Video1_eps_0p0.mp4.The parameters corresponding to Fig. 2 are ε = . , c r = .
01, and l r = .
05 are shown in Video2_eps_0p01.mp4.The video shows that the attractor persists when repulsiveforces are local and weak. Similar behavior is observed whenN=150, which is used in Fig. 1 without repulsion; i.e., c r = ε = . , c r = .
05, and l r = .
05, shown in Video3_eps_0p25.mp4.
ACKNOWLEDGMENTS
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