Dynamics of Decision Making in Animal Group. Motion

Transcription

1 Dynamics of Decision Making in Animal Group Motion Benjamin Nabet, Naomi E Leonard, Iain D Couzin and Simon A Levin June, 007 Abstract We present a continuous model of a multi-agent system motivated by simulation studies on dynamics of decision making in animal groups in motion Each individual moves at constant speed in the plane and adjusts its heading in response to relative headings of others in the population Two subgroups of the population are informed such that individuals in each subgroup have a preferred direction of motion The model exhibits fast and slow time scales allowing for a reduction in the dimension of the problem The stable solutions for the reduced model correspond to compromise by individuals with conflicting preferences We study the global phase space for the proposed reduced model by computing equilibria and proving stability and bifurcations 1 INTRODUCTION Recent research in cooperative control of groups of mobile autonomous agents has led to a growing effort to apply tools from dynamical systems and control theory toward better understanding how B Nabet and NE Leonard are with the Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ, USA {bnabet,naomi}@princetonedu This work is supported in part by ONR grants N and N ID Couzin is with the Department of Zoology, South Parks Road, University of Oxford, Oxford, OX1 3PS, U and the Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ, USA icouzin@princetonedu This work was supported from the Royal Society, Balliol College and EPSRC grants GR/S04765/01 and GR/T1134/01 SA Levin is with the Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ, USA slevin@princetonedu This work was supported in part by DARPA grant HR and NSF grant EF

2 biological systems manage collective tasks such as social foraging or migration In this paper we derive and study the dynamics of a low-dimensional, minimally parameterized, coordinated control system, motivated by an interest in modelling and predicting the behavior of animal groups in motion Many social organisms move in groups when they forage or migrate, and it is thought that the movement decisions they make may depend on social interactions among group members [1,, 3] In Couzin et al [3], the mechanisms of decision-making and leadership are investigated using a discrete simulation of particles moving in the plane In this simulation, each particle represents an individual animal and the motion of each individual is influenced by the state of its neighbors eg, relative position and relative heading Within this group, there are two subgroups of informed individuals and one subgroup of naive individuals; each subgroup of informed individuals has a preferred direction of motion representative of knowledge of location of food or migration route that it can use to make decisions along with the information on its neighbors It is shown in [3] that information can be transferred within groups even when there is no signaling, no identification of the informed individuals, and no evaluation of the information of individuals It was also observed that with two informed subgroups of equal population, the direction of group motion depends on the degree to which the preferred directions differ For low disagreement, the group follows the average preferred direction of all informed individuals, while for large disagreement the group selects one of the two preferred directions The model we propose and study in a simplified form in this paper corresponds to a deterministic set of ordinary differential equations Each agent is modelled as a particle moving in the plane at constant speed with steering rate dependent on inter-particle measurements and, when appropriate, on prior information concerning preferred directions The motivation from a biological point of view is to exploit the provable phase space dynamics of our simplified model to better understand and predict how movement decisions are made in animal groups This model is similar to models used for cooperative control of engineered multi-agent systems For instance, a continuous model of particles moving at constant speed in the plane with steering control heading rate designed to couple the dynamics of the particles has been used for stabilization of circular and parallel collective motion [4, 5] The use of the same kinds of models in the engineered and natural settings is no accident The very efficient and robust ways that an-

3 imals move together and make collective decisions provide inspiration for design in engineering Likewise, tools that have been developed for analysis and synthesis in the engineering context may prove useful for investigation in the natural setting We note that the objectives in engineering applications may be analogous to objectives in the natural setting For example, in the design of mobile sensor networks such as the autonomous ocean sampling network described in [6], the goal is to maximize information intake This has parallels with optimal social foraging The central goal in the present work is to study the global phase space for the proposed simple model by computing equilibria and proving stability and bifurcations Starting from a large-scale particle model, we reduce it to a simple planar model using a time-scale separation Fast dynamics are associated with consensus of individuals with similar information and slow dynamics with the subsequent behavior of these different subgroups In [7], the authors also use time-scale separation to reduce the dimension of consensus dynamics in complex networks There the slow and fast times scales are due to sparse and dense connections among nodes in the network Our planar particle model includes key features of the discrete model of [3]; however, for the purpose of analysis, it is made simpler For example, we first define our model for the full spatial dynamics and then we proceed to study only the dynamics of the headings We prove the time scale separation for the model of the heading dynamics of two informed subgroups and one uninformed subgroup For the bifurcation analysis of the slow dynamics, we focus our study on the two informed subgroups, and discuss the role of the uninformed individuals at the end of the paper We study bifurcations as a function of two bifurcation parameter: 0, the coupling gain that weights the attention paid to neighbors versus the preferred direction, and θ [0,π], the relative angle of the two preferred directions In Section, we present the model We identify fast and slow time scales and prove, for the system with two informed subgroup and one naive group, invariance and attractivity of the reduced slow manifold In Section 3 we classify the equilibria of the reduced-order system with no naive individuals In Section 4 we prove bifurcations in the system as a function of the coupling gain In Sections 5 and 6 we study two specific choices for the parameters and θ for which we can find a closed-form expression for the equilibrium points and compute analytically the bifurcation diagrams In Section 7 we explain how the results change for unevenly sized groups of informed individuals and discuss future directions 3

4 Models and time-scale separation 1 Particle model We consider a population of N individuals each modeled as a particle moving in the plane For the purpose of this paper, we assume that every individual can sense every other individual in the population In the natural setting this all-to-all coupling assumption may be reasonably well justified for tightly clustered groups A future objective will be to apply the theory in the case of limited sensing see eg, [8] The population is classified into three subgroups Let N 1 and N be the number of agents, respectively, in two different subgroups of informed individuals and let N 3 be the number of naive uninformed individuals such that N 1 + N + N 3 = N Let N 1 and N, respectively, be the subset of indices in {1,, N} corresponding to individuals in subgroups 1 and which comprise the two different groups of informed individuals Let N 3 be the remaining subset of indices corresponding to the naive individuals Then the cardinality of N k is N k, k = 1,,3 The preferred heading direction for the individuals in subgroup i is denoted θ i, for i = 1, We model each individual as a particle moving in the plane at constant speed The heading direction of individual j is denoted θ j, and θ j is allowed to take any value in the circle S 1 Let r j R be the position of the j th individual moving at constant speed V 0, then ṙ j = V 0 cos θ j,v 0 sinθ j, j = 1,,N Our simple model describes the dynamics of the heading angles for all individuals in the population independent of their positions This model defines steering terms that depend only on relative heading angles The dynamics are modelled as θ j = sin θ1 θ j + N θ j = sin θ θ j + N θ j = N N sin θ l θ j, j N 1 l=1 N sin θ l θ j, j N l=1 N sin θ l θ j, j N 3 l=1 1 4

5 Individuals with prefered direction 0 rad Individuals with prefered direction rad Uninformed individuals θj time Figure 1: Phase angle of each individual in the group versus time for = 1 For this simulation there are five individuals with preferred direction 0 rad, five individuals with preferred direction rad and twenty individuals with no preferred direction Two time scales in the dynamics can be observed During a short initial transient time, the heading angles of the individuals in each subgroup synchronize Then the three average subgroup directions change slowly to their steady state values We note that the form of the coupling is based on the uramoto model for populations of coupled oscillators [9] The model is similar to that used by Mirollo and Strogatz to represent a group of coupled spins in a random magnetic field [10] In the coupled spin model, there are no subgroups; instead, each individual oscillator has a randomly assigned pinning angle θ j such that the pinning angles are uniformly distributed around the circle The studied system is known in physics as the mean-field theory for the random-field XY model, [11] In [10] it is proven that the system exhibits a jump bifurcation and hysteresis as is varied Model representation with two time scales Now let p k C denote the average of the phasors on the unit circle in the complex plane for the individuals in N k In the coupled oscillator literature, p k is known as the complex order parameter and ρ k := p k provides a measure of synchrony among the phases The average phasor p k is 5

6 computed as p k = ρ k e iψ k = 1 N k l N k e iθl, k = 1,,3 The parameter ρ k takes values in the interval [0,1] It follows that ρ k = 1 if all individuals in N k are heading in the same direction synchronized headings and ρ k = 0 if individuals in N k head in directions such that their averaged velocity is zero The average direction of individuals in N k is ψ k Simulations of the model 1 shown in Figure 1, reveal two time scales in the dynamics First, during a short initial transient time, the heading angles of the individuals in each subgroup synchronize Then we observe a slow drift until the three average subgroup directions reach the steady state Motivated by these observations, we define a new set of independent variables that distinguishes slow and fast variables The average headings ψ 1,ψ and ψ 3 are the slow variables since they characterize the lumped behavior of each of the three subgroups N Following [1], the term 1 N sin θ l θ j, can be rewritten in terms of the complex order parameters as 1 N l=1 N 3 sin θ l θ j = 1 N k ρ k sinψ k θ j 3 N l=1 Using 3, the model 1 becomes k=1 θ j = sin θ1 3 θ j + N k ρ k sin ψ k θ j, j N 1 N k=1 θ j = sin θ 3 θ j + N k ρ k sin ψ k θ j, j N N k=1 3 θ j = N k ρ k sin ψ k θ j, j N 3 N k=1 4 For z 1,z C, let z 1,z = R {z 1 z } We further compute from that p k,ip j = ρ k e iψ k,iρ j e iψ j = 1 e iθl, ie iθm, N k N j l N k m N j which implies ρ k N k ρ j N j sin ψ j ψ k = l N k m N j sin θ m θ l 5 6

7 The identity 5 is equal to zero if j = k Similarly, for k = 1,, ρ k e iψ k,ie i θ k = 1 N k l N k e iθl,ie i θk, which implies ρ k N k sin θk ψ k = l N k sin θk θ l 6 Using 1, 5 and 6 we can compute θj = ρ 1 N 1 sin j N 1 θj = ρ N sin j N θ1 ψ 1 + N ρ 1N 1 ρ N sin ψ ψ 1 + N ρ 1N 1 ρ 3 N 3 sin ψ 3 ψ 1 θ ψ + N ρ 1N 1 ρ N sin ψ 1 ψ + N ρ N ρ 3 N 3 sin ψ 3 ψ j N 3 θj = N ρ 1N 1 ρ 3 N 3 sinψ 1 ψ 3 + N ρ N ρ 3 N 3 sin ψ ψ 3 To represent the fast dynamics, we define unit vectors as complex variables α j C where i N k θ j P θ l l N α j = e k, j N k Then, α j = in k α j θj 1 N k l N k θl, j N k The unit vectors phasors α j represent how much the heading of individual j N k differs from ψ k, the average direction of the subgroup k When all the individuals in the kth subgroup have the same heading, α j = 1, j N k Denote θ = θ 1,,θ N T N and α k = α jk,1,,α jk,nk 1 C N k 1, where N k = { j k,1,,j k,nk }, and consider change of variables θ { α 1,α,α 3,ψ 1,ψ,ψ 3 } Further, suppose N >> 1 and let ǫ = 1/ We assume N 3 is of the same order or smaller than N, and N 1 and N are both smaller than N, such that none of the following are as small as ǫ: 1/N 1, 1/N, N 1 /N, N /N For example, in case = N = 100,N 1 = N = 10,N 3 = 80, then ǫ = 001 and 1/N 1 = 1/N = N 1 /N = N /N = 01 = ǫ Given these assumptions, in the new 7

8 coordinates the coupled multi-agent system dynamics 1 become ǫ α j = in 1 α j ǫ sin θ1 θ j ρ1 sin θ1 ψ 1 + N 1 + k=,3 N k N ρ k sin ψ k θ j ρ 1 sin ψ k ψ 1 N ρ 1 sin ψ 1 θ j 7 =: gj 1 α 1,α,α 3,ψ 1,ψ,ψ 3,ǫ, j N 1, j j 1,N1 ǫ α j = in α j ǫ sin θ θ j ρ sin θ ψ + N + k=1,3 N k N ρ k sin ψ k θ j ρ sin ψ k ψ N ρ sin ψ θ j 8 =: gj α 1,α,α 3,ψ 1,ψ,ψ 3,ǫ, j N, j j,n ǫ α j = in 1 α j N3 N ρ 3 sin ψ 3 θ j + k=,3 N k N ρ k sin ψ k θ j ρ 3 sinψ k ψ 3 9 =: gj 3 α 1,α,α 3,ψ 1,ψ,ψ 3,ǫ, j N 3, j j 3,N3 ψ 1 = 1 1 sin θ1 3 θ j + ρ 1 N 1 N N k N 1 ρ k sin ψ k θ j j N 1 k=1 10 =: f 1 α 1,α,α 3,ψ 1,ψ,ψ 3,ǫ ψ = 1 1 sin θ 3 θ j + ρ N N N k N ρ k sin ψ k θ j cos ψ 1 θ j j N k=1 11 =: f α 1,α,α 3,ψ 1,ψ,ψ 3,ǫ ψ 3 = 1 ρ 3 3 N k ρ k sin ψ k θ j cos ψ 3 θ j N N 3 k=1 cos ψ θ j j N 3 1 =: f 3 α 1,α,α 3,ψ 1,ψ,ψ 3,ǫ for ρ k 0, k = 1,,3 In Appendix A we show that this change of coordinates is well defined The model 7-1 exhibits two time scales where the variables α 1,α,α 3 are the N 3 fast variables and ψ 1,ψ,ψ 3 are the three slow variables The solution α j = 1 for j N k, k = 1,,3, equivalently θ j = ψ k, j N k, k = 1,,3, is an isolated solution of gj k α 1,α,α 3,ψ 1,ψ,ψ 3,0 = 0, k = 1,,3 For this solution ρ k = 1, k = 1,,3 In other words, θ j = ψ k, j N k, k = 1,,3 is an invariant manifold of our system 1 Physically this means that if we start with all individuals 8

9 synchronized within their respective subgroup ie, θ j = ψ k, j N k, k = 1,,3, they will stay like this for all time From the representation of the system dynamics 1 as equations 7-1 with ǫ 1, the corresponding slow dynamics, ie, dynamics on the invariant manifold, are ψ k = f k α 1 = 1,α = 1,α 3 = 1,ψ 1,ψ,ψ 3,0, k = 1,,3, which can be written as ψ 1 = sin θ1 ψ 1 + N N sinψ ψ 1 + N N 3 sin ψ 3 ψ 1 ψ = sin θ ψ + N N 1 sinψ 1 ψ + N N 3 sin ψ 3 ψ 13 ψ 3 = N N 1 sinψ 1 ψ 3 + N N sin ψ ψ 3 In Appendix B, we prove the reduction by proving the stability of the invariant manifold for the boundary layer dynamics Singular perturbation theory see eg [13] guarantees then that solutions to the unreduced dynamics stay close to solutions of the reduced system Consistent with the observations from simulations in [3], the solution of the fast dynamics corresponds to synchronization of all particle headings in subgroup k to common heading ψ k, for k = 1,,3 The slow dynamics, described by the reduced model 13, dictate the behavior of the common heading ψ k of each of the three subgroups, k = 1,,3 This reduced model is one in which all the agents in a subgroup informed subgroups 1 and and naive subgroup 3 behave as a single entity thus the qualifier lumped model and the inter-subgroup coupling term is weighted by the corresponding subgroup population size This grouping of identical individuals, was also observed in the simulation from [3] In that model, the grouping was spatial, each subgroup made a cluster within the group In the remaining sections of this paper, we focus our bifurcation analysis on the reduced dynamic model derived here To further simplify this analysis, we first consider the case that N 1 = N and N 3 = 0 ie equal population for the two informed subgroups and no naive individuals and then extend conclusions to the case N 1 N, N 3 = 0 ie when one informed subgroup is more populated than the other and there are still no naive individuals In the case N 1 = N and N 3 = 0, 13 9

10 becomes ψ 1 = sin θ1 ψ 1 + sin ψ ψ 1 ψ = sin θ ψ + sin ψ 1 ψ This model also corresponds to be the reduced dynamics in the case N 1 = N 1 and 0 not necessarily large Without loss of generality we set θ 1 = 0 The two bifurcation parameters are 0 and θ [0,π] We note that the general reduced system 13 is a gradient system In the case of N 1 = N and N 3 = 0, the dynamics 14 are gradient dynamics such that 14 ψ k = V ψ k, where V is given by V ψ 1,ψ = cos ψ 1 cos θ ψ cos ψ ψ 1 Thus, by LaSalle s Invariance Principle, all solutions converge to the set of critical points of V ψ 1,ψ and there are no periodic solutions 3 Equilibria We first compute the equilibria of the system 14 but note that, in general, we cannot find closed form expressions for all of them The equilibria are given by sinψ 1 + sin ψ ψ 1 = 0 sin θ ψ + sin ψ 1 ψ = 0 There are two sets of solutions, the first set given by ψ 1 = π θ + ψ sin ψ θ = sin θ, 15 10

11 and the second set given by ψ 1 = θ ψ 16 sin θ ψ = sin ψ θ 17 First set of solutions Equation 15 has two solutions: ψ = θ + arcsin sin θ and ψ = π + θ arcsin sin θ These two solutions exist if and only if sin θ 1 Lemma 31 If sin θ < 1, the two equilibria ψ S1 = ψ 1,ψ S1 and ψ S = ψ 1,ψ S satisfying 15 given by ψ S1 = ψ S = π + arcsin sin θ, θ + arcsin sin θ, 18 arcsin sin θ,π + θ arcsin sin θ, 19 are saddle points > 0 and θ [0,π] If sin θ = 1, then ψ S1 = ψ S In this case, if also > 0 and θ 0, π π,π then ψ S1 = ψ S is unstable with one zero eigenvalue and one positive real eigenvalue If θ = π and = then both eigenvalues are zero Proof: We compute the linearization of 14 at each of these two equilibria and show that its eigenvalues are always real and of opposite sign The Jacobian of the system 14 is given by J = cos ψ 1 cos ψ ψ 1 cos ψ ψ 1 cos ψ ψ 1 cos θ ψ 0 cos ψ ψ 1 When we evaluate this matrix at either one of the two equilibria ψ S1 or ψ S, we get J ψsi = cos θ + 1 cos θ 4 sin θ cos θ cos θ 1 4 sin θ Since the Jacobian is symmetric, the eigenvalues are real The product of the two eigenvalues is λ 1 λ = 4 sin θ 1 < 0 for sin θ < 1 11

12 Therefore, for θ [0,π] the eigenvalues of the linearization are real and of opposite sign This implies that equilibria ψ S1 and ψ S, if sin θ < 1, are saddle points > 0 and θ [0,π] In the case sin θ = 1, ψ S1 = ψ S = 3π, π + θ and we get for the Jacobian J ψsi = cos θ cos θ cos θ cos θ The eigenvalues are λ 1 = 0 and λ = cos θ > 0 Therefore for θ 0, π π,π, we have ψ S1 = ψ S = 3π, π + θ and the linearization is unstable with one zero eigenvalue and one strictly positive eigenvalue In case θ = π/ and =, λ 1 = λ = 0 The case in which θ = π is studied further in Section 4 Second set of solutions In order to study we make a change of variables ψ 1,ψ ρ,ψ where ρ [0,1] and Ψ S 1 are defined by ρe iψ = 1 e iψ 1 + e iψ 1 Expanding this out and using 16 we compute ρcos Ψ + isin Ψ = 1 cos ψ 1 + cos ψ + 1 isin ψ 1 + sin ψ ψ1 ψ ψ1 + ψ ψ1 ψ = cos cos + icos θ = cos ψ cos θ + isin θ sin ψ1 + ψ For θ [0,π], implies that Ψ = θ or Ψ = θ + π We can rewrite 17 as sin θ θ cos ψ + cos θ θ θ θ sin ψ + sin ψ cos ψ = 0 3 In Section 6 we study the special case θ = π Here we focus on θ [0,π For Ψ = θ, implies that cos θ ψ = ρ and sin θ ψ = ± 1 ρ Accordingly, 1

13 3 implies that ρ satisfies ρsin θ + 1 ρ cos θ + ρ 1 ρ = 0 4 or ρsin θ 1 ρ cos θ ρ 1 ρ = 0 5 These imply that ρ = 1 if and only if θ = 0, and ρ = 0 if and only if θ = π For θ 0,π, equation 4 does not have any solution for ρ 0,1 since every term on the left is positive, and equation 5 has one solution for ρ 0,1 We call the corresponding equilibrium ψ sync1 := ψ 1,ψ sync1 In the case θ = 0, ψ sync1 = 0,0 Lemma 3 The equilibrium ψ sync1 is a stable node for all, θ [0, [0,π Proof: In order to prove this result, we show that the Jacobian has both eigenvalues real and negative Using cos θ ψ = ρ and sin θ ψ = 1 ρ we can write the Jacobian evaluated at this equilibrium as J ψsync1 = ρcos θ + 1 ρ sin θ + ρ 1 ρ 1 ρ 1 ρcos θ + 1 ρ sin θ + ρ 1 Since the diagonal matrix elements are equal and the off diagonal elements are equal, the eigenvalues are the sum and difference of these elements: λ 1, = ρcos θ + 1 ρ sin θ + ρ 1 ± ρ 1 We find using 5 for all, θ [0, [0,π that 1 ρ sin θ ρ 1 = 1 1 ρ cos θ ρ ρ < 0 6 Thus, for all, θ [0, [0,π, using 6 both eigenvalues are real and negative Hence ψ sync1 is a stable node for all, θ [0, [0,π For Ψ = θ + π, implies that cos θ ψ = ρ and sin θ ψ = ± 1 ρ Hence, 13

14 by 3 ρ has to satisfy ρsin θ + 1 ρ cos θ ρ 1 ρ = 0 7 or ρsin θ 1 ρ cos θ + ρ 1 ρ = 0 8 Equation 7 has one solution for ρ [0,1]; we call the corresponding equilibrium ψ antisync1 := ψ 1,ψ antisync1 Lemma 33 The equilibrium ψ antisync1 is unstable for all, θ [0, [0,π Proof: In order to prove this result, we show that the Jacobian has at least one real, positive eigenvalue Using cos θ ψ = ρ and sin θ ψ = 1 ρ we can write the Jacobian evaluated at this equilibrium as J ψantisync1 = ρcos θ + 1 ρ sin θ ρ 1 ρ 1 ρ 1 ρcos θ + 1 ρ sin θ ρ 1 The matrix has the same symmetry as in Lemma 3 and the eigenvalues can easily be computed to be λ 1, = ρcos θ + 1 ρ sin θ ρ 1 ± ρ 1 One eigenvalue is equal to ρcos θ + 1 ρ sin θ > 0 for all, θ [0, [0,π Hence ψ antisync1 is unstable for all, θ [0, [0,π Equation 8 has between zero and two solutions for ρ [0,1], although we are not able to analytically find in general the range of parameters in which there are solutions nor the nature of their stability The equilibria we get from 8 when they exist are called ψ sync := ψ 1,ψ sync and ψ antisync := ψ 1,ψ antisync For all solutions of the second set, in equations 5, 7 and 8, as gets increasingly large, ρ 1 ρ must approach zero This means that as then ρ 0 or ρ 1 We call an equilibrium synchronized if ψ 1 = ψ mod π and anti-synchronized if ψ 1 ψ = π mod π 14

15 Thus, for very large values of all the equilibria will be either synchronized ρ 1 or antisynchronized ρ 0 For modest values of, the strength of the coupling is less than or equal to the strength of the attraction to the preferred direction, and the equilibria are typically neither fully synchronized nor fully anti-synchronized In this case we call an equilibrium -almost synchronized -almost anti-synchronized if the corresponding equilibrium in the case 1, is synchronized anti-synchronized Thus, almost synchronization occurs at Ψ = θ and Ψ = θ + π Note that these solutions correspond to an exact compromise between the two preferred directions Figure shows two bifurcation diagrams in the cases a θ = 1 rad and b θ = rad with bifurcation parameter The synchrony measure ρ as defined by 1, is plotted as a function of for all equilibria in the second set of solutions There are two equilibria that do not exist for low enough values of ; these two equilibria are solution from 8 We also note in comparing Figures a and b that the stability of these two equilibria changes as a function of and θ, indicating the presence of bifurcations The other two equilibria can be seen to be defined for all values of The stable node is ψ sync1 which is the solution to 5 This equilibrium becomes synchronized as increases, ie, ρ 1 as The unstable node is ψ antisync1 which comes from 7 This equilibrium becomes anti-synchronized as increases, ie, ρ 0 as As predicted above, it can be seen that as increases ρ approaches 0 or 1 also for the two other equilibria 4 Bifurcations in the, ψ i plane As we observed in Section 3, the system 14 undergoes bifurcations as we vary the two bifurcation parameters and θ For example, the two equilibria given by the first set of solutions, ψ S1 and ψ S, are defined if and only if sin θ 1 Also we recall that the equilibria given by equation 8 are not always defined and their stability type is dependent on the values of and θ In Section 5 we study the analytically solvable case = The case θ = π, also solvable analytically, is treated in Section 6 In this section we consider bifurcations in for θ taking fixed value in three different intervals; first for π < θ < π, then for θ = π and finally for 0 < θ < π Figure 3 shows bifurcation diagrams with θ fixed in each of these three intervals The angle ψ 1 is plotted as a function of bifurcation parameter These plots are computed by solving numerically for the 15

16 Stable Equilibrium Unstable Equilibrium Stable Equilibrium Unstable Equilibrium ρ ρ a b Figure : Bifurcation diagrams in cases a θ = 1 rad and b θ = rad The bifurcation parameter is and ρ is plotted as a function of for all equilibria in the second set of solutions We note that two equilibria do not exist for low values of Stability of these same two equilibria changes type between a and b, indicating the presence of bifurcations equilibria and characterizing the stability by computing the eigenvalues of the Jacobian 41 Bifurcations in the, ψ i plane for π < θ < π The bifurcation diagram in the,ψ i plane for θ = 3π 4 is plotted in Figure 3a This is representative of the case π < θ < π There are two bifurcations: one at = 1 when two equilibria appear and one at = 0 > 1 when two equilibria disappear For 1 < < 0 there are two stable equilibria whereas there is only one stable equilibrium when is outside this region The one stable equilibrium that exists for all 0 is ψ sync1 The second stable equilibrium appears through a saddle node bifurcation, although we cannot find an analytic expression for 1, at which this bifurcation occurs We can partially prove that the second stable equilibrium disappears through a hypercritical pitchfork at = 0 From Lemma 31, when = 0 = /sin θ, the two equilibria ψ S1 and ψ S meet and are equal to ψ 0 = ψ 1,ψ 0 = 3π, θ + π For > 0, ψ S1 and ψ S no longer exist With the change of variable ψ 1,ψ ρ,ψ defined by 1 where ρ [0,1] and Ψ S 1, the equilibrium ψ S1 = ψ S = ψ 0 for = 0 becomes ρ,ψ 0 = sin θ, θ + π This equilibrium also solves equation 8 and corresponds to ψ sync at = 0 Hence a third branch of equilibria from the second set of solutions goes through the bifurcation point = 0 It is easy to show that no other branch of equilibria crosses 16

17 ψ1 ψ1 Stable Node Unstable Node Stable Node Saddle Point Saddle Point a b Stable Node Unstable Node Saddle Point ψ1 c Figure 3: Bifurcation diagrams in cases a θ = 3π 4, b θ = π and c θ = π 4 The bifurcation parameter is and ψ 1 is plotted as a function of for all equilibria of the system We observe the hypercritical pitchfork bifurcation for θ = 3π 4 at = 0 For the case θ = π 4, the bifurcation at = 0 only consists of a change in the number of equilibria but does not affect the stability of the system In the case that θ = π, the bifurcation only consists in the disappearance of two saddle points simultaneously with the appearance of two new ones In order to prove that the bifurcation = 0 is a hypercritical pitchfork bifurcation, we use the extension for pitchforks of the general theorem for saddle node bifurcations in [14] However, of the three conditions to check in the theorem, we can verify only the first two Thus, this is a partial proof 1 Non-degeneracy of the linearization The linearization of 14 at ψ = ψ 0 and = 0 is J 0 = f ψ = cot θ 1 1 ψ0,

18 where f is the vector field given by 14 with corresponding state vector ψ = ψ 1,ψ This linearization is non-degenerate since it has a simple zero eigenvalue We set v = 1 and 1 w = 1 1 to be, respectively, the right and left eigenvectors of the linearization for the zero eigenvalue Transversality condition to control non-degeneracy with respect to the parameter For this condition we first check if the eigenvalues cross the imaginary axis with non-zero speed We compute f ψ = 1 ψ0, 0 cos θ which implies that w f ψ v = 0 This means that the velocity with respect to of ψ0, 0 the eigenvalues of the Jacobian evaluated at ψ sync = ψ 0 and = 0 is zero when reaching the value zero at the bifurcation The conditions of this theorem are only sufficient though, we can still prove the bifurcation using the more general form of this condition It remains to show that the equilibrium ψ sync goes from stable to unstable through the bifurcation To do so we look at the eigenvalues of the Jacobian evaluated at ψ sync and show that the stability type of this equilibrium changes as crosses the bifurcation value 0 Using equation 16, we can write the Jacobian evaluated at ψ sync as f ψ = ψsync cos θ ψ cos ψ θ cos ψ θ cos ψ θ cos θ ψ cos ψ θ ψsync 9 The eigenvalues are λ 1 = cos θ ψ and λ = cos θ ψ cos ψ θ Since λ ψ0, 0 = cos θ ψ cos ψ θ ψ0, 0 = cot θ < 0 for θ > π, we look for a change of sign of λ 1 through the bifurcation First we show that along the branch of equilibria corresponding to ψ sync, near the bifurcation at = 0, ψ is an increasing function of, ie, that ψ > 0 Since on this branch ψ0, 0 of equilibria, equation 8 is satisfied, we take the partial derivative of both sides of 8 18

19 with respect to to get ρ sin θ + ρ cos θ 1 ρ + 1 ρ ρ + ρ 1 ρ = ρ Following the proof of Lemma 33, we can express on this branch ρ as a function of ψ as θ ρ = cos ψ 31 Taking partial derivative with respect to of both sides of equation 31 gives ρ θ = sin ψ ψ 3 Substituting 3 into equation 30 gives At the bifurcation, we have ψ = ρ + ρ 3 sin θ ψ 1 ρ sin θ ρcos θ + ρ ψ = 1 ψ0, 0 4 sin θ tan θ > 0 θ > π, 33 ie, about the bifurcation, on the branch of equilibria of ψ sync, for θ > π, ψ is a increasing function of Since λ 1 ψsync1 = 0 at = 0, then we can conclude that λ 1 ψsync = cos θ ψ ψsync is negative for < 0 and positive for > 0 Thus we have proved that ψ sync changes from stable node to saddle point through the bifurcation 3 Transversality condition to control non-degeneracy with respect to the dominant effect of the cubic nonlinear term We first check this condition by computing w i v j v k v l 3 f i ψ j ψ k ψ l = 0, ψ0, 0 for all i,j,k,l {1,} and f i is the ith component of the vector field f given by 14 Since 19

20 these terms are all zero, this isn t sufficient to satisfy the condition Instead, to prove the condition, we suggest to show that the dynamics on the center manifold have a non-degenerate cubic term If this can successfully be carried out, we expect the sign of the cubic term to be positive proving that the bifurcation is a hypercritical pitchfork 4 Bifurcations in the, ψ i plane for θ = π The bifurcation diagram in the,ψ i plane for θ = π is plotted in Figure 3b There is one bifurcation at = when two equilibria disappear and two new ones appear This case is solvable analytically The system 14 becomes ψ 1 = sinψ 1 + sin ψ ψ 1 π ψ = sin ψ + sin ψ 1 ψ From Section 3 we first observe that there is a bifurcation at sin θ = 1, ie, two equilibria in the first set disappear For θ = π, the bifurcation point is at = 41 Equilibria Equation 17 at θ = π becomes cos ψ = cos ψ After some trigonometric manipulation we can rewrite this equation as cos ψ + cos ψ = 0 34 We consider first the case that 0, In this case equation 34 has two solutions ψ = ± arccos This and the solutions of 15 give us a total of four equilibria as follows 1 ψ sync1 = π arccos 1+ 1+,arccos By Lemma 3, the equilibrium ψ sync1 is a stable node for 0, 0

21 ψ antisync1 = π arccos 1+ 1+, arccos By Lemma 33, the equilibrium ψ antisync1 is an unstable node for 0, 3 ψ S1 = π + arccos,arccos By Lemma 31, ψ S1 is a saddle point for all 0, 4 ψ S = π arccos, arccos By Lemma 31, ψ S is a saddle point for all 0, We consider next the case that > The equilibria from the first set of solutions are not defined when > and θ = π Equation 34, in this case, has four solutions 1 ± 1 + ψ = ± arccos This gives us a total of four equilibria as follows 1 ψ sync1 = π arccos 1+ 1+,arccos By Lemma 3, the equilibrium ψ sync1 is a stable node for > ψ antisync1 = π arccos 1+ 1+, arccos By Lemma 33, the equilibrium ψ antisync1 is an unstable node for > 3 ψ antisync = π arccos 1 1+,arccos 1 1+ The Jacobian of the system evaluated at this equilibrium is J = The eigenvalues of this matrix are { , Hence the linearization has its eigenvalues of opposite sign > The equilibrium ψ antisync is a saddle point for all > 4 ψ sync = π + arccos 1 1+, arccos 1 1+ } 1

22 The Jacobian of the system evaluated at this equilibrium is J = The eigenvalues of this matrix are { , Hence the linearization has its eigenvalues of opposite sign > The equilibrium ψ sync is a saddle point for all > } 4 Analysis of the bifurcation diagram The analysis of the previous subsection shows that the bifurcation at = consists in the disappearance of two saddles ψ S1 and ψ S, and the simultaneous appearance of two new saddles ψ antisync and ψ sync At the bifurcation, these four equilibria come together at ψ 1,ψ =, θ = π = 3π,π This equilibrium is highly degenerate; the linearization J is equal to the zero matrix see Lemma 31 This degenerate equilibrium will be encountered again in Section 5 when we set = and study bifurcation in the θ,ψ i plane The θ = π plane studied here and the = plane studied in Section 5 are two orthogonal slices of the full parameter space, θ,ψ i 43 Bifurcation in the, ψ i plane for 0 < θ < π The bifurcation diagram in the,ψ i plane for θ = π 4 is plotted in Figure 3c This is representative of the case 0 < θ < π There are two bifurcations: one at = 1 when two equilibria appear and one at = 0 > 1 when two equilibria disappear For 1 < < 0, there are two additional equilibria but the system still only has one stable equilibrium We are not able to find an analytic expression of 1, at which the additional two equilibria appear from equation 8 However, we know we lose the two equilibria from the first set of solutions when 0 = /sin θ Unlike the case of π < θ < π there is always one and only one stable equilibrium When the two saddles from the first set of solution disappear, there is no pitchfork, rather the equilibrium from equation 8, ψ antisync, switches from being an unstable node to a saddle To prove this we consider the linearization of the system near this bifurcation evaluated on the branch of the equilibria corresponding to ψ antisync The eigenvalues of the Jacobian given by equation 9 are

23 λ 1 = cos θ ψ and λ = cos θ ψ cos ψ θ For θ < π, in some neighborhood of the bifurcation, λ > 0 since λ ψ0, 0 = cot θ > 0 for θ < π The eigenvalue λ 1, as we saw in Section 41, changes sign through the bifurcation In order to determine if the change is from positive to negative or negative to positive, we examine how ψ changes as a function of near the bifurcation Using equation 33, we get ψ = 1 ψ0, 0 4 sin θ tan θ < 0, θ 0, π Hence ψ is a strictly decreasing function of around the bifurcation It is then easy to see that λ 1 ψantisync = cos θ ψ ψantisync becomes negative as crosses the bifurcation value 0 This proves that ψ antisync is an unstable node before the bifurcation and a saddle after the bifurcation Thus, the disappearance of the saddles ψ S1 and ψ S at 0 does not affect the stable equilibria of the system, only the number of unstable equilibria and the type of one unstable equilibrium 5 Bifurcations in the case = In this section we set = and study the bifurcations in the θ,ψ i plane This case is solvable analytically In the model 14, = implies for each subgroup that the strength of the attraction towards the preferred direction is equal to the strength of the attraction to align with the other subgroup The system 14 dynamics become ψ 1 = sin ψ 1 + sinψ ψ 1 ψ = sin θ ψ sin ψ ψ 1 3

24 51 Equilibria For =, equation 17 becomes sin θ ψ = sin ψ θ This equation has four solutions, ψ = 3 θ 3 θ + π 3 3 θ + 4π 3 π The system therefore has a total of six equilibria as follows 1 ψ sync1 = 1 3 θ, 3 θ By Lemma 3, the equilibrium ψ sync1 is a stable node for θ [0,π] ψ sync = 1 θ 3 π 3, θ 3 + π 3 The Jacobian of the system evaluated at this equilibrium is 1 J = cos 3 θ π The eigenvalues of this matrix are { cos 1 θ 3 π 3, 3cos 1 θ 3 π } 3 Both eigenvalues are strictly positive for θ [0, π, and both strictly negative for θ π,π] The equilibrium ψ sync is an unstable node for θ [0, π and a stable node for θ π,π] 3 ψ antisync1 = 1 θ 3 4π 3, θ 3 + 4π 3 By Lemma 33, the equilibrium ψ antisync1 is an unstable node for θ [0,π] 4 ψ antisync = θ π,π The Jacobian of the system evaluated at this equilibrium is J = cos θ The eigenvalues of this matrix are { cos θ,cos θ } which are of opposite sign for all θ [0, π π,π] So the equilibrium ψ antisync is a saddle point for θ [0, π π,π] 4

25 5 ψ S1 = θ + π, θ By Lemma 31, the equilibrium ψ S1 is a saddle point for all θ [0, π π,π] 6 ψ S = θ,π By Lemma 31, the equilibrium ψ S is a saddle point for all θ [0, π π,π] Figure 4 shows an example of the six equilibria in the case = and θ = 1 rad The only stable equilibrium is ψ sync1 which for this example corresponds to motion in the Ψ = 5 rad direction ψ sync1 ψ sync ψ antisync1 ψ antisync ψ S1 ψ S Figure 4: Picture of the six equilibria for = and θ = 1 rad The solid arrow represents ψ 1 on the unit circle, ie, the average heading of the first informed subgroup, and the dashed arrow represents ψ, the average heading of the second informed subgroup 5 Analysis of the bifurcation diagram The analysis of Section 51 shows that the stability type of one of the equilibria, ψ sync, changes at θ = π from an unstable node to a stable node The equilibrium ψ sync for θ = π is a highly degenerate equilibrium; the linearization J is equal to the zero matrix This is the same bifurcation point encountered in Section 4, but approached from an orthogonal direction in the full parameter space, θ,ψ i Figure 5 shows the bifurcation diagram in the θ,ψ 1 plane, ie, ψ1 as a function of bifurcation parameter θ In the bifurcation diagram Figure 5 four equilibria come together at the point in phase space ψ 1,ψ = 3π,π when θ = π This bifurcation is one of Thom s seven 5

26 Unstable Node Saddle Point Stable Node ψ1 : 5 : 7 : 9 Region 1A θ [0, π ] Region 1B θ π, π] π θ Figure 5: Bifurcation diagram in the θ,ψ 1 plane, ie ψ1 as a function of bifurcation parameter θ fixing = Since the equilibria ψ antisync and ψ S1 have the same value for ψ 1 but a different value for ψ, we see on this diagram only five equilibria even though there are six At θ = π there are only three distinct equilibria; this is the degenerate point of the system The multiplicity of the equilibrium 3π,π is four elementary catastrophes; it is called the elliptic umbilic [15] Catastrophe theory applies to gradient systems, and the elementary catastrophes are classified by looking at the form of the potential As discussed in Section, our system obeys gradient dynamics and the associated potential for = is V = cos ψ 1 + cos θ ψ + cos ψ1 ψ 35 To identify the bifurcation as an elliptic umbilic, we examine the unfolding of this potential near the catastrophe ψ 1,ψ, θ = 3π,π, π We write 35 as V = cos u + 3π π + cos + a π + v + cos u + 3π π + v, 36 where u,v and a are respectively the deviation of ψ 1 from 3π, ψ from π and θ from π A Taylor 6

27 expansion of 36, keeping terms up to third order in u and v gives V = cos a 1 v 3 + uv 3! vu sin a v + 1 cos a v + sin a After the following change of variables: x = 1 3 4cos a 1 v y = 3 6 u 1 4cos a 1 6 v, the potential becomes V = x 3 3xy 3 3 sin a 4cos a 1 3 x 31 3 cos a 1 x + sina 37 4cos a In 37 we recognize the standard unfolding of the potential of an elliptic umbilic [16] In the following paragraph we examine the different equilibria in each of the various regions of the bifurcation diagram shown in Figure 5 Region 1A is defined by θ [0, π ] and Region 1B by θ π,π] For each case studied, we draw the pictures of each possible equilibrium stable and unstable on the unit circle, a solid arrow corresponding to ψ 1 and a dashed arrow corresponding to ψ Because = implies equal attraction to the preferred direction as to the other subgroup, equilibria are usually not fully synchronized nor anti-synchronized Instead the equilibria ψ sync1 and ψ sync are -almost synchronized and ψ antisync1 and ψ antisync are -almost anti-synchronized Since ψ S1 and ψ S from 15 are not defined for 1, we cannot use this terminology However, we note that the relative heading of ψ 1 and ψ is equal to π θ for ψ S1 and π + θ for ψ S independent of As θ increases to π, the two saddles become synchronized We call an equilibrium θ -almost synchronized if the corresponding equilibrium in the case θ π is synchronized Region 1A θ [0, π ] The equilibria in the case θ [0, π are shown in Figure 6 Figure 7 shows the equilibria at the bifurcation point θ = π In Figure 6 there are three types of equilibria: the -almost synchronized ψ sync1 and ψ sync, the -almost anti-synchronized ψ antisync1 and ψ antisync and the θ -almost synchronized ψ S1 and ψ S The only stable equilibrium, ψ sync1, 7

28 ψ sync1 ψ sync ψ antisync1 ψ antisync ψ S1 ψ S Figure 6: These diagrams show pictures of all the equilibria for θ = π 4 This is representative of the possible equilibria for the system in Region 1A without its boundary, ie, for θ [0, π The only stable equilibrium is ψ sync1 which for this example corresponds to motion in the Ψ = π 8 direction is the -almost synchronized motion of ψ 1 and ψ in the direction Ψ = θ with each heading remaining on its side nearest its preferred direction of Ψ = θ The unstable equilibria are the two -almost anti-synchronized ψ antisync1 and ψ antisync, the remaining -almost synchronized ψ sync which flanks Ψ = θ + π and the two θ -almost synchronized saddles The first saddle ψ S1 will tend to go closer to the preferred direction θ 1 = 0, and the second saddle ψ S will go closer to θ as θ π As mentioned previously, the case at the boundary θ = π is highly degenerate There are only three distinct equilibria There is still only one stable equilibrium which is -almost synchronized at Ψ = θ = π 4 There is also an unstable -almost anti-synchronized equilibrium ψ antisync1 at Ψ = θ + π = 5π 4 The other equilibrium corresponds to Ψ = θ + π = 5π 4 As can be seen in the bifurcation diagram of Figure 5, there is the superposition of four equilibria ψ sync, ψ S1, ψ S and ψ antisync This equilibrium is called a monkey-saddle in the catastrophe theory literature [16] Region 1B θ π,π] The equilibria in the case θ π,π are shown in Figure 8 Figure 9 shows the equilibria at the boundary θ = π In Figure 8 the equilibria are similar to those from the case where θ [0, π in Figure 6 except that now the -almost synchronized equilibrium 8

29 ψ sync1 ψ sync,ψ antisync,ψ S1,ψ S ψ antisync1 Figure 7: These diagrams show the equilibria of the system at the critical point, ie when both = and θ = π We only have three equilibria The second equilibrium drawn is the superposition of four equilibria ψ sync, ψ antisync, ψ S1 and ψ S ; it has multiplicity four It is called a monkey-saddle in the catastrophe theory literature ψ sync at θ + π is stable Two of the unstable equilibria ψ antisync1,ψ antisync are -almost antisynchronized As mentioned above, for ψ S1 and ψ S, the particles synchronize as θ increases; the saddle ψ S1 is closer to the preferred direction of the first particle and the saddle ψ S is closer to the preferred direction of the second particle In the case θ = π Figure 9, there are still two stable equilibria ψ sync1,ψ sync which are -almost synchronized at Ψ = θ = π and Ψ = θ + π = 3π The unstable equilibria ψ antisync1 and ψ antisync are anti-synchronized The two saddles are synchronized: ψ S1 is synchronized at the preferred direction of the first particle θ 1 = 0 and ψ S is synchronized at the preferred direction of the second particle θ = π 6 Bifurcation in the, ψ i plane for θ = π In this section, we set θ = π, and study the bifurcation in the,ψ i plane This case is solvable analytically For this case, the two preferred headings differ by 180 degrees Since the disagreement is so large, for some range of small values of the group will split without making any compromise This kind of splitting is sometimes observed in swarm-bees [17] The system 14 becomes ψ 1 = sin ψ 1 + sin ψ ψ 1 ψ = sin ψ + sin ψ 1 ψ 38 We note that this system appears in Chapter 8 of [18] 9

30 ψ sync1 ψ sync ψ antisync1 ψ antisync ψ S1 ψ S Figure 8: These diagrams show the pictures of all the equilibria for θ = 3π 4 This is representative of the possible equilibria for the system in Region 1B without its boundary ie for θ π,π The two saddles, ψ S1 and ψ S, tend to be more synchronized than in Figure 6 since θ is closer to π ψ S1 is closer to the preferred direction of the first subgroup and ψ S is closer to the preferred direction of the second subgroup There are two stable equilibria, ψ sync1 and ψ sync 61 Equilibria For θ = π, the equation 17 becomes sin ψ = sin ψ After some trigonometric manipulation we can rewrite this equation as sin ψ 1 + cos ψ = 0 39 We consider first the case that [0,1 In this case equation 39 has two solutions 0 ψ = π This give us a total of four equilibria as follows 1 ψ antisync1 = π,0 By Lemma 33, the equilibrium ψ antisync1 is an unstable node for [0,1] 30

31 ψ sync1 ψ sync ψ antisync1 ψ antisync ψ S1 ψ S Figure 9: These diagrams show the equilibria of the system at the right boundary of Region 1B, ie, for θ = π Only equilibrium ψ sync1 and ψ sync the -almost synchronized equilibria depend on The other equilibria are anti-synchronized ψ antisync1 and ψ antisync or synchronized ψ S1 and ψ S for all ψ antisync = 0,π The Jacobian of the system evaluated at this equilibrium is J = The eigenvalues of this matrix are { 1, 1 + } Hence the linearization has both eigenvalues strictly negative [0,1 The equilibrium ψ antisync is a stable node [0,1 3 ψ S1 = 0,0 By Lemma 31, the equilibrium ψ S1 is a saddle point for all [0,1] 4 ψ S = π,π By Lemma 31, the equilibrium ψ S is a saddle point for all [0,1] 31

32 We consider next the case that > 1 Equation 39, in this case has four solutions ψ = arccos 1 arccos 1 0 π This gives a total of six equilibria as follows 1 ψ sync1 = π arccos 1,arccos 1 By Lemma 3, the equilibrium ψ sync1 is a stable node for > 1 ψ sync = π + arccos 1, arccos 1 The Jacobian of the system evaluated at this equilibrium is J = { The eigenvalues of this matrix are , 1 } Hence the linearization has both eigenvalues strictly negative > 1 The equilibrium ψ sync is a stable node > 1 3 ψ antisync1 = π,0 By Lemma 33, the equilibrium ψ antisync1 is an unstable node for 1 4 ψ antisync = 0,π The Jacobian of the system evaluated at this equilibrium is J = The eigenvalues of this matrix are { 1, 1 + } Hence the linearization has its eigenvalues of opposite sign > 1 The equilibrium ψ antisync is a saddle point > 1 5 ψ S1 = 0,0 By Lemma 31, the equilibrium ψ S1 is a saddle point for all 1 3

33 A A A B < B 6 ψ S = π,π By Lemma 31, the equilibrium ψ S is a saddle point for all 1 6 Analysis of the bifurcation diagram Stable Equilibrium Unstable Equilibrium ψ1 ; < = ; Figure 10: Bifurcation diagram in the,ψ 1 plane, ie, ψ 1 as a function of bifurcation parameter fixing θ = π At = 1 we have a supercritical pitchfork bifurcation We have one stable equilibrium for < 1 and two stable equilibria for > 1 The analysis of the previous subsection, shows that a bifurcation occurs at = 1 The bifurcation diagram Figure 10, suggests that there is a supercritical pitchfork bifurcation To prove this, we use the extension for pitchforks of the general theorem for saddle node bifurcations in [14] There are three conditions to check in the theorem We define ψ 0 = ψ 1,ψ 0 = 0,π, 0 = 1 1 Non-degeneracy of the linearization The linearization of 38 at ψ = ψ 0 and = 0 is J 0 = f ψ = 1 1, ψ0,

34 where f is the vector field given by 38 with corresponding state vector ψ = ψ 1,ψ This linearization is non-degenerate since it has a simple zero eigenvalue We set v = 1 and 1 w = 1 1 to be respectively the right and left eigenvectors of the linearization for the zero eigenvalue Transversality condition to control non-degeneracy with respect to the parameter For this condition we check if the eigenvalues cross the imaginary axis with non-zero speed We compute f ψ = 1 1 1, ψ0, which implies that w f with non-zero speed ψ v = 0 Hence, the eigenvalues cross the imaginary axis ψ0, 0 3 Transversality condition to control non-degeneracy with respect to the dominant effect of the cubic nonlinear term For this condition we compute w i v j v k v l 3 f i ψ j ψ k ψ = 6 < 0, l ψ0, 0 for all i,j,k,l {1,} and f i is the ith component of f Since we get a strictly negative number, the pitchfork is supercritical This last condition completes the proof of the existence of a codimension-one supercritical pitchfork bifurcation at ψ = 0,π, = 1 Before the bifurcation < 1, the only stable equilibrium is ψ antisync = 0,π This corresponds to the case where each informed subgroup follows its own preferred direction; there is no compromise between the individuals and the group splits When < 1 the strength of the coupling force compared to the preferred direction is too weak to influence the stable steady state of the system The motion of the group is the same as if there were no coupling between the two informed subgroups For > 1, there are two stable equilibria, ψ sync1 and ψ sync These 34