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 rate that allows a flock to emerge has been expanded from 1/4 to 1/2.
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 aswhere and is a positive integer, indicates the intensity of the influence between particles, and represents the time lag, which includes the response time of the particle and the communication time between particles and . The communication function can be defined aswhich further satisfies for all , where , . Further, system (1) can be simplified towhere . The initial conditions arewhere and 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 is improved to , that is, the communication rate is expanded from 1/4 to 1/2. (2) Note that with in (1), the communication rate 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 ( and ) are used to analyze the evolution of the aggregation behavior of systems (3) and (4), for :
Thus, for both and , it follows from a few simple calculations that a uniform result can be directly verified about the estimation of influence function as
We still adopt the definition of time-asymptotic flocking proposed in [4].
Definition 1. Let be a solution to systems (3) and (4); a time-asymptotic flocking can be achieved if and only if the solution satisfies and , where and are given in (5).
2. Main Results
This section proposes two different sufficient conditions for systems (3) and (4) with or 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]). Let be the solution of the linear functional differential equation, . If , then
Lemma 2. Let be a solution to systems (3) and (4); then, we havewhere and are defined in (5).
Proof. Making use of system (3) yieldsUsing inequality (6) and the normalization assumptions for communication functions, that is, , we getand this proof is completed.
Lemma 3. Let be a solution of systems (3) and (4); then, the upper Dini derivative of and satisfies
Proof. Without loss of generality, let at time , where . One can obtainSimilarly, without loss of generality, let satisfy at time , where ; it follows from Lemma 2 thatTherefore, (11) is proven.
To establish the flocking conditions, we define a set containing all the initial configurations of asymptotic flocking allowed, that is,where and are shown in (5) and is defined in [2, 3].
Theorem 1. For , suppose that the initial conditions (4) are selected from the set ; then systems (3) and (4) with or can complete the conditional flocking.
Proof. Inspired by the work in [5, 9], we take the following Lyapunov function:Thus, the upper Dini derivative of along with respect to is shown below.Furthermore, combining with Lemma 3, we have , which means that is nonincreasing and then for all . Thus,Since the initial conditions (4) are selected from the set , it follows from (16) thatDue to the fact that has a divergent tail, there must be a constant such that for . Considering inequality (6) yieldsUsing the second inequality in (11) in Lemma 3, we can further derive thatMaking use of Lemma 1, we can show that as 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. For , suppose that the initial configurations (4) are met as follows:and the processing delay satisfieswhere and are defined in (5); then, systems (3) and (4) with or converge to a flock.
Proof. We only need to prove that the initial conditions which satisfy (22) all exist in the set defined in (8). Note thatwhich means that the initial conditions which satisfy (22) all exist in set . 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 with , Theorem 1 and Theorem 2 will degenerate into . If , then the flocking condition is further written as , thereby improving Theorem 3.1 in [4].(2)Comparing Theorem 1 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 .
2.2. Unconditional Flocking
The following theorem describes the implementation of unconditional flocking for systems (3) and (4).
Theorem 3. For , systems (3) and (4) with or converge unconditionally to a flock.
Proof. If , then the interparticle communication function has a tail dissipation . 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 Theorem 3 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:(2)Note that with , the unconditional flocking result in [4] is improved to , which means that the communication rate is expanded from 1/4 to 1/2.(3)Compared with Theorem 3.1 in [5], the range of communication rate has been expanded. Specifically, we promote the results from to , that is, we extend the communication rate 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 . The parameters in (3) are fixed as , and further we have 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. (see Figure 1).
It can be seen from Figure 1 that without a time delay, the particle swarm can converge asymptotically to form a flock under the fixed initial configurations and the above parameters.
Example 2. (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 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 , which satisfies (22).
Example 3. (see Figure 3).
As shown in Figure 3, under the same initial condition with Example 2, this group cannot converge to a flock, and its fatal factor is that , 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 that . It may be selected as and then combined with the discussion in Theorem 3; the system unconditionally converges to a flock. The simulation results are shown in Figure 4.
4. Conclusions
We study the emergence conditions of flocking of multiple particle swarms with processing delays, establish two sufficient conditions for conditional flocking in Theorem 1 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 , we note that , which is obtained from the analysis in Remark 1 and Remark 2, and the flocking results in [4] have been improved from to . It means that the communication rate has been expanded from 1/4 to 1/2. Compared with the work on the flocking results in [5], we have also expanded from 1/4 to 1/2.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study was supported by the National Natural Science Foundation of China (11401577 and 11671011).