
This work formulates the problem of defining a model for opinion dynamics on a general compact Riemannian manifold. Two approaches to modeling opinions on a manifold are explored. The first defines the distance between two points using the projection in the ambient Euclidean space. The second approach defines the distance as the length of the geodesic between two agents. Our analysis focuses on features such as equilibria, the long term behavior, and the energy of the system, as well as the interactions between agents that lead to these features. Simulations for specific manifolds, S1,S2, and T2 , accompany the analysis. Trajectories given by opinion dynamics may resemble n− body Choreography and are called "social choreography". Conditions leading to various types of social choreography are investigated in R2 .
Citation: Aylin Aydoğdu, Sean T. McQuade, Nastassia Pouradier Duteil. Opinion Dynamics on a General Compact Riemannian Manifold[J]. Networks and Heterogeneous Media, 2017, 12(3): 489-523. doi: 10.3934/nhm.2017021
[1] | Aylin Aydoğdu, Sean T. McQuade, Nastassia Pouradier Duteil . Opinion Dynamics on a General Compact Riemannian Manifold. Networks and Heterogeneous Media, 2017, 12(3): 489-523. doi: 10.3934/nhm.2017021 |
[2] | Yuntian Zhang, Xiaoliang Chen, Zexia Huang, Xianyong Li, Yajun Du . Managing consensus based on community classification in opinion dynamics. Networks and Heterogeneous Media, 2023, 18(2): 813-841. doi: 10.3934/nhm.2023035 |
[3] | Mayte Pérez-Llanos, Juan Pablo Pinasco, Nicolas Saintier . Opinion fitness and convergence to consensus in homogeneous and heterogeneous populations. Networks and Heterogeneous Media, 2021, 16(2): 257-281. doi: 10.3934/nhm.2021006 |
[4] | Sergei Yu. Pilyugin, Maria S. Tarasova, Aleksandr S. Tarasov, Grigorii V. Monakov . A model of voting dynamics under bounded confidence with nonstandard norming. Networks and Heterogeneous Media, 2022, 17(6): 917-931. doi: 10.3934/nhm.2022032 |
[5] | Clinton Innes, Razvan C. Fetecau, Ralf W. Wittenberg . Modelling heterogeneity and an open-mindedness social norm in opinion dynamics. Networks and Heterogeneous Media, 2017, 12(1): 59-92. doi: 10.3934/nhm.2017003 |
[6] | Marina Dolfin, Mirosław Lachowicz . Modeling opinion dynamics: How the network enhances consensus. Networks and Heterogeneous Media, 2015, 10(4): 877-896. doi: 10.3934/nhm.2015.10.877 |
[7] | Sabrina Bonandin, Mattia Zanella . Effects of heterogeneous opinion interactions in many-agent systems for epidemic dynamics. Networks and Heterogeneous Media, 2024, 19(1): 235-261. doi: 10.3934/nhm.2024011 |
[8] | Sergei Yu. Pilyugin, M. C. Campi . Opinion formation in voting processes under bounded confidence. Networks and Heterogeneous Media, 2019, 14(3): 617-632. doi: 10.3934/nhm.2019024 |
[9] | Sharayu Moharir, Ananya S. Omanwar, Neeraja Sahasrabudhe . Diffusion of binary opinions in a growing population with heterogeneous behaviour and external influence. Networks and Heterogeneous Media, 2023, 18(3): 1288-1312. doi: 10.3934/nhm.2023056 |
[10] | Rainer Hegselmann, Ulrich Krause . Opinion dynamics under the influence of radical groups, charismatic leaders, and other constant signals: A simple unifying model. Networks and Heterogeneous Media, 2015, 10(3): 477-509. doi: 10.3934/nhm.2015.10.477 |
This work formulates the problem of defining a model for opinion dynamics on a general compact Riemannian manifold. Two approaches to modeling opinions on a manifold are explored. The first defines the distance between two points using the projection in the ambient Euclidean space. The second approach defines the distance as the length of the geodesic between two agents. Our analysis focuses on features such as equilibria, the long term behavior, and the energy of the system, as well as the interactions between agents that lead to these features. Simulations for specific manifolds, S1,S2, and T2 , accompany the analysis. Trajectories given by opinion dynamics may resemble n− body Choreography and are called "social choreography". Conditions leading to various types of social choreography are investigated in R2 .
The emergence of a group's global behavior from local interactions among individual agents is a fascinating feature of opinion dynamics. When local rules imply global patterns in a population, we are observing a phenomenon called self-organization. Traditionally, interest focuses on understanding the complex rules of interacting opinions which lead to certain global configurations, such as classic consensus, alignment, clustering, or the less studied dancing equilibrium [3]. For instance, in bounded-confidence models such as the one proposed by Hegselmann and Krause, the radius of interaction determines the clustering of the system [11]. Motsch and Tadmor studied the influence of the shape of the interaction potential on the convergence to consensus of the Hegselmann-Krause system [15]. Ha, Ha and Kim looked at the Cucker-Smale second-order alignment model and provided a condition on the interaction potential ensuring convergence of the system to alignment [9]. Cristiani, Frasca and Piccoli studied the effect of anisotropic interactions on the behavior of the group [6].
The dynamics of an opinion formation system depend on the state-space [1] and interaction network [11]. Models on the Euclidean space in one dimension (for opinion dynamics) or in two or three dimensions (with applications to groups of animals or robots) have been extensively studied and are well understood. However, such models are locally linear, which may be a limitation when one strives to capture more complex phenomena and better represent reality [21]. In this line of thought, the Kuramoto model on the sphere
The present work defines a general model of opinion dynamics on a Riemannian manifold. We investigate how the manifold on which the model is defined affects the global configurations resulting from opinion dynamics. These are the first steps to build a robust theory of opinion dynamics on general Riemannian manifolds.
There is an inherent difficulty in defining opinion dynamics on a general Riemannian manifold. Using the Riemannian distance, an agent will move towards a point by following the manifold's geodesics, which are well defined only locally. On a larger scale, there might not exist a unique geodesic. Another challenge is the extreme complexity of computing geodesics, even on a relatively simple manifold such as the torus [8]. One way around this issue is to consider the embedding of the manifold into a Euclidean space. Each agent's velocity is defined by projection of the other agents' influence onto the tangent space at that point. This is the choice made in [3].
Other than the mentioned practical aspect, there is an intrinsic rationale for choosing one approach over the other. When evolving along the geodesics of the manifold, one assumes that each agent has a global understanding of the manifold's geometry and is able to choose the shortest path among all possible ones. On the other hand, the approach based on the projection of the desired destination onto the tangent space implies that each agent only holds local information about the space in which it evolves. It chooses to move in the direction which locally seems to bring it closer to the target.
We explore these two specific approaches for our generalized model. The first method, Approach A, uses projections in the Euclidean space in which the manifold is embedded. The second method, Approach B, uses only geodesics defined on the manifold to define strength and direction of interaction. We exhibit properties of the interaction matrix that lead to specific kinds of equilibria. Simulations and examples compare the two methods. Dancing equilibria for Approach B are shown (dancing equilibria were studied for Approach A in [3]).
We use the sphere and torus as sample manifolds to evaluate these approaches. Specifically, we simulate dynamics on the following manifolds:
Opinion dynamics trajectories can resemble
This work will primarily discuss two approaches to define opinion dynamics on a Riemannian manifold. Let
˙xi=N∑j=1aijΨ(d(xi,xj))νij | (1) |
where
•
•
•
•
Each of these terms is further specified in the following.
The evolution of each agent's opinion depends on the opinions of all other agents, with influences weighted by the interaction coefficients
Approach A. Assume that
dP(xi,xj)=‖ΠTxiM(xj−xi)‖ | (2) |
where
νij={ΠTxiM(xj−xi)‖ΠTxiM(xj−xi)‖ if ΠTxiM(xj−xi)≠00 otherwise. | (3) |
With the specific choice
˙xi=N∑j=1aijΠTxiM(xj−xi). | (4) |
This is the approach used in [3], applied to the sphere
Notice that the magnitude of influence,
Supposing that
‖xi(ε)−xj‖2=⟨xi(ε)−xj,xi(ε)−xj⟩=⟨xi(0)−xj,xi(0)−xj⟩+2ε⟨˙xi(0),xi(0)−xj⟩+o(ε) | (5) |
so if
Approach B. This second approach defines
Definition 2.1. The cut locus of a point
Let
dG(xi,xj)=∫10√gγij(s)(˙γij(s),˙γij(s))ds. | (6) |
The direction of influence is determined by the same geodesic:
νij={0 if xj=xi or if xj∈CL(xi)˙γij(0)√gxi(˙γij(0),˙γij(0)) otherwise. | (7) |
Unlike in Approach A, the magnitude of influence is a symmetric function:
Interaction networks. In finite-dimensional systems such as system (1), the set of interacting agents can be described by vertices of a graph. A directed edge exists from a vertex
Sri(x)={j∈{1,…,N},d(xi,xj)≤r}, | (8) |
where
Ski(x)={j∈{1,…,N},αij≤k}. | (9) |
Figures 2 and 3 illustrate differences between the metric and topological networks for the specific example of
Resolution of discontinuities. The definitions of
Approach | A | B | A and B |
Critical points | | | |
Discontinuities | | ||
Condition on |
Firstly, notice that in both approaches,
Ψ(0)=0. | (10) |
In Approach A, we created a discontinuity of
Ψ(0)=0. | (11) |
In Approach B, there is a discontinuity for
Ψ(d)=0 for all d≥ϵ | (12) |
where
Notice that in the case of the geodesics approach (Approach B), the condition
Definition 2.2. The configuration
Proposition 1. The consensus configuration is an equilibrium for system (1).
Proof. In both approaches A and B, if
Proposition 2. Let
Proof. The configuration
ddtˉxi=N∑j=1aijΨ(d(ˉxi,ˉxj))νij=0. |
This is a system of at most
Definition 2.3. The kinetic energy of System (1)-(2)-(3) is the quantity
EP(t):=12N∑i=1‖˙xi(t)‖2. | (13) |
The kinetic energy of System (1)-(6)-(7) is the quantity
EG(t):=12N∑i=1gxi(˙xi(t),˙xi(t)). | (14) |
Proposition 3. Let
limt→∞EP(t)=0. | (15) |
Proof. Let
ddtF(t)=4EP(t). | (16) |
Indeed, notice that
∇xi(N∑j=1aij‖xi−xj‖2)=2ΠTxiMN∑j=1aij(xj−xi)=2˙xi. |
Then we compute
ddtF(t)=N∑k=1⟨∇xk12N∑i,j=1aij(‖xi−xj‖2),˙xk⟩=N∑k=1⟨∇xk[12N∑i=1aik(‖xi−xk‖2)+12N∑j=1akj(‖xk−xj‖2)],˙xk⟩=N∑k=1⟨2∇xk12N∑i=1aik(‖xi−xk‖2),˙xk⟩=N∑k=1⟨2ΠTxkMN∑j=1akj(xj−xk),˙xk⟩=2N∑k=1‖˙xk‖2=4EP(t). | (17) |
where the third equality uses the property:
Since
Now,
This shows that
Remark 1. Propositions 2 and 3 are generalizations of results proven for the case
Remark 2. Proposition 3 assumes that
Definition 2.4. Let
Remark 3. This definition is a generalization of the concept of dancing equilibrium described in [3].
Remark 4. It follows immediately from definition 2.4 that the kinetic energy of a system in dancing equilibrium is constant.
We study both approaches A and B in the case
Approach A. The projection onto an agent's tangent space can be rewritten as:
ΠTxiN∑j=1aij(xj−xi)=N∑j=1aij⟨(cosθjsinθj)−(cosθisinθi),(−sinθicosθi)⟩(−sinθicosθi)=N∑j=1aij(−sinθicosθj+sinθjcosθi)(−sinθicosθi)=N∑j=1aijsin(θj−θi)(−sinθicosθi). | (18) |
So System (1)-(2)-(3) becomes:
for alli∈{1,…,N},˙θi(−sinθicosθi)=N∑j=1aijΨ(|sin(θj−θi)|)sgn(sin(θj−θi))(−sinθicosθi) | (19) |
where
for allx∈R,sgn(x)={1 if x>0−1 if x<00 if x=0. | (20) |
We can then specify:
for all(i,j)∈{1,…,N}2,dP(xi,xj)=|sin(θj−θi)|,νPij=sgn(sin(θj−θi)). | (21) |
This gives the system of scalar equations:
for alli∈{1,…,N},˙θi=N∑j=1aijΨ(|sin(θj−θi)|)sgn(sin(θj−θi)). | (22) |
In particular, in the case
for alli∈{1,…,N},˙θi=N∑j=1aijsin(θj−θi). | (23) |
Approach B. For
dG(xi,xj)=arccos(cos(θj−θi)), νGij=sgn(sin(θj−θi)). | (24) |
System (1)-(6)-(7) is written:
for alli∈{1,…,N}, ˙θi=N∑j=1aijΨ(arccos(cos(θj−θi)))sgn(sin(θj−θi)). | (25) |
In order for the system to be well defined, the interaction function
Ψa(d)={1adford≤ad−πa−πford>a | (26) |
where
Another possible choice is:
We first examine the different equilibria for both approaches.
Theorem 3.1. Consider Approach A, System (22). Let
forallj∈{1,…,N2},θj+N2=θj+π |
is an equilibrium.
Proof. Using the hypotheses from Theorem 3.1, we can easily compute:
˙θi=N∑j=1aijΨ(‖sin(θj−θi)‖)sgn(sin(θj−θi))=N/2∑j=1[aijΨ(‖sin(θj−θi)‖)sgn(sin(θj−θi))+ai(j+N2)Ψ(‖sin(θj+N2−θi)‖)sgn(sin(θj+N2−θi))]=N/2∑j=1[aijΨ(‖sin(θj−θi)‖)sgn(sin(θj−θi))+aijΨ(‖sin(θj+π−θi)‖)sgn(sin(θj+π−θi))]=0. |
Interestingly, Theorem 3.1 is not applicable to Approach B. We illustrate the different behaviors of the two systems by studying the specific example of four agents initially in a rectangular configuration. According to Theorem 3.1, this configuration is an equilibrium for Approach A, independently of the choice of interaction function
This highlights the fundamentally different behaviors of the systems (1)-(2)-(3) and (1)-(6)-(7) in the case
In both approaches A and B, conditions on the interaction matrix
Theorem 3.2. Consider the dynamics on
foralli∈{1,…,N},˙θi=N∑j=1aijΨ(d(xi,xj))νij | (27) |
where
aij={CΨ(d(xi(0),xj(0)))νijifΨ(d(xi(0),xj(0)))≠00otherwise. | (28) |
Then the system is in a dancing equilibrium.
Proof. If the interaction matrix satisfies (28), then at
for alli∈{1,…,N}, ˙θi(0)=N∑j=1C=CN |
so for all
Numerical simulations show the evolution of the system (27) with condition (28) for the projection or the geodesic distance, see Figures 5 and 6.
We study both approaches A and B for
xi=(cosθsinϕ,sinθsinϕ,cosϕ)T. |
Choice of influence function. We choose an influence function
Approach A. On
ΠTxiN∑j=1aij(xj−xi)=ΠTxiN∑j=1aij(xj)=N∑j=1aij(xj−⟨xj,xi⟩xi)=N∑j=1aij((cosθjsinϕjsinθjsinϕjcosϕj)−⟨(cosθjsinϕjsinθjsinϕjcosϕj),(cosθisinϕisinθisinϕicosϕi)⟩(cosθisinϕisinθisinϕicosϕi)). |
Approach B. The geodesic distance
dG(xi,xj)=2arcsin(‖xi−xj‖2), |
and the direction toward
νij=xj−⟨xj,xi⟩xi‖xj−⟨xj,xi⟩xi‖, |
where
Noticing that for
Corollary 1. Consider the dynamics given by geodesic distance (Approach B) on
limt→∞EG(t)=0. | (29) |
Proof. System (1)-(6)-(7), with
˙xi=N∑j=1aijsin(dG(xi,xj))νij. | (30) |
Considering the embedding of the system in
aijsin(dG(xi,xj))νij=aijΠTxiM(xj−xi), | (31) |
thus the system is (4), and by proposition 3,
limt→∞EP(t)=0⟹limt→∞˙xi=0 for all i⟹limt→∞EG(t)=0. | (32) |
We use a fourth order Runge-Kutta scheme to approximate the trajectories. The derivative is calculated as a vector in
For an agent
tθ=(−sinθcosθ0),tϕ=(cosθcosϕsinθcosϕ−sinϕ). |
We express the derivative of an agent as
˙xi=∂xi∂θi˙θ+∂xi∂ϕi˙ϕ. | (33) |
By direct computation, we get:
∂xi∂θi=(−sinθisinϕicosθisinϕi0),∂xi∂ϕi=(cosθicosϕisinθicosϕi−sinϕi). |
We can also express the derivative of
˙xi=⟨˙xi,tθi⟩tθi+⟨˙xi,tϕi⟩tϕi. | (34) |
It follows from (33) and (34) that
˙θi=1sinϕi⟨˙xi,tθi⟩tϕi and ˙ϕi=⟨˙xi,tϕi⟩tϕi. | (35) |
Singularities: In (35), the factor
An additional concern is singularities for agents forming consensus. In equations (3) and (7),
νij=xj−⟨xj,xi⟩xidmin |
We ran simulations using different choices of
Example 4.1. Five agents with a general interaction matrix
Example 4.2. To assess the influence of the curvature of
A=(01−1−1011−10) | (36) |
In Section 6.2, we prove that those dynamics in
We now study how the general dynamics given by equation (1) apply to the specific case of the torus
{x=(R+rcosθ)cosϕy=(R+rcosθ)sinϕz=rsinθ for (ϕ,θ)∈[0,2π)2. |
The angles
We first investigate the behavior of system (1) with Approach B (using the geodesic distance) in the case of
Several challenges arise when defining Approach B on
Secondly, assuming that one is able to efficiently compute the geodesics on
For simplicity, we thus focus on Approach A, where the dynamics are a function of the projection of each vector
Equations (1)-(2) reads:
˙xi=N∑j=1aijΨ(‖ΠTxiT2(xj−xi)‖)νij,i∈{1,…,N}. | (37) |
The vector
νij={ΠTxiT2(xj−xi)‖ΠTxiM(xj−xi)‖ if xj∉Ni0 if xj∈Ni. | (38) |
Let
{⟨xj,tϕi⟩=0⟨xj−xi,tθi⟩=0. |
After computations, we get:
⟨xj,tϕi⟩=0⟺sin(ϕj−ϕi)=0⟺ϕj=ϕi+kπ,k∈Z. |
If
⟨xj−xi,tθi⟩=0⟺sin(θj−θi)=0⟺θj=θi+kπ,k∈Z. |
If
sin(θi+θj)=−2Rrsinθi. |
Notice that this last equation only has a solution if
We now go back to equation (37). We study the specific case where
˙xi=ΠTxiT2(N∑j=1aij(xj−xi)). | (39) |
Hence the velocity reads:
˙xi=N∑j=1aij(xj−xi)−⟨N∑j=1aij(xj−xi),uθi⟩uθi=αi−⟨αi,uθi⟩uθi−(N∑j=1aij)⟨xi,tθi⟩tθi |
where
The velocity of each agent is given by:
˙xi=(−˙ϕisinϕi(R+rcosθi)−r˙θisinθicosϕi˙ϕicosϕi(R+rcosθi)−r˙θisinθisinϕir˙θicosθi)=˙ϕi(R+rcosθi)tϕi+r˙θitθi. | (40) |
From (39) and (40) we get the angular velocities:
{˙ϕi=1(R+rcosθi)⟨N∑j=1aij(xj−xi),tϕi⟩˙θi=1r⟨∑Nj=1aij(xj−xi),tθi⟩. | (41) |
Notice that unlike in the case of
We now analyze the dynamics (1)-(2)-(3) on
Proposition 4. Consider the dynamics (1)-(2)-(3) on
Proof. Suppose that for all
tθi(0)=(00±π)andxj(0)−xi(0)=((R+rcosθj)cosϕj−(R+rcosθi)cosϕi(R+rcosθj)sinϕj−(R+rcosθi)sinϕi0) |
From equation (41) we get:
Remark 5. As a consequence of Proposition 4, if all agents are initially in
Proposition 5. Consider the dynamics (1)-(2)-(3) on
Proof. Suppose without loss of generality that
Remark 6. As a consequence of Proposition 5, if all agents are initially in
To assess the influence of the curvature of the manifold on the dynamics, we compare a simple case involving 3 agents evolving according to the interaction matrix given in equation (36). As in the case of
As seen in Sections 3 and 5.3, when the interaction matrix
In this section, we investigate systems with similar properties of periodicity or symmetry. We use the term social choreography, drawing a parallel with the well-known "n-body choreographies" discovered by Moore [13,14] in the context of point masses subject to gravitational forces. In the n-body problem, the interaction potentials between masses are predetermined, as they depend exclusively on the masses and distances between agents. Hence the conditions for a n-body choreography to occur only depend on the initial state of the system. In the case of social choreography, there are more degrees of freedom, as we design the interaction matrix as well as to set the initial conditions.
We study sufficient conditions on the interaction matrices for the trajectories of the system to be periodic or symmetric by focusing on the Euclidean space
for alli∈{1,…,N},˙xi=N∑j=1aij(xj−xi). | (42) |
We define the kinetic energy
E(t):=12N∑i=1‖˙xi(t)‖2. | (43) |
A simple case of social choreography is that of a system with periodic trajectories, which we define as follows:
Definition 6.1. Let
for alli∈{1,...,N},for allt>0, xi(t+τ)=xi(t). |
We will examine possible periodic behaviors of the system in sections 6.2, 6.3 and 6.4.
We now give sufficient conditions on the interaction matrix and on the initial conditions for the system to be invariant by rotation.
Theorem 6.2. Let
foralli∈{1,…,N},R(2kπN)xi(0)={xi+k(0)ifi+k≤Nxi+k−N(0)ifi+k>N. |
Suppose that the interaction matrix
forallt>0,foralli∈{1,…,N},R(2kπN)xi(t)={xi+k(t)ifi+k≤Nxi+k−N(t)ifi+k>N. |
Proof. Let
From the definition of the matrix
for alli∈{1,…,N},R(2kπN)xi(0)={xi+k(0)ifi+k≤Nxi+k−N(0)ifi+k>N |
can be rewritten as:
˙X=˜AX=P−1k˜APkX. |
From that we compute:
˙Y=Pk˙X=Pk(P−1k˜APkX)=˜APkX=˜AY. |
Similarly,
˙Z=(R(2kπN)˙XT)T=(R(2kπN)(˜AX)T)T=(R(2kπN)XT˜AT)T=˜AZ. |
Since
for alli∈{1,…,N},R(2kπN)xi(t)={xi+k(t)ifi+k≤Nxi+k−N(t)ifi+k>N. |
Another example of social choreography is that of a system in which all agents share one unique orbit. Such choreographies have been discovered in the context of the n-body problem, for instance the "figure 8" orbit for three equal masses [13].
Definition 6.3. Let
for alli,j∈{1,...,N},{z∈M|∃t>0,xi(t)=z}={z∈M|∃t>0,xj(t)=z}. |
To illustrate Theorem 6.2, we study the evolution of N agents initially positioned at regular intervals on a circle, with an interaction matrix and initial conditions given by:
A=(010…0−1−10⋱⋱⋱00⋱⋱⋱⋱⋮⋮⋱⋱⋱⋱00⋱⋱⋱⋱110…0−10)and for alli∈{1,…,N},xi(0)=(cos(2iπN)sin(2iπN)). | (44) |
Notice that
˙x1=x2−xN=R(2πN)x1−R(−2πN)x1. |
This can be written as:
(˙x11˙x12)=(0−2sin(2πN)2sin(2πN)0)(x11x12). |
Solving this linear system yields:
{x11(t)=x11(0)cos(2sin(2πN)t)−x12(0)sin(2sin(2πN)t)=cos(2sin(2πN)t)x12(t)=x11(0)sin(2sin(2πN)t)+x12(0)cos(2sin(2πN)t)=sin(2sin(2πN)t). |
This proves that all agents share one common circular orbit, and their trajectories are periodic of period
Another interesting example is that of 3 agents interacting according to the interaction matrix given previously, which, reduced to
A=(01−1−1011−10). | (45) |
Theorem 6.4. Let
Proof. The
(x1jx2jx3j)(t)=exp(tA)(x01jx02jx03j) |
with
etA=13(1+2cos(√3t)1−cos(√3t)+√3sin(√3t)1−cos(√3t)−√3sin(√3t)1−cos(√3t)−√3sin(√3t)1+2cos(√3t)1−cos(√3t)+√3sin(√3t)1−cos(√3t)+√3sin(√3t)1−cos(√3t)−√3sin(√3t)1+2cos(√3t)). |
Due to the special structure of
(x1jx2jx3j)(t)=13(x01jx02jx03jx02jx03jx01jx03jx01jx02j)(1+2cos(√3t)1−cos(√3t)+√3sin(√3t)1−cos(√3t)−√3sin(√3t)). |
This shows that all three trajectories are periodic, or period
(x1jx2jx3j)(t+2π3√3) |
=13(x01jx02jx03jx02jx03jx01jx03jx01jx02j)(1−cos(√3t)−√3sin(√3t)1+2cos(√3t)1−cos(√3t)+√3sin(√3t))=(x2jx3jx1j)(t). |
This shows that there is one unique shared orbit.
Other conditions on the interaction matrix
Theorem 6.5 (Coupled periodic trajectories). Let
foralli∈{1,…,N},R(4πN)xi(0)={xi+2(0)ifi+2≤Nxi+2−N(0)ifi+2>N. |
Let
A=(0a0…0−b−a0b⋱⋱00−b⋱a⋱⋮⋮⋱⋱⋱⋱00⋱⋱⋱⋱ab0…0−a0). | (46) |
Then the system is periodic of period
forallt>0,foralli∈{1,…,N2},xi(t+τ)=xi+N2(t), |
and the kinetic energy is periodic with period
Proof. First remark that the system satisfies the hypotheses of Theorem 6.2, so
for allt>0,for alli∈{1,…,N},R(4πN)xi(t)={xi+2(t)ifi+2≤Nxi+2−N(t)ifi+2>N. |
Hence the system is entirely known from the positions of the first two agents, since all others can be obtained by simple rotations. We show that this
{˙x1=a(x2−x1)−b(xN−x1)˙x2=b(x3−x2)−a(x1−x2) |
becomes:
{˙x1=(˙x11˙x12)=a[(x21x22)−(x11x12)]−b[(cos(4πN)sin(4πN)−sin(4πN)cos(4πN))(x21x22)−(x11x12)]˙x2=(˙x21˙x22)=b[(cos(4πN)−sin(4πN)sin(4πN)cos(4πN))(x11x12)−(x21x22)]−a[(x11x12)−(x21x22)]. |
This can be rewritten in matrix form as:
(˙x11˙x12˙x21˙x22)=A4(x11x12x21x22) | (47) |
where
A4:=(−a+b0a−bcos(4πN)−bsin(4πN)0−a+bbsin(4πN)a−bcos(4πN)−a+bcos(4πN)−bsin(4πN)a−b0bsin(4πN)−a+bcos(4πN)0a−b)(x11x12x21x22). |
One can easily show that this reduced interaction matrix
τ=2πλ=π√absin(2πN). |
Furthermore, if
As a consequence, the kinetic energy is periodic, of period
Remark 7. Notice that the agents sharing orbits do not interact with one another, as shown in Figure 16.
An example of such a choreography is given in Figure 17.
Remark 8. As a slight generalization, we provide numerical simulations illustrating a similar behavior, but with slightly different conditions: the periodic evolution of 9 agents on three distinct orbits shared three by three, see figures 18 and 19.
In sections 6.2 and 6.3, we provided conditions for the trajectories of the system to be periodic. Here, we explore further the notion of periodicity by studying systems with drift, displaying helical trajectories but periodic kinetic energy.
Definition 6.6. Let
for alli∈{1,...,N},for allt>0,xi(t+τ)=xi(t)+τv. |
Notice that this definition generalizes the notion of periodic trajectories recalled in Definition 6.1, which corresponds to the case
Theorem 6.7. Sufficient conditions for helical trajectories. Let
A=(0a0−d−a0b00−b0cd0−c0). | (48) |
Then the system exhibits helical trajectories.
Proof. First notice that the first and second components
˙xj=(˙x1j˙x2j˙x3j˙x4j)=(d−aa0−d−aa−bb00−bb−ccd0−cc−d)(x1jx2jx3jx4j):=˜A(x1jx2jx3jx4j),for j∈{1,2}. | (49) |
Hence the projections of
λ1=0, λ2=i√(a+c)(b+d) and λ3=−i√(a+c)(b+d). |
There is one eigenvector associated with
ν:=1Δ(ab+bc+Δ,ab−cd+Δ,ab+ad+Δ,Δ)T. |
Let
xj(t)=Cj1v1+Cj2(v1t+ν)+Cj3[vR2cos(λ2t)−vI2sin(λ2t)] +Cj4[vR2sin(λ2t)+vI2cos(λ2t)] |
where
for alli∈{1,…,4},for allt>0,xi(t+τ)=xi(t)+(C12C22)τ. |
Theorem 6.8. A system with helical trajectories has periodic kinetic energy.
Proof. Supose that
An example using Approach A which shows unexpected behavior in the first example (A.1), as well as the interactions matrix and initial positions used for simulations shown in Figure 7 and 9.
Example 7.1. 15 agents with a general interaction matrix
A=(−5−1104−1−5110−20−5−1−2−4−1−53−4−41−221)X=(1.57551.73991.65230.56195.30262.70082.49710.72880.65710.52811.21800.08402.28121.01295.49491.74413.76852.59031.62182.52662.25210.07675.57661.16715.65821.54532.81461.46411.68920.1310) |
Example 7.2. Five agents with a general interaction matrix
A=(−5−1104−1−5110−20−5−1−2−4−1−53−4−41−221)X=(6.17432.84734.58832.76352.16062.56913.66980.81910.6771138672) |
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Approach | A | B | A and B |
Critical points | | | |
Discontinuities | | ||
Condition on |