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FLOCKING AND INVARIANCE OF VELOCITY ANGLES

Motsch and Tadmor considered an extended Cucker-Smale model to investigate the flocking behavior of self-organized systems of interacting species. In this extended model, a cone of the vision was introduced so that outside the cone the influence of one agent on the other is lost and hence the corresponding influence function takes the value zero. This creates a problem to apply the Motsch-Tadmor and Cucker-Smale method to prove the flocking property of the system. Here, we examine the variation of the velocity angles between two arbitrary agents, and obtain a monotonicity property for the maximum cone of velocity angles. This monotonicity permits us to utilize existing arguments to show the flocking property of the system under consideration, when the initial velocity angles satisfy some minor technical constraints.


1.
Introduction. Flocking, where agents in a network adapt by their relative locations to achieve an uniform velocity, is a universal phenomenon in biological, social and economical systems. Examples include bird migration, fish schooling [12,13,4,3,2,8] and emergent economic behavior including common belief in a price system in a complex market environment [5,3].
Reynolds [10] gave three simple rules for flocking: Separation-avoid crowding neighbors (short range repulsion); Alignment-steer towards average heading of neighbors; Cohesion-steer towards average position of neighbors (long range attraction). Later, Vicsek [14] characterized flocking in terms of bounded distanceindividuals stay at bounded distance from each other; and alignment-they all move in the same direction. There were substantial researches including modelling studies on flocking, but with models seemingly too complex to analyze. In 2007, Cucker and Smale [4,3] developed a model, referred to as CS model [4,3] here, that provides a basic framework to describe how agents interact with each other in order to achieve flocking and this model has inspired very intensive activities to explain self-organized behavior in various complex systems. See [1,9,11,6] and references therein.
The CS model describes how agents interact with each other following the simple rule below [10]: where x i ∈ R d and v i ∈ R d are the location and velocity of the agent "i". In the model, α is a positive constant, and φ ij quantifies the pairwise influence of agent "j" on the alignment of agent "i" as a function of their distance. More precisely, in the CS model, we have where φ is given above or, in general, is a strictly positive decreasing function, and β is a parameter. This influence function is symmetric, that is, agent "i" and agent "j" have the same influence on the alignment of each other (φ ij = φ ji ). Motsch and Tadmor [9] later introduced an influence function, which is non-symmetric and takes into account the relative distance between agents, as follows with φ as defined in the CS model. In their celebrated work [9], Motsch and Tadmor also called attention to a more general situation in which signal transmission is via vision. In this configuration, it is possible that agent "i" can see agent "j", but agent "j" may fail to see agent "i" outside a cone of vision.
Here we show that in this revised CS-model and MT-model, flocking is still achieved. We provide a proof based on the "cone invariance" which implies that self-organization does keep all agents within the cone of version and hence the influence remains once initiated. We will formulate the Motsch-Tadmor (MT) model in Section 2, and then establish the cone invariance and flocking in dimension 2 (section 3) and dimension 3 (Section 4). 2. The model and some preliminaries. We consider a self-organized group with N agents. For agent "i", its position is denoted by x i ∈ R d and its velocity by v i ∈ R d , where d > 1 is an integer. Motsch and Tadmor proposed the revised CS and/or MT model which is incorporating a cone of vision [9]: Here, α(α > 0) measures the interaction strength, κ(ω i , x j −x i ) determines whether the agent "j" can be "seen" by the agent "i" who heads in direction ω i := v i / v i : with 2γ being the angle of the cone of vision. The φ ij s determine the pairwise alignment within the cone of vision [9], and is defined similarly to the CS or MT model. Namely, either Here the influence function φ is defined as above.
We can rewrite the model as follows: In the CS model, we let a ii = 1 − j∈Ni a ij and Then, be a solution of system (3), and let d X (t) and d V (t) denote its diameters in position and velocity phase space, which are given by d then we say system (3) converges to a flock. If the above property holds for a particular solution, then we say the solution flocks.
3. Flocking behavior in 2-D spaces. When agents under consideration are animals moving on the land such as wolf packs and elephant herds, we can consider the system posed in a 2-dimensional spaces. In this section, we consider the case where x i ∈ R 2 , v i ∈ R 2 for a solution of the system (3).
For any given t, there exist two agents i 0 = i 0 (t) and j 0 = j 0 (t)" such that where Here and in what follows, θ i0j0 (t) means to fix i 0 and j 0 and take the derivative of the function θ i0j0 (t) with respect to t. In other words, θ i0j0 (t) = θ i,j (t) i=i0,j=j0 .
In the case of the CS-model, we have Hence, we get ). Then In what follows, we try to show that θ(t) ≤ θ(0) for all t by proving that Recall that for any given t, We prove this claim by contradiction. Note that for a given k, we have only two cases: either Case 1: Using the continuity of θ ij (t), we conclude that there must be the first t * Consequently, there must be the first t 1 > 0 and integers i 1 ∈ N and j 1 ∈ N such that θ i1k (t 1 ) + θ kj1 (t 1 ) + θ i1j1 (t 1 ) = 2π.
Then, for any given t, there exists i * ∈ N and j * ∈ N such that θ(t) = θ i * j * (t) and that θ i * j * (t) = θ i * k (t) + θ kj * (t) for every k ∈ N . Using the same argument as above, we conclude that θ i * j * (t) ≤ 0. Now we consider D + θ(t). For any given time t, there exist finitely many i × j ∈ I × J = {i × j|θ ij (t) = θ(t), i ∈ N, j ∈ N } , infinitely h n with h n ≥ 0, lim n→∞ h n = 0 and θ(t + h n ) = θ ij (t + h n ). Then we can find a subsequence {h mn } of {h n } ∞ n=1 and a fixed i * × j * ∈ I × J such that θ(t + h mn ) = θ i * j * (t + h mn ) holds. Therefore, This concludes that θ(t) ≤ θ(0) for all t > 0, completing the proof.
Lemma 3.1 ensures that monotonicity of the maximal cone of vision. As will be shown below, this ensures that agents will eventually move towards the same direction. To describe our arguments, for the ith-agent's position x i ∈ R 2 and its velocity v i ∈ R 2 , we introduce the rectangular coordinates: (6). Letθ(0) = θ ij (0), and assume θ ij (0) = θ ik (0) + θ kj (0) for every other agent k ∈ N . Ifθ(0) < 2γ − π with π 2 ≤ γ < π, then we have Proof. By Lemma 3.1, for any t ≥ 0, we havē This implies that any agent "i" and agent "j" satisfy one of the following three situations When n = 1, we can choose the agents p and q which satisfy d V 1 = v 1p − v 1q for any given t. We now consider φ ij defined in the MT model [9].
If case 1) occurs, we have a pq = 0 and a qp = 0. Then If case 2) occurs, then we have a pq = 0 and a qp = 0, and . For the CS model, using a similar argument as that for MT model above, we have d dt When φ ij is defined in the CS model, we have a ij = for any any pair (i, j). Thus, we have
Proof. From Lemma 3.1 and Lemma 3.2 we have x l for some k and l. Then, The argument below is similar to that in [9]. Namely, we introduce an energy Ψ(r)dr is nonincreasing along the pathway (d X (t), d V 1 (t), d V 2 (t)), and we deduce (7) can be rewritten as Obviously, we have d X(t) ≤ d * for all t ≥ 0. As φ(r) is decreasing on (0, +∞), we obtain By using the Gronwall's inequality, we easily get lim t→∞ d V n (t) = 0, n = 1, 2.
4. Flocking behavior in 3-D spaces and remarks. Some of the arguments can be adopted to the case of phase spaces, as we outlined below. So, we now consider agent i, with its position x i ∈ R 3 and its velocity v i ∈ R 3 .
, v j (t) > as the angle of velocity v i and v j and defineθ(t) = max Then we have θ(t) ≤θ(0), for any t > 0.
Proof. We consider the case where the initial condition satisfies i), and a similar argument applies to case ii). From Lemma 4.1 and Lemma 4.2, we have