# Flocking Behavior of Cucker–Smale Model with Processing Delay

Authors: Jianfei Cheng; Maoli Chen; Xiao Wang
Journal: Complexity
 (2020)

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## Abstract

The dynamics of a delay multiparticle swarm, which contains symmetric and asymmetric pairwise influence functions, are analyzed. Two different sufficient conditions to achieve conditional flocking are obtained. One does not have a clear relationship with this delay, and the other proposes a range of processing delays that affect the emergence of a flock. It is also pointed out that if the interparticle communication function has tail dissipation, unconditional flocking can be guaranteed. Compared with the previous results, the range of the communication rateβ that allows a flock to emerge has been expanded from 1/4 to 1/2.


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## Body

## 1. Introduction

There are many survival-oriented clusters in nature, such as ant colonies that coordinate food transportation, birds that increase the success rate of foraging, and fish that unite against danger, and so on. The research on the colony of biological groups should be traced back to Reynolds’ simulation experiments on birds in [1]; further some scholars have proposed many motion models to mathematically characterize them. Among them, a second-order model proposed by Cucker and Smale in [2, 3] to explain self-organizing behavior in complex adaptive systems has been favored by researchers and continuously improved. For example, Motsch and Tadmor in [4] modified the symmetry of the influence intensity between particles to be asymmetric to explore the aggregation behavior of nonuniformly distributed particle swarms. Some scholars have carried out the impact of the time delay on flocking or consensus of the system in [5–7] and the references therein.Liu and Wu in [5] proposed a model with processing delay, which is described as(1)x˙it=vit,i=1,2,…,N,v˙it=α∑j=1NIxit−τ−xjt−τvjt−τ−vit,where xi,vi∈Rd and d is a positive integer, α>0 indicates the intensity of the influence between particles, and τ>0 represents the time lag, which includes the response time of the particle i and the communication time between particles i and j. The communication function can be defined as(2)Irij=ICSrij≔ψrijNorIrij=IMTrij≔ψrij∑k=1Nψrik,which further satisfies ∑l=1NIrkl=1 for all k∈Γ, where ψr=1+r2−β,β≥0, rijt=xit−xjt,i,j∈Γ. Further, system (1) can be simplified to(3)x˙it=vit,i∈Γ,v˙it=αv¯it−vit,where v¯it=∑j=1NIrijt−τvjt−τ. The initial conditions are(4)xiθ=φiθ,viθ=ϕiθ,θ∈−τ,0,where φi,ϕi∈C2=C×C and C≔C−τ,0,R2 is the Banach space of all continuous functions.In this study, we further consider the flocking conditions of the delayed model proposed in [5]. The significant contributions of our results are reflected in the following three aspects. (1) Compared with Theorem 3.1 in [5], the unconditional flocking condition ∫∞ψ2rdr=∞ is improved to ∫∞ψrdr=∞, that is, the communication rate β is expanded from 1/4 to 1/2. (2) Note that with τ=0 in (1), the communication rate β of unconditional flocking in [4] has also been expanded from 1/4 to 1/2. (3) It is clearly pointed out that processing delay can affect the occurrence of aggregation behavior, which is specifically manifested in the controllable range of the delay in flocking conditions.The following two variables (Dx and Dv) are used to analyze the evolution of the aggregation behavior of systems (3) and (4), for t≥−τ:(5)Dxt=maxi,j∈Γxit−xjt,Dvt=maxi,j∈Γvit−vjt.Thus, for bothICSr and IMTr, it follows from a few simple calculations that a uniform result can be directly verified about the estimation of influence function Ir as(6)Irijt−τ≥ψDxt−τN,for alli,j∈Γ.We still adopt the definition of time-asymptotic flocking proposed in [4].Definition 1.
Letxit,viti=1N be a solution to systems (3) and (4); a time-asymptotic flocking can be achieved if and only if the solution satisfies supt>0Dxt<+∞ and limt⟶+∞Dvt=0, where Dxt and Dvt are given in (5).


## 2. Main Results

This section proposes two different sufficient conditions for systems (3) and (4) with ICS or IMT to achieve the conditional flocking in Theorem 1 and Theorem 2. We have also established certain conditions for the completion of unconditional flocking in Theorem 3.


### 2.1. Conditional Flocking

To establish the flocking solution of systems (3) and (4), the following important auxiliary lemmas are introduced first.Lemma 1 (see [8]).
Letxt be the solution of the linear functional differential equation, x˙t=λ−δ1xt+δ2xt−τ. If δ2<δ1, then(7)limt⟶+∞xt=x∗=λδ1−δ2.Lemma 2.
Letxit,viti=1N be a solution to systems (3) and (4); then, we have(8)vit−vjt,v¯it−v¯jt≤1−ψDxt−τNDvtDvt−τ,where Dx and Dv are defined in (5).Proof.
Making use of system (3) yields(9)vit−vjt,v¯it−v¯jt=∑p=1NIript−τ∑q≠pIrjqt−τvit−vjt,vpt−τ−vqt−τ≤∑p=1NIript−τ∑q≠pIrjqt−τDvtDvt−τ.
Using inequality (6) and the normalization assumptions for communication functions, that is, ∑j=1NIrijt−τ=1, we get(10)vit−vjt,v¯it−v¯jt≤∑p=1NIript−τ∑q≠pIrjqt−τDvtDvt−τ=∑p=1NIript−τ1−Irjpt−τDvtDvt−τ≤1−ψDxt−τNDvtDvt−τ,and this proof is completed.Lemma 3.
Letxit,viti=1N be a solution of systems (3) and (4); then, the upper Dini derivative of Dxt and Dvt satisfies(11)D+Dxt≤Dvt,a.e.t≥−τ,D+Dvt≤α1−ψDxt−τNDvt−τ−αDvt.Proof.
Without loss of generality, letDxt=xpt−xqt at time t, where p,q∈Γ. One can obtain(12)D+Dxt≤x˙pt−x˙qt=vpt−vqt≤Dvt.
Similarly, without loss of generality, letDvt satisfy Dvt=vpt−vqt at time t, where p,q∈1,2,…N; it follows from Lemma 2 that(13)D+Dv2t=2αvpt−vqt,v˙pt−v˙qt=2αvpt−vqt,v¯pt−v¯qt−2αDv2t≤2α1−ψDxt−τNDvtDvt−τ−2αDv2t.
Therefore, (11) is proven.
To establish the flocking conditions, we define a set containing all the initial configurations of asymptotic flocking allowed, that is,(14)S≔xiθ,viθ,θ∈−τ,0,i=1,2,…,N.Dv0+α∫−τ0Dvsds<αN∫Dx−τ∞ψrdr,where Dx and Dv are shown in (5) and ψr=1+r2−β,β≥0 is defined in [2, 3].Theorem 1.
Forβ>1/2, suppose that the initial conditions (4) are selected from the set S; then systems (3) and (4) with IMTr or ICSr can complete the conditional flocking.Proof.
Inspired by the work in [5, 9], we take the following Lyapunov function:(15)EDx,Dvt=Dvt+αN∫Dx−τDxt−τψrdr+α∫−τ0Dvt+sds.
Thus, the upper Dini derivative ofEDx,Dvt along Dx,Dv with respect to t is shown below.(16)D+EDxt,Dvt=D+Dvt+αND+∫Dx−τDxt−τψrdr+αD+∫−τ0Dvt+sds.
Furthermore, combining with Lemma3, we have D+EDxt,Dvt≤0, which means that EDx,Dvt is nonincreasing and then EDx,Dvt≤EDx,Dv0 for all t>0. Thus,(17)αN∫Dx−τDxt−τψrdr≤Dv0+α∫−τ0Dvsds.
Since the initial conditions (4) are selected from the set S, it follows from (16) that(18)αN∫Dx−τDxt−τψrdr<αN∫Dx−τ∞ψrdr.
Due to the fact thatψ has a divergent tail, there must be a constant D∗<∞ such that Dxt−τ≤D∗ for t≥0. Considering inequality (6) yields(19)Irijt−τ≥ψDxt−τN≥ψD∗N,for allj∈Γ.
Using the second inequality in (11) in Lemma 3, we can further derive that(20)D+Dvt≤α1−ψD∗NDvt−τ−αDvt.
Making use of Lemma1, we can show that Dvt⟶0 as t⟶∞ and systems (3) and (4) converge to a flock as shown in Definition 1. The proof is completed.
Another flocking condition closely related to processing delay is proposed in the following theorem.Theorem 2.
Forβ>1/2, suppose that the initial configurations (4) are met as follows:(21)0<Dv0<αN∫Dx−τ∞ψrdr,and the processing delay τ satisfies(22)0<τ<τ0≔1αRταN∫Dx−τ∞ψrdr−Dv0,where Rτ≔maxθ∈−τ,0Dvθ>0 and Dx,Dv are defined in (5); then, systems (3) and (4) with IMTr or ICSr converge to a flock.Proof.
We only need to prove that the initial conditions which satisfy (22) all exist in the set S defined in (8). Note that(23)αN∫Dx−τ∞ψrdr>Dv0+ατRτ>Dv0+α∫−τ0Dvsds,which means that the initial conditions which satisfy (22) all exist in set S. Consequently, systems (3) and (4) converge to a flock.Remark 1.
The following notes are listed for the above two different results of conditional flocking.(1)
Note that withτ=0, Theorem 1 and Theorem 2 will degenerate into Dv0<α/N∫∞ψrdr. If α=N, then the flocking condition is further written as Dv0<∫∞ψrdr, thereby improving Theorem 3.1 in [4].(2)
Comparing Theorem1 and Theorem 2, we can get the following two points worthy of attention. First, it is clear from the set of allowed initial conditions that the former is larger than the latter. Second, the latter helps us realize that the occurrence of aggregation behavior is indeed affected by the size of τ.


### 2.2. Unconditional Flocking

The following theorem describes the implementation of unconditional flocking for systems (3) and (4).Theorem 3.
Forβ∈0,1/2, systems (3) and (4) with IMTr or ICSr converge unconditionally to a flock.Proof.
Ifβ∈0,1/2, then the interparticle communication function has a tail dissipation ∫∞ψrdr=∞. Similar to the proof of Theorem 1, it can directly verify the unconditional flocking result. It will be omitted here.Remark 2.
The two annotations for Theorem3 are described below.(1)
The fundamental reason for the unconditional flocking of systems (3) and (4) is that the following condition always holds for any initial configuration:(24)Dv0+α∫−τ0Dvsds<αN∫Dx−τ∞ψrdr=∞.(2)
Note that withτ=0, the unconditional flocking result in [4] is improved to ∫∞ψrdr=∞, which means that the communication rate β is expanded from 1/4 to 1/2.(3)
Compared with Theorem 3.1 in [5], the range of communication rate β has been expanded. Specifically, we promote the results from ∫∞ψ2rdr=∞ to ∫∞ψrdr=∞, that is, we extend the communication rate β from 1/4 to 1/2. Thus, the results in [5] have been improved.


## 2.1. Conditional Flocking

To establish the flocking solution of systems (3) and (4), the following important auxiliary lemmas are introduced first.Lemma 1 (see [8]).
Letxt be the solution of the linear functional differential equation, x˙t=λ−δ1xt+δ2xt−τ. If δ2<δ1, then(7)limt⟶+∞xt=x∗=λδ1−δ2.Lemma 2.
Letxit,viti=1N be a solution to systems (3) and (4); then, we have(8)vit−vjt,v¯it−v¯jt≤1−ψDxt−τNDvtDvt−τ,where Dx and Dv are defined in (5).Proof.
Making use of system (3) yields(9)vit−vjt,v¯it−v¯jt=∑p=1NIript−τ∑q≠pIrjqt−τvit−vjt,vpt−τ−vqt−τ≤∑p=1NIript−τ∑q≠pIrjqt−τDvtDvt−τ.
Using inequality (6) and the normalization assumptions for communication functions, that is, ∑j=1NIrijt−τ=1, we get(10)vit−vjt,v¯it−v¯jt≤∑p=1NIript−τ∑q≠pIrjqt−τDvtDvt−τ=∑p=1NIript−τ1−Irjpt−τDvtDvt−τ≤1−ψDxt−τNDvtDvt−τ,and this proof is completed.Lemma 3.
Letxit,viti=1N be a solution of systems (3) and (4); then, the upper Dini derivative of Dxt and Dvt satisfies(11)D+Dxt≤Dvt,a.e.t≥−τ,D+Dvt≤α1−ψDxt−τNDvt−τ−αDvt.Proof.
Without loss of generality, letDxt=xpt−xqt at time t, where p,q∈Γ. One can obtain(12)D+Dxt≤x˙pt−x˙qt=vpt−vqt≤Dvt.
Similarly, without loss of generality, letDvt satisfy Dvt=vpt−vqt at time t, where p,q∈1,2,…N; it follows from Lemma 2 that(13)D+Dv2t=2αvpt−vqt,v˙pt−v˙qt=2αvpt−vqt,v¯pt−v¯qt−2αDv2t≤2α1−ψDxt−τNDvtDvt−τ−2αDv2t.
Therefore, (11) is proven.
To establish the flocking conditions, we define a set containing all the initial configurations of asymptotic flocking allowed, that is,(14)S≔xiθ,viθ,θ∈−τ,0,i=1,2,…,N.Dv0+α∫−τ0Dvsds<αN∫Dx−τ∞ψrdr,where Dx and Dv are shown in (5) and ψr=1+r2−β,β≥0 is defined in [2, 3].Theorem 1.
Forβ>1/2, suppose that the initial conditions (4) are selected from the set S; then systems (3) and (4) with IMTr or ICSr can complete the conditional flocking.Proof.
Inspired by the work in [5, 9], we take the following Lyapunov function:(15)EDx,Dvt=Dvt+αN∫Dx−τDxt−τψrdr+α∫−τ0Dvt+sds.
Thus, the upper Dini derivative ofEDx,Dvt along Dx,Dv with respect to t is shown below.(16)D+EDxt,Dvt=D+Dvt+αND+∫Dx−τDxt−τψrdr+αD+∫−τ0Dvt+sds.
Furthermore, combining with Lemma3, we have D+EDxt,Dvt≤0, which means that EDx,Dvt is nonincreasing and then EDx,Dvt≤EDx,Dv0 for all t>0. Thus,(17)αN∫Dx−τDxt−τψrdr≤Dv0+α∫−τ0Dvsds.
Since the initial conditions (4) are selected from the set S, it follows from (16) that(18)αN∫Dx−τDxt−τψrdr<αN∫Dx−τ∞ψrdr.
Due to the fact thatψ has a divergent tail, there must be a constant D∗<∞ such that Dxt−τ≤D∗ for t≥0. Considering inequality (6) yields(19)Irijt−τ≥ψDxt−τN≥ψD∗N,for allj∈Γ.
Using the second inequality in (11) in Lemma 3, we can further derive that(20)D+Dvt≤α1−ψD∗NDvt−τ−αDvt.
Making use of Lemma1, we can show that Dvt⟶0 as t⟶∞ and systems (3) and (4) converge to a flock as shown in Definition 1. The proof is completed.
Another flocking condition closely related to processing delay is proposed in the following theorem.Theorem 2.
Forβ>1/2, suppose that the initial configurations (4) are met as follows:(21)0<Dv0<αN∫Dx−τ∞ψrdr,and the processing delay τ satisfies(22)0<τ<τ0≔1αRταN∫Dx−τ∞ψrdr−Dv0,where Rτ≔maxθ∈−τ,0Dvθ>0 and Dx,Dv are defined in (5); then, systems (3) and (4) with IMTr or ICSr converge to a flock.Proof.
We only need to prove that the initial conditions which satisfy (22) all exist in the set S defined in (8). Note that(23)αN∫Dx−τ∞ψrdr>Dv0+ατRτ>Dv0+α∫−τ0Dvsds,which means that the initial conditions which satisfy (22) all exist in set S. Consequently, systems (3) and (4) converge to a flock.Remark 1.
The following notes are listed for the above two different results of conditional flocking.(1)
Note that withτ=0, Theorem 1 and Theorem 2 will degenerate into Dv0<α/N∫∞ψrdr. If α=N, then the flocking condition is further written as Dv0<∫∞ψrdr, thereby improving Theorem 3.1 in [4].(2)
Comparing Theorem1 and Theorem 2, we can get the following two points worthy of attention. First, it is clear from the set of allowed initial conditions that the former is larger than the latter. Second, the latter helps us realize that the occurrence of aggregation behavior is indeed affected by the size of τ.


## 2.2. Unconditional Flocking

The following theorem describes the implementation of unconditional flocking for systems (3) and (4).Theorem 3.
Forβ∈0,1/2, systems (3) and (4) with IMTr or ICSr converge unconditionally to a flock.Proof.
Ifβ∈0,1/2, then the interparticle communication function has a tail dissipation ∫∞ψrdr=∞. Similar to the proof of Theorem 1, it can directly verify the unconditional flocking result. It will be omitted here.Remark 2.
The two annotations for Theorem3 are described below.(1)
The fundamental reason for the unconditional flocking of systems (3) and (4) is that the following condition always holds for any initial configuration:(24)Dv0+α∫−τ0Dvsds<αN∫Dx−τ∞ψrdr=∞.(2)
Note that withτ=0, the unconditional flocking result in [4] is improved to ∫∞ψrdr=∞, which means that the communication rate β is expanded from 1/4 to 1/2.(3)
Compared with Theorem 3.1 in [5], the range of communication rate β has been expanded. Specifically, we promote the results from ∫∞ψ2rdr=∞ to ∫∞ψrdr=∞, that is, we extend the communication rate β from 1/4 to 1/2. Thus, the results in [5] have been improved.


## 3. Numerical Simulations

Some numerical simulations will be enumerated to illustrate the effect of processing delay on the aggregation behavior of systems (3) and (4). For the convenience of calculation, consider a particle swarm composed of 8 particles and its initial conditions are set as φiθ,ϕiθ=8−i,4−i,i=1,…,8,θ∈−τ,0. The parameters in (3) are fixed as α=15,β=0.51>0.5, and further we have τ0=1.0116 from (22). To understand the aggregation behavior of the groups (3) and (4) intuitively and conveniently, we used two features in all experiments, namely, the velocity of each particle and the maximum of the relative position between particles.Example 1.
α=15,β=0.51,τ=0 (see Figure 1).
It can be seen from Figure1 that without a time delay, the particle swarm can converge asymptotically to form a flock under the fixed initial configurations and the above parameters.Figure 1
α=15,β=0.51>0.5,τ=0. Systems (3) and (4) converge to a flock.Example 2.
α=15,β=0.51,τ=1<τ0 (see Figure 2).
Considering the delay range (22) established in Theorem 2, we can claim that the flocking of the particle population (3) and (4) with τ=1<τ0 can be maintained. As we can see in Figure 2, systems (3) and (4) can still aggregate and form a flock after introducing processing time lag τ, which satisfies (22).Figure 2
α=15,β=0.51,τ=1<τ0. Systems (3) and (4) converge asymptotically to form a flock. The convergence rate is slower than that in Example 1 due to processing delays. The diameter of the population is larger than that in Example 1, which means that the cohesion and aggregation density between particles are not as good as the case in Example 1.Example 3.
α=15,β=0.51,τ=12>τ0 (see Figure 3).
As shown in Figure3, under the same initial condition with Example 2, this group cannot converge to a flock, and its fatal factor is that τ=12>τ0, that is, (22) in Theorem 2 is broken.
To realize the emergence of flocking in this group, the following method can be adopted, that is, to appropriately adjust the communication rate between particles so thatβ<0.5. It may be selected as β=0.2<0.5 and then combined with the discussion in Theorem 3; the system unconditionally converges to a flock. The simulation results are shown in Figure 4.Figure 3
α=15,β=0.51,τ=12>τ0. Systems (3) and (4) fail to the emergence of flocking due to the destruction of (22) in Theorem 2.Figure 4
α=15,β=0.2<0.5,τ=12. Systems (3) and (4) form a flock unconditionally.


## 4. Conclusions

We study the emergence conditions of flocking of multiple particle swarms with processing delays, establish two sufficient conditions for conditional flocking in Theorem1 and Theorem 2, and give an unconditional flocking result in Theorem 3. In particular, Theorem 2 intuitively explains the fact that processing delays affect the emergence of flocking, which is reflected in the time-lag range (22) that affects the emergence of flocking. For 0≤β≤1/2, we note that τ=0, which is obtained from the analysis in Remark 1 and Remark 2, and the flocking results in [4] have been improved from Dv0<∫∞ψ2rdr to Dv0<∫∞ψrdr. It means that the communication rate β has been expanded from 1/4 to 1/2. Compared with the work on the flocking results in [5], we have also expanded β from 1/4 to 1/2.



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