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Stability analysis of a swarm model with rooted leadership
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Stability analysis of a swarm model with rooted leadership Chun-Hsien Li Department of Mathematics, National Kaohsiung Normal University, Yanchao District, Kaohsiung City 82444, Taiwan
a r t i c l e
i n f o
Article history: Received 1 July 2018 Received in revised form 4 September 2018 Accepted 5 September 2018 Available online xxxx Communicated by C.R. Doering Keywords: Stability Swarm Cohesion Rooted leadership
a b s t r a c t In this paper, we consider the stability of a swarm model, which is typically associated with the phenomenon of maintaining cohesion. In the swarm model, each individual has its own intrinsic nonlinear dynamics and the interaction between individuals follows a rooted leadership topology. We prove that, if the corresponding real symmetric matrix is negative definite, then the swarm is stable in the sense that all individuals will eventually form a cohesive swarm. In addition, we obtain the bounds of the swarm size. Numerical simulations were conducted to validate the theoretical results. © 2018 Elsevier B.V. All rights reserved.
1. Introduction The emergent behavior of collective motion can be observed in many biological systems and is a truly fascinating natural phenomenon. It is widely believed that one of the key features of collective animal behavior is the way animal groups successfully maintain cohesion in the face of predatory attacks and environmental perturbations. Common examples include the flocking of birds, schooling of fish, swarming of bacteria, and so on [31]. In recent years, two emergent patterns of collective behavior have been the focus of a substantial body of research. The first is the velocity consensus of a moving population of individuals, which is referred to as flocking behavior. The second is whether a population can reach cohesion, that is, when the diameter of the population remains bounded by a fixed constant. This is often referred to as swarming behavior. Due to its practical and theoretical importance, much research has been conducted on flocking and swarming behaviors over the past few decades. For the flocking behavior, a celebrated model is proposed by Cucker and Smale [7]. This model provides a framework to examine the emergent properties of flocks. Later, the Cucker–Smale model has been extensively studied and generalized in several directions, e.g., the stochastic effects [13,17], bonding forces [29], collision avoidance [1,5], and leadership effects [8,12,20,22–25,30]. For the swarming behavior, stability of flock and mill ring solutions for second-order swarming models have been studied in [2,3,19]. A swarming model with collision avoidance has been considered in [6]. The emergence of chaos in social foraging swarms has been investigated in [9,11] and
a novel swarm dynamics with applications in automated multiagent systems is present in [10]. Although an increasing number of mathematical models have been proposed and analyzed for use in studying swarming behavior, in this paper, we are interested in a first-order swarming model. Gazi and Passino introduced a first-order model for analyzing swarm cohesion [14,16]. The dynamics of their model, which is composed of N individuals in an n-dimensional Euclidean space, is described by:
xi (t ) =
N
https://doi.org/10.1016/j.physleta.2018.09.010 0375-9601/© 2018 Elsevier B.V. All rights reserved.
i = 1, 2, · · · , N ,
(1)
where xi ∈ Rn is the position of the ith individual, and the continuous function g : Rn → Rn represents the function of attraction and repulsion between individuals. This function has the form:
g ( y ) = − y ( ga ( y ) − gr ( y )),
(2)
R+
where ga : → R+ represents the magnitude of the attraction + term, gr : R → R+ represents the magnitude of the repulsion
term, and y is the usual Euclidean norm for y ∈ Rn . The function g is assumed to satisfy the condition that attraction dominates over large distances, repulsion dominates over short distances, and there is a distance at which the attraction and repulsion are balanced. A simple example of such a function is as follows:
E-mail address: [email protected].
g (xi (t ) − x j (t )),
j =1, j =i
g ( y ) = − y a − b exp −
y 2 c
,
(3)
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where the positive parameters a, b, and c satisfy b > a. The first term −ay describes the attraction effect and the second term b exp − y 2 /c y represents the repulsion effect. Based on the features of function g, it is shown that all individuals will reach cohesion in a finite time and then derive a bound on the swarm size. It should be noted that the results in [14,16] indicate that all swarm members are stationary whenever they reach cohesion. In other words, the center of cohesion for all individuals is static. However, the center of the individuals may actually change with time. For example, when animals search for food in groups, they may move together toward a region with higher nutrient concentration; thus, the center of the swarm may be time-varying. The process of searching for food is usually associated with the problem of social foraging. In [15], a model for the social foraging of swarms in a profile of nutrients was proposed. In this model, the equation of motion of each individual i is described by
xi (t ) = −∇xi σ (xi ) +
N
g (xi (t ) − x j (t )),
i = 1, 2, · · · , N ,
j =1, j =i
(4) where σ : R → R represents the environment function, which can be a profile of nutrients, or attractant or repellent substances; and the function g represents the mutual interactions between individuals and is of the form of (2). The authors study the stability properties of the collective behavior of the swarm for different profiles, and provide conditions for collective convergence toward more favorable regions of the profile. The effect of communication time delays was considered in [27], and the results show that for the delayed model, swarming behavior occurs under certain conditions. In addition, the swarm was found to diverge when the time delay increased to a certain value. Note that in the aforementioned models, the coupling topologies are globally coupled. Nevertheless, it is well-known that the coupling topology has specific effects on the dynamics of coupled systems. Thus, a more complex coupling topology should be employed when modeling realistic problems. In [26], Liu et al. studied a foraging swarm model given by: n
xi (t ) = −∇xi σ (xi ) +
N
w i j g (xi (t ) − x j (t )),
i = 1, 2, · · · , N ,
j =1, j =i
(5) where the functions σ and g have the same meanings as mentioned above; and the N × N real matrix W = ( w i j ), w i j ≥ 0 for all i = j is the coupling weight matrix representing the topological structure of the system. Based on the detailed balance condition of the weights, there exist some scalars ξi > 0 (i = 1, 2, · · · , N ) such that ξi w i j = ξ j w ji for all i , j. The authors then showed that the individuals will form a cohesive swarm of finite size and will converge to more favorable areas of the profile. In [21], Li considered a more general coupling topology and demonstrated that, if the coupling topology is strongly connected, then the swarm will achieve cohesion in a finite time. Recently, a swarm model with nonlinear profiles was investigated in [32]. The model is governed by the following system of differential equations:
xi (t ) = f (xi (t )) +
N
g (xi (t ) − x j (t )),
i = 1, 2, · · · , N ,
(6)
j =1, j =i
where the nonlinear function f : Rn → Rn defined by f (xi ) = ( f 1 (xi ), f 2 (xi ), · · · , f n (xi )) describes the intrinsic dynamics of
each individual within the swarm and the function g is of the form given in (3). The authors derive a set of conditions that can ensure the occurrence of swarming within a finite time. In addition, the authors consider a swarm model with an attraction function that has a limited sensing range, and found that in this situation, the coupling topology may change from time to time. Even though the effects of the coupling topology are taken into account in the study, the underlying topological structure is still assumed to be strongly connected. However, other results indicate that a leader– follower relationship is present in real biological systems [4,28], and the corresponding coupling topologies may therefore not be strongly connected. Mathematical models of flocking behavior have been developed to study leader–follower relationships. For instance, Shen [30] considered a hierarchical leadership structure in which each individual is influenced only by its superiors in a specified hierarchy. Later, Li and Xue [24,25] studied a more general case, called rooted leadership, which involves a global leader that directly or indirectly influences all other individuals. Various other leadership structures were considered in [8,12,20,22,23]. However, to the best of our knowledge, little research has been done on swarm models with leadership structures. Therefore, the purpose of this work is to study the stability of a swarm model with a rooted leadership structure. By utilizing the Lyapunov function method, we prove that if a real symmetric matrix is negative definite, then the swarm is stable in the sense that all individuals will eventually form a cohesive swarm. Since a hierarchical leadership structure is a special case of a rooted leadership structure, the swarm criterion can be directly applied to swarm models with hierarchical leadership. The remainder of this paper is organized as follows. In Section 2, we propose a swarm model with rooted leadership. In Section 3, based on certain assumptions, we derive the conditions that ensure the occurrence of swarm cohesion. In Section 4, the results of numerical experiments are provided to confirm the theoretical results. Finally, our conclusions are presented in Section 5. 2. Model formulation In this section, we propose a swarm model with a rooted leadership topology. Consider a swarm consisting of N individuals whose motion dynamics are described by the following system of ordinary differential equations:
xi (t ) = f (xi (t )) +
N
w i j g (xi − x j ),
i = 1, 2, · · · , N ,
(7)
j =1, j =i
where xi ∈ Rn is the position of the ith individual; and f : Rn → Rn defined by f (xi ) = ( f 1 (xi ), f 2 (xi ), · · · , f n (xi )) describes the intrinsic dynamics of the ith individual. Throughout this paper, we assume that f satisfies the following Lipschitz condition: Assumption (H1). For all x, y ∈ Rn , there exists a constant θ such that
f (x) − f ( y ) ≤ θx − y . The function g : R → R is a continuous function with linear long-range attraction and nonlinear short-range bounded repulsion. More precisely,
g ( y ) = − y ( ga ( y ) − gr ( y )),
(8)
where ga ( y ) = a > 0 represents the magnitude of the attraction term, and gr ( y ) represents the magnitude of the repulsion term that satisfies the following assumption:
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Assumption (H2). gr ( y ) y ≤ b for any y ∈ Rn and for some finite positive constant b.
Assumption (H3). (7) possesses a rooted leadership structure if there is a rooted individual, say, v 1 , which is not influenced by other individuals, but directly or indirectly influences other individuals. In other words, there exists a directed path from v 1 to any of the other individuals v i for i = 1 in the corresponding diagraph.
0 ⎜ w 21 ⎜ ⎜ w 31
⎜ W =⎜ ⎜ w 41 ⎜ ⎜ . ⎝ .. w N1
0 0 w 32
0 w 23 0
··· ··· ···
0 w 24 w 34
..
.
w 42
w 43
.. .
.. .
···
w N2
w N3
w N4
··· .. . ···
⎞
0 w 2N ⎟ ⎟ w 3N ⎟ w 4N
.. .
⎟ ⎟, ⎟ ⎟ ⎟ ⎠
+
0
j =1, j =i
wij
w i j gr (xi (t ) − x j (t ))(xi (t ) − x j (t )),
j =1, j =i
i = 2, 3, · · · , N .
(10)
3. Swarm cohesion analysis In this section, we study the stability of the swarm model described in (9) in terms of its cohesiveness. To accomplish this, we first define an ( N − 1) × ( N − 1) real symmetric matrix H = (h i j ), where the entries h i j are given by:
1 ≤ i = j ≤ N − 1, θ − adi +1 , a ( w i +1, j +1 + w j +1,i +1 ) 1 ≤ i = j ≤ N − 1. 2
h i j :=
By constructing a suitable Lyapunov function, we can derive the following (note that this is the main result of this paper).
N
xi (t ) − x1 (t )2 ≤
b2 dˆ 2 ( N − 1)
i =2
λ2max ( H )
,
1 e i −1 (t )e i −1 (t ), 2 N
V (t ) =
Taking the derivative of V (t ) along the trajectories of the error system (10) yields
V (t ) =
N i =2
=
w i j gr (xi (t ) − x j (t ))(xi − x j ),
N i =2
(9)
for t ≥ 0.
i =2
+a
e i −1 (t )e i −1 (t )
e i −1 (t ) N
f (xi (t )) − f (x1 (t )) − adi e i −1 (t )
w i j e j −1 (t )
j =2, j =i
We can then write:
di :=
N
+
Proof. Consider the following Lyapunov function candidate:
j =1, j =i
N
and
w i j e j −1 (t )
where λmax ( H ) is the largest eigenvalue of the matrix H .
w i j ( xi − x j )
i = 2, 3, · · · , N .
N j =2, j =i
j =1, j =i N
w i j gr (xi (t ) − x j (t ))(xi (t ) − x j (t ))
= f (xi (t )) − f (x1 (t )) − adi e i −1 (t ) + a
t →∞
x1 (t ) = f (x1 (t )), xi (t ) = f (xi (t )) − a
w i j (xi (t ) − x j (t ))
j =1, j =i
j =1, j =i
lim
where the matrix W is neither symmetric nor irreducible. In addition, the first row indicates that all entries w 1 j = 0 for j ≥ 2 so that the rooted individual, i.e., individual 1, is not influenced by any other member of the swarm. Consequently, we can reformulate (7) as
N
N
+
N
Theorem 3.1. Consider the swarm model described in (9) and assume that Assumptions (H1), (H2), and (H3) hold. If the real symmetric matrix H is negative definite, that is, y H y < 0 for all nonzero vectors y ∈ R N −1 , then the swarm will be stable in the sense that
By Assumption (H3), the coupling matrix W is given by
⎛
e i −1 (t ) = f (xi (t )) − f (x1 (t )) − a
An example of this type of function is gr ( y ) = exp − y 2 /c , where c is a positive constant. One can verify √ that gr ( y ) y is a bounded function with a maximum value c /2 exp(−1/2) that is √ attained when y = c /2. The real N × N matrix W = ( w i j ) is a coupling weight matrix representing the topological structure of the swarm. It follows from (8) that g (xi − x j ) = 0 ∈ Rn for 1 ≤ i = j ≤ N, and hence the value w ii can be arbitrary in (7). In our case, we used w ii = 0 for i = 1, 2, · · · , N. In mathematical terms, the coupling topology of a swarm can be described by a digraph. Each vertex of the digraph represents an individual, and each directed arc between two vertices represents the relationship between the two individuals. Now, suppose that the digraph is composed of a vertex set V = { v 1 , v 2 , · · · , v N } and a directed arc set E ⊆ V × V . Then, the coupling matrix W = ( w i j ) assigns a positive weight w i j for an arc ( v j , v i ) ∈ E . In other words, v j influences v i if and only if w i j > 0. In this paper, we assume that (7) is equipped with the following rooted leadership structure [25].
3
dˆ = max di 2≤ i ≤ N
to simplify the notation. One can verify that di > 0 for all i ≥ 2 since every individual, except individual 1, is influenced by at least one individual. If we define the error vectors e i −1 (t ) = xi (t ) − x1 (t ) for i = 2, 3, · · · , N. Then, we can obtain the following error system from (9):
+
N
w i j gr (xi (t ) − x j (t ))(xi (t ) − x j (t )) ,
j =1, j =i
for t > 0.
(11)
Utilizing Assumption (H1), we have 2 e i −1 (t )( f (xi (t )) − f (x1 (t ))) ≤ θe i −1 (t ) ,
for i = 2, 3, · · · , N . (12)
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From Assumption (H2), one can derive that N i =2
N
e i −1 (t )
≤
N
w i j gr (xi (t ) − x j (t ))(xi (t ) − x j (t ))
j =1, j =i
bdi e i −1 (t ) ≤ bdˆ
N
i =2
e i −1 (t ).
(13)
i =2 Fig. 1. Rooted leadership structure for Example 4.1.
Substituting (12) and (13) into (11), we obtain for t > 0
V (t ) ≤
N i =2
N
+a
j =2, j =i
≤
Remark 3.1. In Theorem 3.1, we require that the real symmetric matrix H is negative definite. This implies that all main diagonal elements of H are negative, i.e., θ − adi +1 < 0 for i = 1, 2, · · · , N − 1. According to Assumption (H3), we can obtain di > 0 for i ≥ 2 and hence, the diagonal elements of H could be negative. Besides, we can see that the parameters θ and a also play important role. Specifically, large values of θ and small values of a could prevent H from being a negative definite matrix. In other words, a small θ and large a favor swarm stability, which is consistent with biological reality.
θe i −1 (t )2 − adi e i −1 (t )2 w i j e i −1 (t )e j −1 (t )
+ bdˆ
N
e i −1 (t )
i =2
N
(θ − adi )e i −1 (t )2
i =2
N
+a
w i j e i −1 (t )e j −1 (t ) + bdˆ
j =2, j =i
N
e i −1 (t )
i =2
≤ e (t ) H e (t ) + bdˆ
N
e i −1 (t ),
(14)
i =2
where e (t ) = (e 1 (t ), e 2 (t ), · · · , e N −1 (t )) . Note that
N
2 e i −1 (t )
=
i =2
N N
e i −1 (t )e j −1 (t )
i =2 j =2
2
N
(e i −1 (t )2 + e j −1 (t )2 )
N
N −1
|hii | >
i =2 j =2
= ( N − 1)
|hi j |,
for all
i = 1, 2, · · · , N − 1,
j =1, j =i
e i −1 (t )2 .
(15)
i =2
Substituting (15) into (14), we can obtain for t > 0
N N V (t ) ≤ λmax ( H ) e i −1 (t )2 + bdˆ ( N − 1) e i −1 (t )2 N = e
There are many ways to determine if a real symmetric matrix is negative definite, one of which is to calculate the eigenvalues and check to see if they are all negative. However, as it is not trivial to compute the eigenvalues, we adopt an alternate method to determine whether matrix H is negative definite. More precisely, according to Corollary 7.2.3 in [18], if H is strictly diagonally dominant, i.e.,
1 N
≤
Remark 3.2. We can see that the bound of the swarm depends on the size N, the repulsive force, and the coupling topology of the swarm.
i =2
i =2
and if h ii < 0 for all i = 1, 2, · · · , N − 1, then H is negative definite. Thus, we have the following criterion: Corollary 3.1. Consider the swarm model described in (9) and assume that Assumptions (H1), (H2), and (H3) hold. If the symmetric matrix H is strictly diagonally dominant and if h ii < 0 for all i = 1, 2, · · · , N − 1, or equivalently,
adi +1 − θ >
2 i −1 (t )
i =2
⎞ N × ⎝λmax ( H ) e i −1 (t )2 + bdˆ ( N − 1)⎠ , ⎛
for all
a 2
N −1
( w i +1, j +1 + w j +1,i +1 ),
j =1, j =i
i = 1, 2, · · · , N − 1,
then the individuals will eventually form a cohesive swarm of finite size.
i =2
which implies that V (t ) < 0 is satisfied as long as
N λmax ( H ) e i −1 (t )2 + bdˆ ( N − 1) < 0.
(16)
i =2
Since matrix H is symmetric negative definite, we know that λmax ( H ) < 0. Therefore, (16) implies that the inequality N i =2
e i −1 (t )2 ≤
b2 dˆ 2 ( N − 1)
As a special case of a rooted leadership structure, hierarchical leadership is an important coupling topology that has been considered in the context of flocking behavior. Here, we study the swarm stability of (9) for the following hierarchical leadership structure [30], which causes some of w i j to disappear. Assumption (H4). (9) possesses the following hierarchical leadership structure: (i) w i j > 0 implies that j < i; and (ii) if the leader set of each individual i is defined by L(i ) := { j | w i j > 0}, then for any i > 1, L(i ) is non-empty.
λ2max ( H )
will be satisfied as t → ∞. This completes the proof.
2
According to Assumption (H4), the coupling matrix W becomes the following lower triangular matrix:
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Fig. 2. Two-dimensional plots of xi (t ), 1 ≤ i ≤ 11 in Example 4.1 at time t = 0, 10, 20, 30, 40, 50. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)
Fig. 3. Phase trajectories of xi (t ), 1 ≤ i ≤ 11 in Example 4.1 with (left) b = 0.8 and (right) b = 1.6. The red circle and red square indicate the initial and final positions of the leader (i.e., individual 1), respectively.
e i −1 (t ) = f (xi (t )) − f (x1 (t )) − adi e i −1 (t ) + a
+
w i j e j −1 (t )
j
w i j gr (xi (t ) − x j (t ))(xi (t ) − x j (t )),
j
i = 2, 3, · · · , N . Fig. 4. Rooted leadership structure for Example 4.2.
⎛
0 ⎜ w 21 ⎜ ⎜ w 31
⎜ W =⎜ ⎜ w 41 ⎜ ⎜ . ⎝ .. w N1
0 0 w 32
0 0 0
w 42
w 43
.. .
.. .
w N2
w N3
0 0 0
.. ..
.
. ···
··· ··· ··· ··· .. . w N , N −1
⎞
0 0⎟ ⎟ 0⎟
⎟ ⎟. 0⎟ ⎟ .. ⎟ .⎠
(17)
We now define a new ( N − 1) × ( N − 1) real symmetric matrix H = ( h i j ), where the entries h i j are defined as:
h i j :=
θ − adi +1 , 1 ≤ i = j ≤ N − 1, a w j +1 , i +1 1 ≤ i < j ≤ N − 1. 2
We can similarly derive the next two results. Corollary 3.2. Consider the swarm model described in (9) and assume that Assumptions (H1), (H2), and (H4) hold. If the symmetric matrix H is negative definite, then the swarm will be stable in the sense that
0
Because L(i ) is non-empty for i = 2, 3, · · · , N, we can still obtain di > 0 for all i ≥ 2, and the error system (10) can be rewritten as
lim
t →∞
N i =2
xi (t ) − x1 (t )2 ≤
b2 dˆ 2 ( N − 1)
λ2max ( H)
.
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Fig. 5. Two-dimensional plots of xi (t ), 1 ≤ i ≤ 11 in Example 4.2 at time t = 0, 10, 20, 30, 40, 50.
Fig. 7. Hierarchical leadership structure for Examples 4.3 and 4.4.
Fig. 6. Phase trajectories of xi (t ), 1 ≤ i ≤ 11 in Example 4.2. The red circle and red square indicate the initial and final positions of the leader (i.e., individual 1), respectively.
≤
N
(θ − adi )e i −1 (t )2
i =2
+a Proof. Consider the same Lyapunov function as defined in Theorem 3.1. Taking the derivative of V (t ) along the trajectories of the error system (17) yields
V (t ) =
N i =2
+a
w i j e j −1 (t )
w i j gr (xi (t ) − x j (t ))(xi (t ) − x j (t )) ,
for t > 0.
N
j
ˆ w i j e i −1 (t )e j −1 (t ) + bd
Corollary 3.3. Consider the swarm model described in (9) and assume that Assumptions (H1), (H2), and (H4) hold. If the symmetric matrix H is strictly diagonally dominant, and if h ii < 0 for all i = 1, 2, · · · , N − 1, or equivalently,
adi +1 − θ >
θe i −1 (t )2 − adi e i −1 (t )2
i =2
e i −1 (t ),
where e (t ) = (e 1 (t ), e 2 (t ), · · · , e N −1 (t )) . By using an argument similar to that in the proof of Theorem 3.1, this claim can be proven. 2
Based on Assumptions (H1) and (H2), we can derive for t > 0
+a
≤ e (t ) H e (t ) + bdˆ
i =2 N
e i −1 (t ) f (xi (t )) − f (x1 (t )) − ad i e i −1 (t )
j
V (t ) ≤
j
i =2
j
+
N w i j e i −1 (t )e j −1 (t ) + bdˆ e i −1 (t )
N i =2
for all
e i −1 (t )
a 2
⎛ ⎞ i −1 N −1 ⎝ w i +1 , j +1 + w j +1 , i +1 ⎠ , j =1
j = i +1
i = 1, 2, · · · , N − 1,
then the individuals will eventually form a cohesive swarm of finite size.
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Fig. 8. Two-dimensional plots of xi (t ), 1 ≤ i ≤ 11 in Example 4.3 at time t = 0, 10, 20, 30, 40, 50.
Fig. 9. Phase trajectories of xi (t ), 1 ≤ i ≤ 11 in Example 4.3 with (left) c = 0.2 and (right) c = 0.8. The red circle and red square indicate the initial and final positions of the leader (i.e., individual 1), respectively.
with b = 0.8. One can verify that λmax ( H ) = −0.0845 < 0, and hence H is negative definite. Note that
4. Numerical experiments In this section, we present the results of several numerical simulations to illustrate the theoretical analysis. In the following examples, we consider swarm systems composed of 11 individuals (i.e., N = 11) in 2D Euclidean space. Example 4.1. We first consider the condition in Theorem 3.1. The corresponding rooted leadership structure is shown in Fig. 1, where the arrow i → j means that ith individual influences jth individual. The positive entries of W are given by w 21 = w 31 = w 41 = 2, w 26 = w 32 = w 52 = w 62 = w 49 = w 73 = w 83 = w 78 = w 94 = w 10,4 = w 11,4 = w 87 = w 11,10 = 1. Let a = 0.5. The nonlinear function f describing the intrinsic dynamics is given by
f (xi ) = (sin(0.1xi1 ), cos(0.1xi2 )) , and the Lipschitz constant is given by θ = 0.1. The repulsive function gr ( y ) is chosen as
gr ( y ) =
b
y
(18)
|h55 | = ad6 − θ = 0.4 <
a 2
⎛ ⎞ 4 10 ⎝ w 6 , j +1 + w j +1,6 ⎠ = 0.5. j =1
j =6
In this example, the symmetric matrix H is not strictly diagonally dominant. Thus, it follows from Theorem 3.1 that (9) will reach cohesion. Two-dimensional (2D) plots of xi (t ) for 1 ≤ i ≤ 11 versus time t = 0, 10, 20, 30, 40, 50 are depicted in Fig. 2. And the trajectory of each individual for 0 ≤ t ≤ 50 is recorded in left panel of Fig. 3. In these figures, we can see that all individuals indeed eventually formed a cohesive swarm. Besides, as we can see in Theorem 3.1, the repulsive term can have a specific effect on the size of the swarm bound. That is, the swarm bound increases as the parameter b increases, which can be observed in Fig. 3. Example 4.2. We now consider the condition in Corollary 3.1. The coupling topology is depicted in Fig. 4, and the positive entries of W are given by w 41 = 3, w 21 = w 31 = w 10,4 = 2, and
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Fig. 10. Two-dimensional plots of xi (t ), 1 ≤ i ≤ 11 in Example 4.4 at time t = 0, 10, 20, 30, 40, 50.
w 32 = w 52 = w 62 = w 73 = w 83 = w 78 = w 94 = w 11,4 = w 87 = w 11,10 = 1. The nonlinear function f is now given by
f (xi ) = (cos(0.1xi1 ), 0.1 tanh(xi2 )) . The Lipschitz constant is therefore θ = 0.1. Let a = 0.5 and the repulsive function gr ( y ) is the same as the one given in (18) with b = 0.5. In such settings, it can be shown that the symmetric matrix H is strictly diagonally dominant and the main diagonal entries are all negative. In fact, we can compute λmax ( H ) = −0.1929 < 0, which implies that H is negative definite. Hence, by Corollary 3.1, (9) will reach cohesion. Two-dimensional plots of xi (t ) for 1 ≤ i ≤ 11 versus time and phase trajectories for 0 ≤ t ≤ 50 are, respectively, plotted in Figs. 5 and 6, in which the swarming behavior can be clearly observed. Example 4.3. In this and the next example, we consider the hierarchical leadership structure. The coupling topology is illustrated in Fig. 7. In the present example, the positive entries of W are w 21 = w 31 = w 41 = w 52 = w 62 = w 73 = w 83 = w 94 = w 10,4 = w 11,4 = 1. The function f is of the form
f (xi ) = (cos(0.1xi1 ), sin(0.1xi2 )) , and so θ = 0.1. Let a = 1, and the repulsive function gr ( y ) is chosen as
gr ( y ) = exp(−
y 2 c
)
(19)
with c = 0.2. It can be verified that the symmetric matrix H is negative definite since λmax ( H ) = −0.0340 < 0. Consequently, by Corollary 3.2, (9) will eventually form a cohesive swarm. Twodimensional plots of xi (t ) for 1 ≤ i ≤ 11 versus time and phase trajectories for 0 ≤ t ≤ 50 are, respectively, depicted in Figs. 8 and 9, in which we can see that swarming behavior indeed occurs. In analogy to Example 4.1 the swarm bound increases as the parameter c increases, which can be seen in Fig. 9.
Fig. 11. Phase trajectories of xi (t ), 1 ≤ i ≤ 11 in Example 4.4. The red circle and red square indicate the initial and final positions of the leader (i.e., individual 1), respectively.
Example 4.4. In the final example, the coupling topology is the same as the one shown in Example 4.3 and the positive entries of W are the same as those in Example 4.3 except for w 21 = w 31 = w 41 = 2. The nonlinear function f is given by
f (xi ) = (sin(0.2xi1 ), cos(0.2xi2 )) , which means θ = 0.2. Let a = 1 and let gr ( y ) be in the form of (19) with c = 0.5. It can be verified that the symmetric matrix H is strictly diagonally dominant. Therefore, by Corollary 3.3, (9) will eventually form a cohesive swarm. Two-dimensional plots of xi (t ) for 1 ≤ i ≤ 11 and 0 ≤ t ≤ 50 are depicted in Figs. 10 and 11 in which the swarming behavior can be observed. 5. Conclusions In this paper, we presented the results of our investigation into the stability properties of a swarm model in which each individual has its own intrinsic nonlinear dynamics and the interaction between individuals within the swarm system follows a rooted leadership structure. In our analysis, we proved that, if a symmetric matrix is negative definite, then the swarm system will
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eventually acquire cohesion and the bound of the swarm can be obtained. Our results can also be applied to hierarchical leadership structures, which is a special case of a rooted leadership topology. We conducted numerical experiments to demonstrate the theoretical results, and showed that swarming behavior indeed eventually occurs. Finally, we conclude this paper with the following remark. Most of the works that conduct research on swarming behaviors mainly focus on homogeneous agents. However, swarming behaviors for heterogeneous agents is not a rare sight, but frequently observed in nature. Thus far, to the best of our knowledge, limited efforts have been made towards studying heterogeneous swarming and we will leave it for our future work.
[11] S. Das, Chaotic patterns in the discrete-time dynamics of social foraging swarms with attractant-repellent profiles: an analysis, Nonlinear Dyn. 82 (2015) 1399–1417.
Acknowledgements
[18] R.A. Horn, C.R. Johnson, Matrix Analysis, second edition, Cambridge University Press, Cambridge, 2013.
The author would like to thank the anonymous referee for his/her helpful comments and suggestions that improved the paper. This work was supported by the Ministry of Science and Technology of Taiwan under the grant MOST 105-2115-M-017-001.
[19] T. Kolokolnikov, H. Sun, D. Uminsky, A.L. Bertozzi, Stability of ring patterns arising from two-dimensional particle interactions, Phys. Rev. E 84 (2011) 015203.
References [1] S. Ahn, H. Choi, S.-Y. Ha, H. Lee, On collision-avoiding initial configurations to Cucker–Smale type flocking models, Commun. Math. Sci. 10 (2012) 625–643. [2] G. Albi, D. Balagué, J.A. Carrillo, J. Von Brecht, Stability analysis of flock and mill rings for second order models in swarming, SIAM J. Appl. Math. 74 (2014) 794–818. [3] J.A. Carrillo, Y. Huang, S. Martin, Nonlinear stability of flock solutions in second-order swarming models, Nonlinear Anal., Real World Appl. 17 (2014) 332–343. [4] I.D. Couzin, J. Krause, N.R. Franks, S. Levin, Effective leadership and decision making in animal groups on the move, Nature 433 (2005) 513–516. [5] F. Cucker, J.-G. Dong, Avoiding collisions in flocks, IEEE Trans. Autom. Control 55 (2010) 1238–1243. [6] F. Cucker, J.-G. Dong, A conditional, collision-avoiding, model for swarming, Discrete Contin. Dyn. Syst. 34 (2014) 1009–1020. [7] F. Cucker, S. Smale, Emergent behavior in flocks, IEEE Trans. Autom. Control 52 (2007) 852–862. [8] F. Dalmao, E. Mordecki, Cucker–Smale flocking under hierarchical leadership and random interactions, SIAM J. Appl. Math. 71 (2011) 1307–1316. [9] S. Das, U. Halder, D. Maity, Chaotic dynamics in social foraging swarms—an analysis, IEEE Trans. Syst. Man Cybern., Part B, Cybern. 42 (2012) 1288–1293. [10] S. Das, D. Goswami, S. Chatterjee, S. Mukherjee, Stability and chaos analysis of a novel swarm dynamics with applications to multi-agent systems, Eng. Appl. Artif. Intell. 30 (2014) 189–198.
[12] J.-G. Dong, Flocking under hierarchical leadership with a free-will leader, Int. J. Robust Nonlinear Control 23 (2013) 1891–1898. [13] R. Erban, J. Haškovec, Y. Sun, A Cucker–Smale model with noise and delay, SIAM J. Appl. Math. 76 (2016) 1535–1557. [14] V. Gazi, K.M. Passino, Stability analysis of swarms, IEEE Trans. Autom. Control 48 (2003) 692–697. [15] V. Gazi, K.M. Passino, Stability analysis of social foraging swarms, IEEE Trans. Syst. Man Cybern., Part B, Cybern. 34 (2004) 539–557. [16] V. Gazi, K.M. Passino, A class of attractions/repulsion functions for stable swarm aggregations, Int. J. Control 77 (2004) 1567–1579. [17] S.-Y. Ha, K. Lee, D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker–Smale system, Commun. Math. Sci. 7 (2009) 453–469.
[20] C.-H. Li, S.-Y. Yang, A new discrete Cucker–Smale flocking model under hierarchical leadership, Discrete Contin. Dyn. Syst., Ser. B 21 (2016) 2587–2599. [21] W. Li, Stability analysis of swarms with general topology, IEEE Trans. Syst. Man Cybern., Part B, Cybern. 38 (2008) 1084–1097. [22] Z. Li, Effectual leadership in flocks with hierarchy and individual preference, Discrete Contin. Dyn. Syst. 34 (2014) 3683–3702. [23] Z. Li, S.-Y. Ha, X. Xue, Emergent phenomena in an ensemble of Cucker–Smale particles under joint rooted leadership, Math. Models Methods Appl. Sci. 24 (2014) 1389–1419. [24] Z. Li, X. Xue, Cucker–Smale flocking under rooted leadership with fixed and switching topologies, SIAM J. Appl. Math. 70 (2010) 3156–3174. [25] Z. Li, X. Xue, Cucker–Smale flocking under rooted leadership with free-will agents, Physica A 410 (2014) 205–217. [26] B. Liu, T. Chu, L. Wang, Z. Wang, Collective motion of a class of social foraging swarms, Chaos Solitons Fractals 38 (2008) 277–292. [27] Q. Liu, L. Wang, Z. Liao, Stability analysis of swarms with interaction time delays, Inf. Sci. 192 (2012) 244–254. [28] M. Nagy, Z. Ákos, D. Biro, T. Vicsek, Hierarchical group dynamics in pigeon flocks, Nature 464 (2007) 890–893. [29] J. Park, H. Kim, S.-Y. Ha, Cucker–Smale flocking with inter-particle bonding forces, IEEE Trans. Autom. Control 55 (2010) 2617–2623. [30] J. Shen, Cucker–Smale flocking under hierarchical leadership, SIAM J. Appl. Math. 68 (2007) 694–719. [31] T. Vicsek, A. Zafeiris, Collective motion, Phys. Rep. 517 (2012) 71–140. [32] W. Yu, G. Chen, M. Cao, J. Lü, H.-T. Zhang, Swarming behaviors in multi-agent systems with nonlinear dynamics, Chaos 23 (2013) 043118.
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Physics Letters A www.elsevier.com/locate/pla
Stability analysis of a swarm model with rooted leadership Chun-Hsien Li Department of Mathematics, National Kaohsiung Normal University, Yanchao District, Kaohsiung City 82444, Taiwan
a r t i c l e
i n f o
Article history: Received 1 July 2018 Received in revised form 4 September 2018 Accepted 5 September 2018 Available online xxxx Communicated by C.R. Doering Keywords: Stability Swarm Cohesion Rooted leadership
a b s t r a c t In this paper, we consider the stability of a swarm model, which is typically associated with the phenomenon of maintaining cohesion. In the swarm model, each individual has its own intrinsic nonlinear dynamics and the interaction between individuals follows a rooted leadership topology. We prove that, if the corresponding real symmetric matrix is negative definite, then the swarm is stable in the sense that all individuals will eventually form a cohesive swarm. In addition, we obtain the bounds of the swarm size. Numerical simulations were conducted to validate the theoretical results. © 2018 Elsevier B.V. All rights reserved.
1. Introduction The emergent behavior of collective motion can be observed in many biological systems and is a truly fascinating natural phenomenon. It is widely believed that one of the key features of collective animal behavior is the way animal groups successfully maintain cohesion in the face of predatory attacks and environmental perturbations. Common examples include the flocking of birds, schooling of fish, swarming of bacteria, and so on [31]. In recent years, two emergent patterns of collective behavior have been the focus of a substantial body of research. The first is the velocity consensus of a moving population of individuals, which is referred to as flocking behavior. The second is whether a population can reach cohesion, that is, when the diameter of the population remains bounded by a fixed constant. This is often referred to as swarming behavior. Due to its practical and theoretical importance, much research has been conducted on flocking and swarming behaviors over the past few decades. For the flocking behavior, a celebrated model is proposed by Cucker and Smale [7]. This model provides a framework to examine the emergent properties of flocks. Later, the Cucker–Smale model has been extensively studied and generalized in several directions, e.g., the stochastic effects [13,17], bonding forces [29], collision avoidance [1,5], and leadership effects [8,12,20,22–25,30]. For the swarming behavior, stability of flock and mill ring solutions for second-order swarming models have been studied in [2,3,19]. A swarming model with collision avoidance has been considered in [6]. The emergence of chaos in social foraging swarms has been investigated in [9,11] and
a novel swarm dynamics with applications in automated multiagent systems is present in [10]. Although an increasing number of mathematical models have been proposed and analyzed for use in studying swarming behavior, in this paper, we are interested in a first-order swarming model. Gazi and Passino introduced a first-order model for analyzing swarm cohesion [14,16]. The dynamics of their model, which is composed of N individuals in an n-dimensional Euclidean space, is described by:
xi (t ) =
N
https://doi.org/10.1016/j.physleta.2018.09.010 0375-9601/© 2018 Elsevier B.V. All rights reserved.
i = 1, 2, · · · , N ,
(1)
where xi ∈ Rn is the position of the ith individual, and the continuous function g : Rn → Rn represents the function of attraction and repulsion between individuals. This function has the form:
g ( y ) = − y ( ga ( y ) − gr ( y )),
(2)
R+
where ga : → R+ represents the magnitude of the attraction + term, gr : R → R+ represents the magnitude of the repulsion
term, and y is the usual Euclidean norm for y ∈ Rn . The function g is assumed to satisfy the condition that attraction dominates over large distances, repulsion dominates over short distances, and there is a distance at which the attraction and repulsion are balanced. A simple example of such a function is as follows:
E-mail address: [email protected].
g (xi (t ) − x j (t )),
j =1, j =i
g ( y ) = − y a − b exp −
y 2 c
,
(3)
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where the positive parameters a, b, and c satisfy b > a. The first term −ay describes the attraction effect and the second term b exp − y 2 /c y represents the repulsion effect. Based on the features of function g, it is shown that all individuals will reach cohesion in a finite time and then derive a bound on the swarm size. It should be noted that the results in [14,16] indicate that all swarm members are stationary whenever they reach cohesion. In other words, the center of cohesion for all individuals is static. However, the center of the individuals may actually change with time. For example, when animals search for food in groups, they may move together toward a region with higher nutrient concentration; thus, the center of the swarm may be time-varying. The process of searching for food is usually associated with the problem of social foraging. In [15], a model for the social foraging of swarms in a profile of nutrients was proposed. In this model, the equation of motion of each individual i is described by
xi (t ) = −∇xi σ (xi ) +
N
g (xi (t ) − x j (t )),
i = 1, 2, · · · , N ,
j =1, j =i
(4) where σ : R → R represents the environment function, which can be a profile of nutrients, or attractant or repellent substances; and the function g represents the mutual interactions between individuals and is of the form of (2). The authors study the stability properties of the collective behavior of the swarm for different profiles, and provide conditions for collective convergence toward more favorable regions of the profile. The effect of communication time delays was considered in [27], and the results show that for the delayed model, swarming behavior occurs under certain conditions. In addition, the swarm was found to diverge when the time delay increased to a certain value. Note that in the aforementioned models, the coupling topologies are globally coupled. Nevertheless, it is well-known that the coupling topology has specific effects on the dynamics of coupled systems. Thus, a more complex coupling topology should be employed when modeling realistic problems. In [26], Liu et al. studied a foraging swarm model given by: n
xi (t ) = −∇xi σ (xi ) +
N
w i j g (xi (t ) − x j (t )),
i = 1, 2, · · · , N ,
j =1, j =i
(5) where the functions σ and g have the same meanings as mentioned above; and the N × N real matrix W = ( w i j ), w i j ≥ 0 for all i = j is the coupling weight matrix representing the topological structure of the system. Based on the detailed balance condition of the weights, there exist some scalars ξi > 0 (i = 1, 2, · · · , N ) such that ξi w i j = ξ j w ji for all i , j. The authors then showed that the individuals will form a cohesive swarm of finite size and will converge to more favorable areas of the profile. In [21], Li considered a more general coupling topology and demonstrated that, if the coupling topology is strongly connected, then the swarm will achieve cohesion in a finite time. Recently, a swarm model with nonlinear profiles was investigated in [32]. The model is governed by the following system of differential equations:
xi (t ) = f (xi (t )) +
N
g (xi (t ) − x j (t )),
i = 1, 2, · · · , N ,
(6)
j =1, j =i
where the nonlinear function f : Rn → Rn defined by f (xi ) = ( f 1 (xi ), f 2 (xi ), · · · , f n (xi )) describes the intrinsic dynamics of
each individual within the swarm and the function g is of the form given in (3). The authors derive a set of conditions that can ensure the occurrence of swarming within a finite time. In addition, the authors consider a swarm model with an attraction function that has a limited sensing range, and found that in this situation, the coupling topology may change from time to time. Even though the effects of the coupling topology are taken into account in the study, the underlying topological structure is still assumed to be strongly connected. However, other results indicate that a leader– follower relationship is present in real biological systems [4,28], and the corresponding coupling topologies may therefore not be strongly connected. Mathematical models of flocking behavior have been developed to study leader–follower relationships. For instance, Shen [30] considered a hierarchical leadership structure in which each individual is influenced only by its superiors in a specified hierarchy. Later, Li and Xue [24,25] studied a more general case, called rooted leadership, which involves a global leader that directly or indirectly influences all other individuals. Various other leadership structures were considered in [8,12,20,22,23]. However, to the best of our knowledge, little research has been done on swarm models with leadership structures. Therefore, the purpose of this work is to study the stability of a swarm model with a rooted leadership structure. By utilizing the Lyapunov function method, we prove that if a real symmetric matrix is negative definite, then the swarm is stable in the sense that all individuals will eventually form a cohesive swarm. Since a hierarchical leadership structure is a special case of a rooted leadership structure, the swarm criterion can be directly applied to swarm models with hierarchical leadership. The remainder of this paper is organized as follows. In Section 2, we propose a swarm model with rooted leadership. In Section 3, based on certain assumptions, we derive the conditions that ensure the occurrence of swarm cohesion. In Section 4, the results of numerical experiments are provided to confirm the theoretical results. Finally, our conclusions are presented in Section 5. 2. Model formulation In this section, we propose a swarm model with a rooted leadership topology. Consider a swarm consisting of N individuals whose motion dynamics are described by the following system of ordinary differential equations:
xi (t ) = f (xi (t )) +
N
w i j g (xi − x j ),
i = 1, 2, · · · , N ,
(7)
j =1, j =i
where xi ∈ Rn is the position of the ith individual; and f : Rn → Rn defined by f (xi ) = ( f 1 (xi ), f 2 (xi ), · · · , f n (xi )) describes the intrinsic dynamics of the ith individual. Throughout this paper, we assume that f satisfies the following Lipschitz condition: Assumption (H1). For all x, y ∈ Rn , there exists a constant θ such that
f (x) − f ( y ) ≤ θx − y . The function g : R → R is a continuous function with linear long-range attraction and nonlinear short-range bounded repulsion. More precisely,
g ( y ) = − y ( ga ( y ) − gr ( y )),
(8)
where ga ( y ) = a > 0 represents the magnitude of the attraction term, and gr ( y ) represents the magnitude of the repulsion term that satisfies the following assumption:
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Assumption (H2). gr ( y ) y ≤ b for any y ∈ Rn and for some finite positive constant b.
Assumption (H3). (7) possesses a rooted leadership structure if there is a rooted individual, say, v 1 , which is not influenced by other individuals, but directly or indirectly influences other individuals. In other words, there exists a directed path from v 1 to any of the other individuals v i for i = 1 in the corresponding diagraph.
0 ⎜ w 21 ⎜ ⎜ w 31
⎜ W =⎜ ⎜ w 41 ⎜ ⎜ . ⎝ .. w N1
0 0 w 32
0 w 23 0
··· ··· ···
0 w 24 w 34
..
.
w 42
w 43
.. .
.. .
···
w N2
w N3
w N4
··· .. . ···
⎞
0 w 2N ⎟ ⎟ w 3N ⎟ w 4N
.. .
⎟ ⎟, ⎟ ⎟ ⎟ ⎠
+
0
j =1, j =i
wij
w i j gr (xi (t ) − x j (t ))(xi (t ) − x j (t )),
j =1, j =i
i = 2, 3, · · · , N .
(10)
3. Swarm cohesion analysis In this section, we study the stability of the swarm model described in (9) in terms of its cohesiveness. To accomplish this, we first define an ( N − 1) × ( N − 1) real symmetric matrix H = (h i j ), where the entries h i j are given by:
1 ≤ i = j ≤ N − 1, θ − adi +1 , a ( w i +1, j +1 + w j +1,i +1 ) 1 ≤ i = j ≤ N − 1. 2
h i j :=
By constructing a suitable Lyapunov function, we can derive the following (note that this is the main result of this paper).
N
xi (t ) − x1 (t )2 ≤
b2 dˆ 2 ( N − 1)
i =2
λ2max ( H )
,
1 e i −1 (t )e i −1 (t ), 2 N
V (t ) =
Taking the derivative of V (t ) along the trajectories of the error system (10) yields
V (t ) =
N i =2
=
w i j gr (xi (t ) − x j (t ))(xi − x j ),
N i =2
(9)
for t ≥ 0.
i =2
+a
e i −1 (t )e i −1 (t )
e i −1 (t ) N
f (xi (t )) − f (x1 (t )) − adi e i −1 (t )
w i j e j −1 (t )
j =2, j =i
We can then write:
di :=
N
+
Proof. Consider the following Lyapunov function candidate:
j =1, j =i
N
and
w i j e j −1 (t )
where λmax ( H ) is the largest eigenvalue of the matrix H .
w i j ( xi − x j )
i = 2, 3, · · · , N .
N j =2, j =i
j =1, j =i N
w i j gr (xi (t ) − x j (t ))(xi (t ) − x j (t ))
= f (xi (t )) − f (x1 (t )) − adi e i −1 (t ) + a
t →∞
x1 (t ) = f (x1 (t )), xi (t ) = f (xi (t )) − a
w i j (xi (t ) − x j (t ))
j =1, j =i
j =1, j =i
lim
where the matrix W is neither symmetric nor irreducible. In addition, the first row indicates that all entries w 1 j = 0 for j ≥ 2 so that the rooted individual, i.e., individual 1, is not influenced by any other member of the swarm. Consequently, we can reformulate (7) as
N
N
+
N
Theorem 3.1. Consider the swarm model described in (9) and assume that Assumptions (H1), (H2), and (H3) hold. If the real symmetric matrix H is negative definite, that is, y H y < 0 for all nonzero vectors y ∈ R N −1 , then the swarm will be stable in the sense that
By Assumption (H3), the coupling matrix W is given by
⎛
e i −1 (t ) = f (xi (t )) − f (x1 (t )) − a
An example of this type of function is gr ( y ) = exp − y 2 /c , where c is a positive constant. One can verify √ that gr ( y ) y is a bounded function with a maximum value c /2 exp(−1/2) that is √ attained when y = c /2. The real N × N matrix W = ( w i j ) is a coupling weight matrix representing the topological structure of the swarm. It follows from (8) that g (xi − x j ) = 0 ∈ Rn for 1 ≤ i = j ≤ N, and hence the value w ii can be arbitrary in (7). In our case, we used w ii = 0 for i = 1, 2, · · · , N. In mathematical terms, the coupling topology of a swarm can be described by a digraph. Each vertex of the digraph represents an individual, and each directed arc between two vertices represents the relationship between the two individuals. Now, suppose that the digraph is composed of a vertex set V = { v 1 , v 2 , · · · , v N } and a directed arc set E ⊆ V × V . Then, the coupling matrix W = ( w i j ) assigns a positive weight w i j for an arc ( v j , v i ) ∈ E . In other words, v j influences v i if and only if w i j > 0. In this paper, we assume that (7) is equipped with the following rooted leadership structure [25].
3
dˆ = max di 2≤ i ≤ N
to simplify the notation. One can verify that di > 0 for all i ≥ 2 since every individual, except individual 1, is influenced by at least one individual. If we define the error vectors e i −1 (t ) = xi (t ) − x1 (t ) for i = 2, 3, · · · , N. Then, we can obtain the following error system from (9):
+
N
w i j gr (xi (t ) − x j (t ))(xi (t ) − x j (t )) ,
j =1, j =i
for t > 0.
(11)
Utilizing Assumption (H1), we have 2 e i −1 (t )( f (xi (t )) − f (x1 (t ))) ≤ θe i −1 (t ) ,
for i = 2, 3, · · · , N . (12)
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From Assumption (H2), one can derive that N i =2
N
e i −1 (t )
≤
N
w i j gr (xi (t ) − x j (t ))(xi (t ) − x j (t ))
j =1, j =i
bdi e i −1 (t ) ≤ bdˆ
N
i =2
e i −1 (t ).
(13)
i =2 Fig. 1. Rooted leadership structure for Example 4.1.
Substituting (12) and (13) into (11), we obtain for t > 0
V (t ) ≤
N i =2
N
+a
j =2, j =i
≤
Remark 3.1. In Theorem 3.1, we require that the real symmetric matrix H is negative definite. This implies that all main diagonal elements of H are negative, i.e., θ − adi +1 < 0 for i = 1, 2, · · · , N − 1. According to Assumption (H3), we can obtain di > 0 for i ≥ 2 and hence, the diagonal elements of H could be negative. Besides, we can see that the parameters θ and a also play important role. Specifically, large values of θ and small values of a could prevent H from being a negative definite matrix. In other words, a small θ and large a favor swarm stability, which is consistent with biological reality.
θe i −1 (t )2 − adi e i −1 (t )2 w i j e i −1 (t )e j −1 (t )
+ bdˆ
N
e i −1 (t )
i =2
N
(θ − adi )e i −1 (t )2
i =2
N
+a
w i j e i −1 (t )e j −1 (t ) + bdˆ
j =2, j =i
N
e i −1 (t )
i =2
≤ e (t ) H e (t ) + bdˆ
N
e i −1 (t ),
(14)
i =2
where e (t ) = (e 1 (t ), e 2 (t ), · · · , e N −1 (t )) . Note that
N
2 e i −1 (t )
=
i =2
N N
e i −1 (t )e j −1 (t )
i =2 j =2
2
N
(e i −1 (t )2 + e j −1 (t )2 )
N
N −1
|hii | >
i =2 j =2
= ( N − 1)
|hi j |,
for all
i = 1, 2, · · · , N − 1,
j =1, j =i
e i −1 (t )2 .
(15)
i =2
Substituting (15) into (14), we can obtain for t > 0
N N V (t ) ≤ λmax ( H ) e i −1 (t )2 + bdˆ ( N − 1) e i −1 (t )2 N = e
There are many ways to determine if a real symmetric matrix is negative definite, one of which is to calculate the eigenvalues and check to see if they are all negative. However, as it is not trivial to compute the eigenvalues, we adopt an alternate method to determine whether matrix H is negative definite. More precisely, according to Corollary 7.2.3 in [18], if H is strictly diagonally dominant, i.e.,
1 N
≤
Remark 3.2. We can see that the bound of the swarm depends on the size N, the repulsive force, and the coupling topology of the swarm.
i =2
i =2
and if h ii < 0 for all i = 1, 2, · · · , N − 1, then H is negative definite. Thus, we have the following criterion: Corollary 3.1. Consider the swarm model described in (9) and assume that Assumptions (H1), (H2), and (H3) hold. If the symmetric matrix H is strictly diagonally dominant and if h ii < 0 for all i = 1, 2, · · · , N − 1, or equivalently,
adi +1 − θ >
2 i −1 (t )
i =2
⎞ N × ⎝λmax ( H ) e i −1 (t )2 + bdˆ ( N − 1)⎠ , ⎛
for all
a 2
N −1
( w i +1, j +1 + w j +1,i +1 ),
j =1, j =i
i = 1, 2, · · · , N − 1,
then the individuals will eventually form a cohesive swarm of finite size.
i =2
which implies that V (t ) < 0 is satisfied as long as
N λmax ( H ) e i −1 (t )2 + bdˆ ( N − 1) < 0.
(16)
i =2
Since matrix H is symmetric negative definite, we know that λmax ( H ) < 0. Therefore, (16) implies that the inequality N i =2
e i −1 (t )2 ≤
b2 dˆ 2 ( N − 1)
As a special case of a rooted leadership structure, hierarchical leadership is an important coupling topology that has been considered in the context of flocking behavior. Here, we study the swarm stability of (9) for the following hierarchical leadership structure [30], which causes some of w i j to disappear. Assumption (H4). (9) possesses the following hierarchical leadership structure: (i) w i j > 0 implies that j < i; and (ii) if the leader set of each individual i is defined by L(i ) := { j | w i j > 0}, then for any i > 1, L(i ) is non-empty.
λ2max ( H )
will be satisfied as t → ∞. This completes the proof.
2
According to Assumption (H4), the coupling matrix W becomes the following lower triangular matrix:
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Fig. 2. Two-dimensional plots of xi (t ), 1 ≤ i ≤ 11 in Example 4.1 at time t = 0, 10, 20, 30, 40, 50. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)
Fig. 3. Phase trajectories of xi (t ), 1 ≤ i ≤ 11 in Example 4.1 with (left) b = 0.8 and (right) b = 1.6. The red circle and red square indicate the initial and final positions of the leader (i.e., individual 1), respectively.
e i −1 (t ) = f (xi (t )) − f (x1 (t )) − adi e i −1 (t ) + a
+
w i j e j −1 (t )
j
w i j gr (xi (t ) − x j (t ))(xi (t ) − x j (t )),
j
i = 2, 3, · · · , N . Fig. 4. Rooted leadership structure for Example 4.2.
⎛
0 ⎜ w 21 ⎜ ⎜ w 31
⎜ W =⎜ ⎜ w 41 ⎜ ⎜ . ⎝ .. w N1
0 0 w 32
0 0 0
w 42
w 43
.. .
.. .
w N2
w N3
0 0 0
.. ..
.
. ···
··· ··· ··· ··· .. . w N , N −1
⎞
0 0⎟ ⎟ 0⎟
⎟ ⎟. 0⎟ ⎟ .. ⎟ .⎠
(17)
We now define a new ( N − 1) × ( N − 1) real symmetric matrix H = ( h i j ), where the entries h i j are defined as:
h i j :=
θ − adi +1 , 1 ≤ i = j ≤ N − 1, a w j +1 , i +1 1 ≤ i < j ≤ N − 1. 2
We can similarly derive the next two results. Corollary 3.2. Consider the swarm model described in (9) and assume that Assumptions (H1), (H2), and (H4) hold. If the symmetric matrix H is negative definite, then the swarm will be stable in the sense that
0
Because L(i ) is non-empty for i = 2, 3, · · · , N, we can still obtain di > 0 for all i ≥ 2, and the error system (10) can be rewritten as
lim
t →∞
N i =2
xi (t ) − x1 (t )2 ≤
b2 dˆ 2 ( N − 1)
λ2max ( H)
.
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Fig. 5. Two-dimensional plots of xi (t ), 1 ≤ i ≤ 11 in Example 4.2 at time t = 0, 10, 20, 30, 40, 50.
Fig. 7. Hierarchical leadership structure for Examples 4.3 and 4.4.
Fig. 6. Phase trajectories of xi (t ), 1 ≤ i ≤ 11 in Example 4.2. The red circle and red square indicate the initial and final positions of the leader (i.e., individual 1), respectively.
≤
N
(θ − adi )e i −1 (t )2
i =2
+a Proof. Consider the same Lyapunov function as defined in Theorem 3.1. Taking the derivative of V (t ) along the trajectories of the error system (17) yields
V (t ) =
N i =2
+a
w i j e j −1 (t )
w i j gr (xi (t ) − x j (t ))(xi (t ) − x j (t )) ,
for t > 0.
N
j
ˆ w i j e i −1 (t )e j −1 (t ) + bd
Corollary 3.3. Consider the swarm model described in (9) and assume that Assumptions (H1), (H2), and (H4) hold. If the symmetric matrix H is strictly diagonally dominant, and if h ii < 0 for all i = 1, 2, · · · , N − 1, or equivalently,
adi +1 − θ >
θe i −1 (t )2 − adi e i −1 (t )2
i =2
e i −1 (t ),
where e (t ) = (e 1 (t ), e 2 (t ), · · · , e N −1 (t )) . By using an argument similar to that in the proof of Theorem 3.1, this claim can be proven. 2
Based on Assumptions (H1) and (H2), we can derive for t > 0
+a
≤ e (t ) H e (t ) + bdˆ
i =2 N
e i −1 (t ) f (xi (t )) − f (x1 (t )) − ad i e i −1 (t )
j
V (t ) ≤
j
i =2
j
+
N w i j e i −1 (t )e j −1 (t ) + bdˆ e i −1 (t )
N i =2
for all
e i −1 (t )
a 2
⎛ ⎞ i −1 N −1 ⎝ w i +1 , j +1 + w j +1 , i +1 ⎠ , j =1
j = i +1
i = 1, 2, · · · , N − 1,
then the individuals will eventually form a cohesive swarm of finite size.
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Fig. 8. Two-dimensional plots of xi (t ), 1 ≤ i ≤ 11 in Example 4.3 at time t = 0, 10, 20, 30, 40, 50.
Fig. 9. Phase trajectories of xi (t ), 1 ≤ i ≤ 11 in Example 4.3 with (left) c = 0.2 and (right) c = 0.8. The red circle and red square indicate the initial and final positions of the leader (i.e., individual 1), respectively.
with b = 0.8. One can verify that λmax ( H ) = −0.0845 < 0, and hence H is negative definite. Note that
4. Numerical experiments In this section, we present the results of several numerical simulations to illustrate the theoretical analysis. In the following examples, we consider swarm systems composed of 11 individuals (i.e., N = 11) in 2D Euclidean space. Example 4.1. We first consider the condition in Theorem 3.1. The corresponding rooted leadership structure is shown in Fig. 1, where the arrow i → j means that ith individual influences jth individual. The positive entries of W are given by w 21 = w 31 = w 41 = 2, w 26 = w 32 = w 52 = w 62 = w 49 = w 73 = w 83 = w 78 = w 94 = w 10,4 = w 11,4 = w 87 = w 11,10 = 1. Let a = 0.5. The nonlinear function f describing the intrinsic dynamics is given by
f (xi ) = (sin(0.1xi1 ), cos(0.1xi2 )) , and the Lipschitz constant is given by θ = 0.1. The repulsive function gr ( y ) is chosen as
gr ( y ) =
b
y
(18)
|h55 | = ad6 − θ = 0.4 <
a 2
⎛ ⎞ 4 10 ⎝ w 6 , j +1 + w j +1,6 ⎠ = 0.5. j =1
j =6
In this example, the symmetric matrix H is not strictly diagonally dominant. Thus, it follows from Theorem 3.1 that (9) will reach cohesion. Two-dimensional (2D) plots of xi (t ) for 1 ≤ i ≤ 11 versus time t = 0, 10, 20, 30, 40, 50 are depicted in Fig. 2. And the trajectory of each individual for 0 ≤ t ≤ 50 is recorded in left panel of Fig. 3. In these figures, we can see that all individuals indeed eventually formed a cohesive swarm. Besides, as we can see in Theorem 3.1, the repulsive term can have a specific effect on the size of the swarm bound. That is, the swarm bound increases as the parameter b increases, which can be observed in Fig. 3. Example 4.2. We now consider the condition in Corollary 3.1. The coupling topology is depicted in Fig. 4, and the positive entries of W are given by w 41 = 3, w 21 = w 31 = w 10,4 = 2, and
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Fig. 10. Two-dimensional plots of xi (t ), 1 ≤ i ≤ 11 in Example 4.4 at time t = 0, 10, 20, 30, 40, 50.
w 32 = w 52 = w 62 = w 73 = w 83 = w 78 = w 94 = w 11,4 = w 87 = w 11,10 = 1. The nonlinear function f is now given by
f (xi ) = (cos(0.1xi1 ), 0.1 tanh(xi2 )) . The Lipschitz constant is therefore θ = 0.1. Let a = 0.5 and the repulsive function gr ( y ) is the same as the one given in (18) with b = 0.5. In such settings, it can be shown that the symmetric matrix H is strictly diagonally dominant and the main diagonal entries are all negative. In fact, we can compute λmax ( H ) = −0.1929 < 0, which implies that H is negative definite. Hence, by Corollary 3.1, (9) will reach cohesion. Two-dimensional plots of xi (t ) for 1 ≤ i ≤ 11 versus time and phase trajectories for 0 ≤ t ≤ 50 are, respectively, plotted in Figs. 5 and 6, in which the swarming behavior can be clearly observed. Example 4.3. In this and the next example, we consider the hierarchical leadership structure. The coupling topology is illustrated in Fig. 7. In the present example, the positive entries of W are w 21 = w 31 = w 41 = w 52 = w 62 = w 73 = w 83 = w 94 = w 10,4 = w 11,4 = 1. The function f is of the form
f (xi ) = (cos(0.1xi1 ), sin(0.1xi2 )) , and so θ = 0.1. Let a = 1, and the repulsive function gr ( y ) is chosen as
gr ( y ) = exp(−
y 2 c
)
(19)
with c = 0.2. It can be verified that the symmetric matrix H is negative definite since λmax ( H ) = −0.0340 < 0. Consequently, by Corollary 3.2, (9) will eventually form a cohesive swarm. Twodimensional plots of xi (t ) for 1 ≤ i ≤ 11 versus time and phase trajectories for 0 ≤ t ≤ 50 are, respectively, depicted in Figs. 8 and 9, in which we can see that swarming behavior indeed occurs. In analogy to Example 4.1 the swarm bound increases as the parameter c increases, which can be seen in Fig. 9.
Fig. 11. Phase trajectories of xi (t ), 1 ≤ i ≤ 11 in Example 4.4. The red circle and red square indicate the initial and final positions of the leader (i.e., individual 1), respectively.
Example 4.4. In the final example, the coupling topology is the same as the one shown in Example 4.3 and the positive entries of W are the same as those in Example 4.3 except for w 21 = w 31 = w 41 = 2. The nonlinear function f is given by
f (xi ) = (sin(0.2xi1 ), cos(0.2xi2 )) , which means θ = 0.2. Let a = 1 and let gr ( y ) be in the form of (19) with c = 0.5. It can be verified that the symmetric matrix H is strictly diagonally dominant. Therefore, by Corollary 3.3, (9) will eventually form a cohesive swarm. Two-dimensional plots of xi (t ) for 1 ≤ i ≤ 11 and 0 ≤ t ≤ 50 are depicted in Figs. 10 and 11 in which the swarming behavior can be observed. 5. Conclusions In this paper, we presented the results of our investigation into the stability properties of a swarm model in which each individual has its own intrinsic nonlinear dynamics and the interaction between individuals within the swarm system follows a rooted leadership structure. In our analysis, we proved that, if a symmetric matrix is negative definite, then the swarm system will
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eventually acquire cohesion and the bound of the swarm can be obtained. Our results can also be applied to hierarchical leadership structures, which is a special case of a rooted leadership topology. We conducted numerical experiments to demonstrate the theoretical results, and showed that swarming behavior indeed eventually occurs. Finally, we conclude this paper with the following remark. Most of the works that conduct research on swarming behaviors mainly focus on homogeneous agents. However, swarming behaviors for heterogeneous agents is not a rare sight, but frequently observed in nature. Thus far, to the best of our knowledge, limited efforts have been made towards studying heterogeneous swarming and we will leave it for our future work.
[11] S. Das, Chaotic patterns in the discrete-time dynamics of social foraging swarms with attractant-repellent profiles: an analysis, Nonlinear Dyn. 82 (2015) 1399–1417.
Acknowledgements
[18] R.A. Horn, C.R. Johnson, Matrix Analysis, second edition, Cambridge University Press, Cambridge, 2013.
The author would like to thank the anonymous referee for his/her helpful comments and suggestions that improved the paper. This work was supported by the Ministry of Science and Technology of Taiwan under the grant MOST 105-2115-M-017-001.
[19] T. Kolokolnikov, H. Sun, D. Uminsky, A.L. Bertozzi, Stability of ring patterns arising from two-dimensional particle interactions, Phys. Rev. E 84 (2011) 015203.
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