Dynamics of Collective Decisions in a Time ... - World Scientific
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International Journal of Bifurcation and Chaos, Vol. 24, No. 8 (2014) 1440010 (9 pages) c World Scientific Publishing Company DOI: 10.1142/S0218127414400100
Dynamics of Collective Decisions in a Time-Dependent Environment Stamatios C. Nicolis Mathematics Department, Uppsala University, Uppsala 751 06, Sweden [email protected] Gr´egoire Nicolis Centre for Nonlinear Phenomena and Complex Systems, Universit´e Libre de Bruxelles, Campus Plaine, C.P. 231, B-1050 Brussels, Belgium [email protected] Received February 2, 2014 The field of dynamical systems had been revolutionized by the seminal work of Leonid Shil’nikov. As a tribute to his genius we analyze in this paper the response of dynamical systems to systematic variations of a control parameter in time, using a normal form approach. Explicit expressions of the normal forms and of their parameter dependences are derived for a class of systems possessing multiple steady-states associated to collective choices between several options in group-living organisms, giving rise to bifurcations of the pitchfork and of the limit point type. Depending on the conditions, delays in the transitions between states, stabilization of metastable states, or on the contrary enhancement of the choice of the most rewarding option induced by the time dependence of the parameter are identified. Keywords: Normal forms; dynamical bifurcations; collective behavior; mathematical biology.
1. Introduction The time evolution of a natural system is typically described by a set of coupled differential equations of the form dyi = vi ({yi }, µ) dt
i = 1, . . . , n
(1)
where {yi } is the set of state variables, the vector field {vi } is typically a nonlinear function of {yi } and µ is a set of control parameters accounting for the environmental constraints. It is well known that as long as the control parameters µ remain fixed in time, there exist critical values µc at which the behavior of a
certain reference solution changes qualitatively. New branches of solutions then take over, leading to such phenomena as multiple simultaneously stable steady states, periodic or quasi-periodic oscillations, and deterministic chaos. In general, there exists no universal classification of all transitions and all different behaviors in a given dynamical system. However, a fundamental result of nonlinear dynamics is that in the vicinity of certain types of criticalities, the dynamics as described by the full set of Eq. (1) simplifies considerably. Specifically, there exists a reduced set of variables {xi } (i = 1, . . . , m, m < n) related to combinations of the original {yi }’s to which one refers to as order parameters, satisfying a set of evolution equations
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whose structure is universal in the sense that it only depends on the type of criticality considered, to which one refers to as normal forms [Guckenheimer & Holmes, 1983]: dxi = fi ({xj }, λ), dt
i = 1, . . . , m, m < n
(2)
where λ’s are combinations of the original parameters. On the other hand, in most situations of interest the environment in which a system is embedded is subjected to variability. New constraints may be switched on and old ones disrupted, suddenly or gradually, at some stage of the evolution following, for instance, the action of an external timedependent field, an extreme event or a systematic temperature drift due to climatic change. At the level of the dynamical system of interest this will be reflected by the presence of control parameters that are no longer fixed but become now time-dependent. One is then led to replace the dynamical system of Eq. (1) by the nonautonomous system [Benoˆıt, 1991; Nicolis & Nicolis, 2014; Nicolis & Dussutour, 2008] dyi = vi ({yi }, µ(t)), dt
i = 1, . . . , n.
2. Mathematical Model of Collective Decision-Making in the Presence of Several Options Let ci , i = 1, . . . , n be the number of individuals in a group adopting option i — for instance, a group of ants presented with food sources of different nutrient values. A generic model capturing the competition between the options can then be written in the form of Eq. (1), where vi = Φσi
i = 1, . . . , n.
(4)
j=1
The first, positive term, corresponds to the attractiveness of option i over the others. Here Φ is the flux of individuals toward the resource area (related to the size of the group), σi measures the quality of the option and k is a threshold beyond which the presence of an option becomes effective. Finally the second, negative term describes the rate at which option i is abandoned [Camazine et al., 2001; Beckers et al., 1992]. Introducing scaled variables and parameters through the transformation
(3)
Our objective in the present work is to identify some signatures arising from the presence of timedependent control parameters. In particular, we will inquire whether the universality characterizing the reduction of system in Eq. (1) to the normal form in Eq. (2) prevails in the presence of time-dependent control parameters [Eq. (3)] in as much as the transformations leading to the normal form are likely to affect the parameters in a system-dependent manner. We will illustrate these points on a class of dynamical systems arising in biology, in the context of collective choices that group-living organisms are led to make when confronted with several options. The model is presented in Sec. 2, where the bifurcation diagram of the steady-state solutions for fixed parameter values is constructed. Sections 3 and 4 are devoted to the effect of time-dependent parameters and to the construction of the explicit forms of the associated normal forms in two representative cases corresponding to a pitchfork and to a limit point bifurcation. The main conclusions are summarized in Sec. 5.
(k + ci )2 − νci , n 2 (k + cj )
yi =
ci , k
qi =
σi , k
φ=
Φ ν
(5)
and rescaling the time variable accordingly, one may reduce Eqs. (1) and (4) to the following system, written here explicitly for the simplest case of n = 2 options: dy1 = v1 (y1 , y2 , φq1 ) dt = φq1
(1 + y1 )2 − y1 (1 + y1 )2 + (1 + y2 )2
dy2 = v2 (y1 , y2 , φq2 ) dt = φq2
(6)
(1 + y2 )2 − y2 . (1 + y1 )2 + (1 + y2 )2
We first consider the case where the parameters do not vary in time and seek for the steady-state solutions of this system. Adding the two equations one obtains y1 y2 + = φ. (7) q1 q2
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Dividing the two equations and substituting y2 in terms of y1 from Eq. (7) one further obtains a closed equation for y1 , q 22 q 22 q2 3 1 + 2 y1 − q1 φ 1 + 2 2 + 2 − 1 y 21 q1 q1 q1 + [2 + q 22 φ2 + 2φ(q2 − q1 )]y1 − q1 φ = 0.
(8)
Two qualitatively different situations may arise, according to whether q1 and q2 are equal or take different values [Nicolis & Deneubourg, 1999].
2.1. The two options have the same quality, q1 = q2 = q We take q = 1, without loss of generality. Substituting into Eq. (8) one finds two kinds of solutions. • A “homogeneous” solution in which the two options are chosen on equal footing, y0 =
φ . 2
(9a)
• A pair of “inhomogeneous” solutions in which one of the two options is preferred: φ φ2 − 1. (9b) y± = ± 2 4 The corresponding bifurcation diagram is depicted in Fig. 1. Branch y0 loses its stability at a value φc = 2 of the control parameter, beyond which stable branches y± emerge through a pitchfork bifurcation.
2.2. One of the two options has a higher quality We choose q2 = 1, q1 = 1 + η (η > 0). To sort out the effect of the “imperfection” η we consider the case η 1. Keeping dominant terms in η one may then cast the solutions of Eq. (8) in the form y1 =
φ +w 2
where w satisfies the cubic equation η φ2 w − φ(φ + 2) = 0. w3 + 1 − 4 4
(10a)
(10b)
As long as η is not strictly zero this equation admits a single real solution for φ less than a critical
Fig. 1. Bifurcation diagram of steady-state solutions of system (6) for q1 = q2 = 1.
value φ∗ and three real solutions for φ larger than φ∗. At φ = φ∗ two of these latter solutions merge. The criticality φ∗ is determined by the vanishing of the discriminant of the cubic, 2 3 φ∗ −1 27η 2 4 . (10c) = 64 φ∗2 (φ∗ + 2)2 The corresponding bifurcation diagram is depicted in Fig. 2, showing the presence of a limit point bifurcation at φ = φ∗. As can be seen, the upper branch exists for all φ values and is always stable. Among the two other branches, which exist for φ > φ∗, the lower branch is stable and the upper one is unstable. To determine the relative attractivity of the stable solutions, we integrate the full Eqs. (6) for a range of initial conditions distributed uniformly in phase space [Nicolis et al., 2011]. As expected, when the two options are equal, y1 is present 50% of the time (case of pitchfork bifurcation, not shown). The mean fraction of y1 within the population as a function of the flux φ when q1 > q2 (case of limit point bifurcation) is shown in Fig. 3. As can be seen, y1 always dominates before criticality, reaching values up to 90%. Subsequently its selectivity is decreasing before saturating for large φ to a value greater than 0.5 (dashed line in Fig. 3). If on the other hand, the initial conditions are sampled in a narrow
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3. Effect of Time-Dependent Control Parameters: Pitchfork Bifurcation
Fig. 2.
As in Fig. 1 but for q2 = 1, q1 = 1 + η with η = 0.1.
phase space band surrounding a state of equipartition y1 ≈ y2 with y1 and y2 both small, one obtains (plain line in Fig. 3) a monotonic increase of the mean ratio. This is due to the fact that this way of sampling favors the attractor associated to option 1.
We now angment the analysis of the preceding section to account for a time dependence of φ, the principal control parameter present in the problem, first limiting ourselves to the case q1 = q2 = 1. The type of time dependence of φ(t) chosen is that of a slow drift, corresponding to a systematic change of environmental conditions occurring on a long time scale compared to the system’s intrinsic time scales. This will be accounted for by the presence of a scaling factor in front of the t-argument of the function φ(t), with 0 < 1. Rather than study the full set of Eqs. (6) with φ a time-dependent function of the above kind we outline, in the spirit of the Introduction, the reduction to the normal form description. We first observe that summing Eqs. (6), one obtains a closed equation for the sum u = y1 + y2 , du = φ(t) − u. dt
(11)
On the other hand, using symmetry arguments one is led to anticipate that the difference x = y1 − y2 should play the role of the order parameter. To determine the conditions under which this variable satisfies a closed equation — the normal form — we subtract the two Eqs. (6). After some elementary manipulations we obtain x(u(t) + 2) dx = 2φ(t) − x. dt (u(t) + 2)2 + x2
(12)
We see that in addition to its intrinsic dynamics, variable x is subjected to a forcing by the timedependent control parameter φ(t) and by the quantity u(t), which satisfies Eq. (11). Now since φ depends on time through the combination t, one sees immediately that the solution of Eq. (11) can be written in the form u(t) = φ(t) + O() Fig. 3. Mean fraction of y1 within a population as a function of the flux φ for q2 = 1, q1 = 1 + η and η = 0.1 (case of limit point bifurcation). (Dashed line) Initial conditions distributed uniformly in phase space. (Full line) Initial conditions limited to a narrow band around a state of equipartition with y1 and y2 = 0.1. Number of initial conditions is 2000.
+ (exponentially decaying terms). Keeping the dominant contribution and substituting into Eq. (12) we obtain
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φ(t) + 2 dx = 2φ(t) x − x. dt (φ(t) + 2)2 + x2
(13)
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This relation provides us with the extension of the center manifold associated to the pitchfork bifurcation, to the case where the control parameter varies slowly on time. To arrive at the generalized normal form, we expand in x and keep terms up to third order: φ(t) − 2 2φ(t) dx = x− x3 . dt φ(t) + 2 (φ(t) + 2)3
(14)
For later use, we also write out the linearized part of Eqs. (13) and (14) around the trivial solution x = 0, dx = λ(t)x dt with
x(t) = x0 exp
t 0
(15a)
dt λ(t )
(15b)
where we have set λ(t) =
φ(t) − 2 . φ(t) + 2
(15c)
Consider first, as a reference, the case where the time dependence of φ is switched off ( = 0). Equation (15c) reproduces then the threshold value φc = 2 determined in Sec. 2.1 beyond which the critical eigenvalue λ of the system’s Jacobian matrix is changing sign, x(t) starts increasing exponentially and a pitchfork bifurcation is taking place, while Eqs. (13) and (14) reproduce the full bifurcation diagram of Fig. 1 or its approximate form in the vicinity of criticality. This situation changes radically in the case where φ(t) depends on time, since [cf. Eq. (15b)] it is the sign of the integral of λ(t) from zero to t rather than of λ(t) itself that controls the exponential decay or the exponential growth of x(t) and hence the transition to the instability of the trivial state x = 0. As an example, for a variation of φ in the form of a ramp, φ = φ0 + t one obtains t 4 φ0 + 2 + t . dt λ(t ) = t − ln I(t) = φ0 + 2 0
Fig. 4. Times at which the functions λ(t) and I(t) in Eqs. (15c) and (17) change sign, as a function of the ramp parameter for φ0 = 0.
behavior of x(t) beyond this value as obtained from Eqs. (15a)–(15c). We conclude that the time variation of φ enhances the stability of the trivial (homogeneous) state and delays the bifurcation.
(16)
(17)
Figure 4 depicts λ(t) and I(t) as functions of t. We see that while λ(t) changes sign at time t0 = |φ0 − 2|/, I(t) changes sign at a quite different, delayed value td . Figure 5 depicts the
Fig. 5. Order parameter x versus time for q1 = q2 = 1, = 0.01, φ0 = 0 as given by the full equation (13) (bold dashed line), the normal form truncated to the third order (Eq. (14), full line) and the linearized version (Eq. (15a), bold line). The dashes follow the branches of the adiabatic bifurcation diagram [Eqs. (9)], with φ given by Eq. (16).
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For reference the “adiabatic” bifurcation diagram for x as obtained by setting the time derivative in Eq. (13) or (14) equal to zero is also drawn. We notice that beyond the aforementioned delay x(t) tends to merge asymptotically with the upper and lower branches of this diagram. We refer to this behavior as the adiabatic approximation. Significantly, the effective control parameter λ(t) identified in Eq. (15a) depends on time in a system-specific fashion, in the sense that its dependence is different from the one of the control parameter of the original equations as it carries the signature of the transformations leading from these equations to the normal form. In particular, while φ(t) increases without bound for t → ∞ [Eq. (16)], λ(t) saturates at a parameter-independent finite value equal to unity. We close this section by extending the numerical experiment described at the end of Sec. 2 to the case where parameter φ is time dependent. Specifically, we consider an ensemble of initial conditions randomly distributed in phase space around the state of equipartition y1 = y2 and an initial value φ0 less than φc . We subsequently integrate the full set of equations (6) for times up to a value for which φ attains the largest value plotted in the diagram of Fig. 1. As expected, the two options remain, in the mean, equivalent up to sampling errors (not shown). It is worth noting that a small portion (about 3%) of realizations remains stuck to the unstable branch, owing presumably to the aforementioned enhancement of its stability induced by the time dependence of φ.
4. Effect of Time-Dependent Control Parameters: Limit Point Bifurcation In this section, we analyze the effect of a time dependence of parameter φ in the case where one of the two options has a higher quality: q2 = 1, q1 = 1 + η, η > 0 (Sec. 2.2). Our starting point are again Eqs. (6), which we write in the form dy1 = v1 (y1 , y2 , φ, η) dt = φ(t)
(1 + y1 )2 − y1 (1 + y1 )2 + (1 + y2 )2
+ φ(t)η
(1 + y1 )2 (1 + y1 )2 + (1 + y2 )2
dy2 = v2 (y1 , y2 , φ) dt = φ(t)
(1 + y2 )2 − y2 (1 + y1 )2 + (1 + y2 )2 (18)
where φ(t) is given by Eq. (16). Contrary to the case of Sec. 3 where a global analysis was possible owing to the a priori knowledge of the order parameter associated to the problem, we here need to resort to a local description around the critical value φ∗ given by Eq. (10c). To this end we express the variables y1 , y2 and the parameter φ (keeping η fixed) as [Nicolis, 1995; Guckenheimer & Holmes, 1983] y1 = y ∗1 + w1 y2 = y ∗2 + w2
(19)
φ = φ∗ + (φ − φ∗ ) where y ∗1 , y ∗2 are given by the steady-state solutions of (18) at the criticality. We next expand Eqs. (18) around the criticality keeping terms up to the second order, ∂vi ∗ ∂vi ∗ ∂vi ∗ dwi = w1 + w2 + (φ − φ∗ ) dt ∂y1 ∂y2 ∂φ ∗ 2 ∗ 2 ∂ vi ∂ vi 1 2 w1 + 2 w1 w2 + 2 ∂y1 ∂y2 ∂y 21 2 ∗ ∂ vi + w22 (i = 1, 2). (20) ∂y 22 By construction, the Jacobian matrix {Lij } of the vector field (v1 , v2 ) ∂vi Lij = ∂yj possesses a zero eigenvalue at criticality. Let u1 u= u2 be the associated eigenvector. We define the order parameter x associated to the problem by the relation
u1 w1 = x(t) . (21) w2 u2 Multiplying both sides of Eq. (20) by the null eigenvector
+ u1 u+ = u+ 2
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of the transposed Jacobian matrix {Lji } we arrive then after same straightforward calculations at the following (closed) equation for x(t) in the vicinity of the criticality: dx = λ(t) + bx2 dt
(22a)
λ(t) = a(φ(t) − φ∗ ) = a(φ0 − φ∗ ) + at.
u+ 1
As an example, for η = 0.01 one obtains u1 = 0.711, u2 = −0.706 and the following explicit form of Eq. (22a): dx = −0.112(φ(t) − φ∗ ) + 0.163x2 dt = −0.112(φ0 − φ∗ ) − 0.00112t + 0.163x2 . (24) To complete the bifurcation diagram we need the third branch of solutions, which exists on both sides of the limit point φ∗ . Proceeding as in Sec. 2.2 we obtain the diagram of Fig. 6, where full and dashed lines denote stable and unstable solutions, respectively. For reference, the dots denote values of the solutions of the full system. The agreement is quite satisfactory, extending over a sizable range of values of φ. Notice that in the terminology of Sec. 3 the diagram in Fig. 6 is to be qualified as “adiabatic”, since the branches drawn are given by the quasi steady-state solutions of Eqs. (22) or (24) in which t and φ are merely related through Eq. (16). Interestingly, the effective control parameter λ(t) appearing in the normal form depends on time in a way similar to the dependence of the control parameter φ, except for a system-dependent scaling factor a multiplying the original ramp parameter . This reflects, once again — albeit in a milder form compared to the case of pitchfork bifurcation — the fact that the explicit form of λ(t) depends on the various transformations carried out in the derivation of the normal form.
(22b)
The coefficients a and b in Eq. (22), which is nothing but the normal form of a limit point bifurcation, are given by the following relations ∗
∂v1 ∗ ∂v2 + u+ 2 ∂φ ∂φ a= + + (u1 u1 + u2 u2 ) 2 2 ∗ 2 ∗ ∗ ∂ v1 ∂ v1 ∂ v1 + 2 u1 + 2 u1 u2 + u22 u1 2 ∂y1 ∂y2 ∂y 1 ∂y 22 b= + (u1 u+ 1 + u2 u2 ) 2 2 ∗ 2 ∗ ∗ ∂ v2 ∂ v2 ∂ v2 + 2 u1 + 2 u1 u2 + u22 u2 2 ∂y1 ∂y2 ∂y 1 ∂y 22 . + + (u1 u+ 1 + u2 u2 )
where the effective control parameter λ(t) is given by [cf. also Eq. (16)]
(23a)
(23b)
We now turn to the time-dependent behavior beyond the adiabatic limit. The objective is to identify the new effects induced by the time dependence of the control parameter. To get some insight on this point we start with an initial value of the control parameter close, but prior to the criticality
Fig. 6. Bifurcation diagrams as obtained from the full system [Eq. (18)] in the adiabatic approximation (dots) and from the normal form equation [Eq. (22)] for η = = 0.01. Full and dashed lines denote stable and unstable branches, respectively.
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and with an initial condition for x(t) close to, and slightly below the limit point. At the level of Eqs. (22) or (24) this implies that φ0 is close to φ∗ , λ0 is a small positive number and x(0) is a negative number of small absolute value (we anticipate that a < 0, b > 0). Linearizing Eq. (22a) around this state we obtain the following equation for the short time behavior of the deviation δx = x − x(0): dδx = λ0 + at + 2bx(0)δx. dt
(25)
By construction, for λ fixed and equal to λ0 , x(0) would evolve toward the stable branch of solutions which lies at a finite distance from the limit point (Figs. 2 and 7). On the other hand, for sufficiently small , and given that λ0 and x(0) are also small (see above), the initial stages of the evolution of δx as given by Eq. (25) will be very slow. It is thus possible that when the ramp is switched on the value attained by the control parameter after some lapse of time t, λ(t) = λ0 + at, will bring the system into the range of attraction of the lower (stable) branch of the bifurcation diagram, before the escape to the upper branch takes place. In other words, the time variation of the control parameter could in principle contribute to the stabilization of the less favorable state for a certain range of values of λ, and x(0) [Davies & Krishna, 1996; Nicolis & Nicolis, 2014]. To substantiate this conjecture, the full equations (18) are integrated for a range of initial conditions distributed in a narrow band around a state of equipartition (y1 ≈ y2 ) corresponding to two different φ0 values, one much smaller than and one close to, but slightly less than the critical value φ∗. Figure 7 summarizes the behavior of the mean fraction of y1 within the population versus the instantaneous value of φ, for two different values of the parameter . In both cases, the majority of trajectories are eventually attracted toward the more favorable option. Still, while the dominance of state 1 is quite clearcut for = 0.01, it is less so for = 0.1. This reflects the fact that for this latter value of there exists a time interval during which a non-negligible fraction of trajectories is attracted by the less favorable option, in agreement with the predictions drawn earlier from Eq. (25). This trend becomes more pronounced for initial conditions such that the mean ratio is less than 0.5, which tend to eventually evolve toward the attraction basin (in the sense of the adiabatic approximation) of the less favorable state.
Fig. 7. Mean fraction of y1 as a function of time covering a range of φ values from 0 to 5, for = 0.01 (full line) and = 0.1 (dashed line) and for initial φ much smaller than the critical value φ∗ (left part) and slightly less than φ∗ (right part). In both cases initial conditions are in a narrow band around a state of equipartition. Number of initial conditions as in Fig. 3.
5. Conclusions In this paper, we outlined the derivation of the normal form of a dynamical system giving rise to pitchfork and to limit point bifurcations associated to the choice between different options in a population of group-living organisms, in the presence of time-dependent control parameters. We have shown that the general structure of the normal forms is not affected by the time dependence of the parameters. On the other hand the effective control parameter appearing in the normal form depends on time in a system-specific fashion, typically different from the one in the original equations, reflecting the specific transformations that lead from these equations to the normal form. A second conclusion emerging from our study is that, far from being driven passively by the variations in the control parameter, the system responds in many cases in a highly nontrivial manner. A first example is provided by the transient stabilization of the unstable state and the associated delay in the occurrence of the transition to the new states in the case of the pitchfork bifurcation.
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A second example is provided by the important role of the ramp parameter in determining the relative attractivities of the states associated to the different options in the case of the limit point bifurcation. Depending on the magnitude of this parameter the choice of the more favorable option may be further optimized or, on the contrary, the system may be attracted transiently toward the less favorable option and remain trapped in this state thereafter. In nature, among the mechanisms at the origin of systematic variations of control parameters on a slow time scale is climatic variability. The flexible responses identified in the present work provide some clues on how a population subjected to cooperative interactions may cope with changes of this kind. In this respect it would be interesting to extend our study to account for the variability of parameters other than the flux φ present in the original equations, such as the rate ν [Eq. (4)] at which an option may be abandoned. In this latter case, the competition between the adverse actions of the two parameters may bring out some unsuspected effects. Finally, it would be interesting to extend the analysis to an arbitrary number of options, as in this case some of the bifurcations occurring will be associated to eigenvalues of increasing degeneracy [Nicolis & Deneubourg, 1999].
Acknowledgment S. C. Nicolis acknowledges support from the European Research Council Grant (Ref. IDCAB 220/ 104702003).
References Beckers, R., Deneubourg, J.-L. & Goss, S. [1992] “Trails and u-turns in the selection of a path by the ant Lasius-niger,” J. Theoret. Biol. 159, 397–415. Benoˆıt, E. [1991] Dynamic Bifurcations (Springer, Berlin). Camazine, S., Deneubourg, J.-L., Franks, N. R., Sneyd, J., Theraulaz, G. & Bonabeau, E. [2001] Self-Organization in Biological Systems (Princeton University Press). Davies, H. G. & Krishna, R. [1996] “Nonstationary response near generic bifurcations,” Nonlin. Dyn. 10, 235–250. Guckenheimer, J. & Holmes, P. [1983] Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, NY). Nicolis, G. [1995] Introduction to Nonlinear Science (Cambridge University Press). Nicolis, S. C. & Deneubourg, J.-L. [1999] “Emerging patterns and food recruitment in ants: An analytical study,” J. Theoret. Biol. 198, 575–592. Nicolis, S. C. & Dussutour, A. [2008] “Self-organization, collective decision making and resource exploitation strategies in social insects,” Eur. Phys. J. B-Cond. Matt. Compl. Syst. 65, 379–385. Nicolis, S. C., Zabzina, N., Latty, T. & Sumpter, D. J. T. [2011] “Collective irrationality and positive feedback,” PloS One 6, e18901. Nicolis, C. & Nicolis, G. [2014] “Dynamical responses to time-dependent control parameters in the presence of noise: A normal form approach,” Phys. Rev. E 89, 022903.
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International Journal of Bifurcation and Chaos, Vol. 24, No. 8 (2014) 1440010 (9 pages) c World Scientific Publishing Company DOI: 10.1142/S0218127414400100
Dynamics of Collective Decisions in a Time-Dependent Environment Stamatios C. Nicolis Mathematics Department, Uppsala University, Uppsala 751 06, Sweden [email protected] Gr´egoire Nicolis Centre for Nonlinear Phenomena and Complex Systems, Universit´e Libre de Bruxelles, Campus Plaine, C.P. 231, B-1050 Brussels, Belgium [email protected] Received February 2, 2014 The field of dynamical systems had been revolutionized by the seminal work of Leonid Shil’nikov. As a tribute to his genius we analyze in this paper the response of dynamical systems to systematic variations of a control parameter in time, using a normal form approach. Explicit expressions of the normal forms and of their parameter dependences are derived for a class of systems possessing multiple steady-states associated to collective choices between several options in group-living organisms, giving rise to bifurcations of the pitchfork and of the limit point type. Depending on the conditions, delays in the transitions between states, stabilization of metastable states, or on the contrary enhancement of the choice of the most rewarding option induced by the time dependence of the parameter are identified. Keywords: Normal forms; dynamical bifurcations; collective behavior; mathematical biology.
1. Introduction The time evolution of a natural system is typically described by a set of coupled differential equations of the form dyi = vi ({yi }, µ) dt
i = 1, . . . , n
(1)
where {yi } is the set of state variables, the vector field {vi } is typically a nonlinear function of {yi } and µ is a set of control parameters accounting for the environmental constraints. It is well known that as long as the control parameters µ remain fixed in time, there exist critical values µc at which the behavior of a
certain reference solution changes qualitatively. New branches of solutions then take over, leading to such phenomena as multiple simultaneously stable steady states, periodic or quasi-periodic oscillations, and deterministic chaos. In general, there exists no universal classification of all transitions and all different behaviors in a given dynamical system. However, a fundamental result of nonlinear dynamics is that in the vicinity of certain types of criticalities, the dynamics as described by the full set of Eq. (1) simplifies considerably. Specifically, there exists a reduced set of variables {xi } (i = 1, . . . , m, m < n) related to combinations of the original {yi }’s to which one refers to as order parameters, satisfying a set of evolution equations
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whose structure is universal in the sense that it only depends on the type of criticality considered, to which one refers to as normal forms [Guckenheimer & Holmes, 1983]: dxi = fi ({xj }, λ), dt
i = 1, . . . , m, m < n
(2)
where λ’s are combinations of the original parameters. On the other hand, in most situations of interest the environment in which a system is embedded is subjected to variability. New constraints may be switched on and old ones disrupted, suddenly or gradually, at some stage of the evolution following, for instance, the action of an external timedependent field, an extreme event or a systematic temperature drift due to climatic change. At the level of the dynamical system of interest this will be reflected by the presence of control parameters that are no longer fixed but become now time-dependent. One is then led to replace the dynamical system of Eq. (1) by the nonautonomous system [Benoˆıt, 1991; Nicolis & Nicolis, 2014; Nicolis & Dussutour, 2008] dyi = vi ({yi }, µ(t)), dt
i = 1, . . . , n.
2. Mathematical Model of Collective Decision-Making in the Presence of Several Options Let ci , i = 1, . . . , n be the number of individuals in a group adopting option i — for instance, a group of ants presented with food sources of different nutrient values. A generic model capturing the competition between the options can then be written in the form of Eq. (1), where vi = Φσi
i = 1, . . . , n.
(4)
j=1
The first, positive term, corresponds to the attractiveness of option i over the others. Here Φ is the flux of individuals toward the resource area (related to the size of the group), σi measures the quality of the option and k is a threshold beyond which the presence of an option becomes effective. Finally the second, negative term describes the rate at which option i is abandoned [Camazine et al., 2001; Beckers et al., 1992]. Introducing scaled variables and parameters through the transformation
(3)
Our objective in the present work is to identify some signatures arising from the presence of timedependent control parameters. In particular, we will inquire whether the universality characterizing the reduction of system in Eq. (1) to the normal form in Eq. (2) prevails in the presence of time-dependent control parameters [Eq. (3)] in as much as the transformations leading to the normal form are likely to affect the parameters in a system-dependent manner. We will illustrate these points on a class of dynamical systems arising in biology, in the context of collective choices that group-living organisms are led to make when confronted with several options. The model is presented in Sec. 2, where the bifurcation diagram of the steady-state solutions for fixed parameter values is constructed. Sections 3 and 4 are devoted to the effect of time-dependent parameters and to the construction of the explicit forms of the associated normal forms in two representative cases corresponding to a pitchfork and to a limit point bifurcation. The main conclusions are summarized in Sec. 5.
(k + ci )2 − νci , n 2 (k + cj )
yi =
ci , k
qi =
σi , k
φ=
Φ ν
(5)
and rescaling the time variable accordingly, one may reduce Eqs. (1) and (4) to the following system, written here explicitly for the simplest case of n = 2 options: dy1 = v1 (y1 , y2 , φq1 ) dt = φq1
(1 + y1 )2 − y1 (1 + y1 )2 + (1 + y2 )2
dy2 = v2 (y1 , y2 , φq2 ) dt = φq2
(6)
(1 + y2 )2 − y2 . (1 + y1 )2 + (1 + y2 )2
We first consider the case where the parameters do not vary in time and seek for the steady-state solutions of this system. Adding the two equations one obtains y1 y2 + = φ. (7) q1 q2
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Dividing the two equations and substituting y2 in terms of y1 from Eq. (7) one further obtains a closed equation for y1 , q 22 q 22 q2 3 1 + 2 y1 − q1 φ 1 + 2 2 + 2 − 1 y 21 q1 q1 q1 + [2 + q 22 φ2 + 2φ(q2 − q1 )]y1 − q1 φ = 0.
(8)
Two qualitatively different situations may arise, according to whether q1 and q2 are equal or take different values [Nicolis & Deneubourg, 1999].
2.1. The two options have the same quality, q1 = q2 = q We take q = 1, without loss of generality. Substituting into Eq. (8) one finds two kinds of solutions. • A “homogeneous” solution in which the two options are chosen on equal footing, y0 =
φ . 2
(9a)
• A pair of “inhomogeneous” solutions in which one of the two options is preferred: φ φ2 − 1. (9b) y± = ± 2 4 The corresponding bifurcation diagram is depicted in Fig. 1. Branch y0 loses its stability at a value φc = 2 of the control parameter, beyond which stable branches y± emerge through a pitchfork bifurcation.
2.2. One of the two options has a higher quality We choose q2 = 1, q1 = 1 + η (η > 0). To sort out the effect of the “imperfection” η we consider the case η 1. Keeping dominant terms in η one may then cast the solutions of Eq. (8) in the form y1 =
φ +w 2
where w satisfies the cubic equation η φ2 w − φ(φ + 2) = 0. w3 + 1 − 4 4
(10a)
(10b)
As long as η is not strictly zero this equation admits a single real solution for φ less than a critical
Fig. 1. Bifurcation diagram of steady-state solutions of system (6) for q1 = q2 = 1.
value φ∗ and three real solutions for φ larger than φ∗. At φ = φ∗ two of these latter solutions merge. The criticality φ∗ is determined by the vanishing of the discriminant of the cubic, 2 3 φ∗ −1 27η 2 4 . (10c) = 64 φ∗2 (φ∗ + 2)2 The corresponding bifurcation diagram is depicted in Fig. 2, showing the presence of a limit point bifurcation at φ = φ∗. As can be seen, the upper branch exists for all φ values and is always stable. Among the two other branches, which exist for φ > φ∗, the lower branch is stable and the upper one is unstable. To determine the relative attractivity of the stable solutions, we integrate the full Eqs. (6) for a range of initial conditions distributed uniformly in phase space [Nicolis et al., 2011]. As expected, when the two options are equal, y1 is present 50% of the time (case of pitchfork bifurcation, not shown). The mean fraction of y1 within the population as a function of the flux φ when q1 > q2 (case of limit point bifurcation) is shown in Fig. 3. As can be seen, y1 always dominates before criticality, reaching values up to 90%. Subsequently its selectivity is decreasing before saturating for large φ to a value greater than 0.5 (dashed line in Fig. 3). If on the other hand, the initial conditions are sampled in a narrow
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3. Effect of Time-Dependent Control Parameters: Pitchfork Bifurcation
Fig. 2.
As in Fig. 1 but for q2 = 1, q1 = 1 + η with η = 0.1.
phase space band surrounding a state of equipartition y1 ≈ y2 with y1 and y2 both small, one obtains (plain line in Fig. 3) a monotonic increase of the mean ratio. This is due to the fact that this way of sampling favors the attractor associated to option 1.
We now angment the analysis of the preceding section to account for a time dependence of φ, the principal control parameter present in the problem, first limiting ourselves to the case q1 = q2 = 1. The type of time dependence of φ(t) chosen is that of a slow drift, corresponding to a systematic change of environmental conditions occurring on a long time scale compared to the system’s intrinsic time scales. This will be accounted for by the presence of a scaling factor in front of the t-argument of the function φ(t), with 0 < 1. Rather than study the full set of Eqs. (6) with φ a time-dependent function of the above kind we outline, in the spirit of the Introduction, the reduction to the normal form description. We first observe that summing Eqs. (6), one obtains a closed equation for the sum u = y1 + y2 , du = φ(t) − u. dt
(11)
On the other hand, using symmetry arguments one is led to anticipate that the difference x = y1 − y2 should play the role of the order parameter. To determine the conditions under which this variable satisfies a closed equation — the normal form — we subtract the two Eqs. (6). After some elementary manipulations we obtain x(u(t) + 2) dx = 2φ(t) − x. dt (u(t) + 2)2 + x2
(12)
We see that in addition to its intrinsic dynamics, variable x is subjected to a forcing by the timedependent control parameter φ(t) and by the quantity u(t), which satisfies Eq. (11). Now since φ depends on time through the combination t, one sees immediately that the solution of Eq. (11) can be written in the form u(t) = φ(t) + O() Fig. 3. Mean fraction of y1 within a population as a function of the flux φ for q2 = 1, q1 = 1 + η and η = 0.1 (case of limit point bifurcation). (Dashed line) Initial conditions distributed uniformly in phase space. (Full line) Initial conditions limited to a narrow band around a state of equipartition with y1 and y2 = 0.1. Number of initial conditions is 2000.
+ (exponentially decaying terms). Keeping the dominant contribution and substituting into Eq. (12) we obtain
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φ(t) + 2 dx = 2φ(t) x − x. dt (φ(t) + 2)2 + x2
(13)
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This relation provides us with the extension of the center manifold associated to the pitchfork bifurcation, to the case where the control parameter varies slowly on time. To arrive at the generalized normal form, we expand in x and keep terms up to third order: φ(t) − 2 2φ(t) dx = x− x3 . dt φ(t) + 2 (φ(t) + 2)3
(14)
For later use, we also write out the linearized part of Eqs. (13) and (14) around the trivial solution x = 0, dx = λ(t)x dt with
x(t) = x0 exp
t 0
(15a)
dt λ(t )
(15b)
where we have set λ(t) =
φ(t) − 2 . φ(t) + 2
(15c)
Consider first, as a reference, the case where the time dependence of φ is switched off ( = 0). Equation (15c) reproduces then the threshold value φc = 2 determined in Sec. 2.1 beyond which the critical eigenvalue λ of the system’s Jacobian matrix is changing sign, x(t) starts increasing exponentially and a pitchfork bifurcation is taking place, while Eqs. (13) and (14) reproduce the full bifurcation diagram of Fig. 1 or its approximate form in the vicinity of criticality. This situation changes radically in the case where φ(t) depends on time, since [cf. Eq. (15b)] it is the sign of the integral of λ(t) from zero to t rather than of λ(t) itself that controls the exponential decay or the exponential growth of x(t) and hence the transition to the instability of the trivial state x = 0. As an example, for a variation of φ in the form of a ramp, φ = φ0 + t one obtains t 4 φ0 + 2 + t . dt λ(t ) = t − ln I(t) = φ0 + 2 0
Fig. 4. Times at which the functions λ(t) and I(t) in Eqs. (15c) and (17) change sign, as a function of the ramp parameter for φ0 = 0.
behavior of x(t) beyond this value as obtained from Eqs. (15a)–(15c). We conclude that the time variation of φ enhances the stability of the trivial (homogeneous) state and delays the bifurcation.
(16)
(17)
Figure 4 depicts λ(t) and I(t) as functions of t. We see that while λ(t) changes sign at time t0 = |φ0 − 2|/, I(t) changes sign at a quite different, delayed value td . Figure 5 depicts the
Fig. 5. Order parameter x versus time for q1 = q2 = 1, = 0.01, φ0 = 0 as given by the full equation (13) (bold dashed line), the normal form truncated to the third order (Eq. (14), full line) and the linearized version (Eq. (15a), bold line). The dashes follow the branches of the adiabatic bifurcation diagram [Eqs. (9)], with φ given by Eq. (16).
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For reference the “adiabatic” bifurcation diagram for x as obtained by setting the time derivative in Eq. (13) or (14) equal to zero is also drawn. We notice that beyond the aforementioned delay x(t) tends to merge asymptotically with the upper and lower branches of this diagram. We refer to this behavior as the adiabatic approximation. Significantly, the effective control parameter λ(t) identified in Eq. (15a) depends on time in a system-specific fashion, in the sense that its dependence is different from the one of the control parameter of the original equations as it carries the signature of the transformations leading from these equations to the normal form. In particular, while φ(t) increases without bound for t → ∞ [Eq. (16)], λ(t) saturates at a parameter-independent finite value equal to unity. We close this section by extending the numerical experiment described at the end of Sec. 2 to the case where parameter φ is time dependent. Specifically, we consider an ensemble of initial conditions randomly distributed in phase space around the state of equipartition y1 = y2 and an initial value φ0 less than φc . We subsequently integrate the full set of equations (6) for times up to a value for which φ attains the largest value plotted in the diagram of Fig. 1. As expected, the two options remain, in the mean, equivalent up to sampling errors (not shown). It is worth noting that a small portion (about 3%) of realizations remains stuck to the unstable branch, owing presumably to the aforementioned enhancement of its stability induced by the time dependence of φ.
4. Effect of Time-Dependent Control Parameters: Limit Point Bifurcation In this section, we analyze the effect of a time dependence of parameter φ in the case where one of the two options has a higher quality: q2 = 1, q1 = 1 + η, η > 0 (Sec. 2.2). Our starting point are again Eqs. (6), which we write in the form dy1 = v1 (y1 , y2 , φ, η) dt = φ(t)
(1 + y1 )2 − y1 (1 + y1 )2 + (1 + y2 )2
+ φ(t)η
(1 + y1 )2 (1 + y1 )2 + (1 + y2 )2
dy2 = v2 (y1 , y2 , φ) dt = φ(t)
(1 + y2 )2 − y2 (1 + y1 )2 + (1 + y2 )2 (18)
where φ(t) is given by Eq. (16). Contrary to the case of Sec. 3 where a global analysis was possible owing to the a priori knowledge of the order parameter associated to the problem, we here need to resort to a local description around the critical value φ∗ given by Eq. (10c). To this end we express the variables y1 , y2 and the parameter φ (keeping η fixed) as [Nicolis, 1995; Guckenheimer & Holmes, 1983] y1 = y ∗1 + w1 y2 = y ∗2 + w2
(19)
φ = φ∗ + (φ − φ∗ ) where y ∗1 , y ∗2 are given by the steady-state solutions of (18) at the criticality. We next expand Eqs. (18) around the criticality keeping terms up to the second order, ∂vi ∗ ∂vi ∗ ∂vi ∗ dwi = w1 + w2 + (φ − φ∗ ) dt ∂y1 ∂y2 ∂φ ∗ 2 ∗ 2 ∂ vi ∂ vi 1 2 w1 + 2 w1 w2 + 2 ∂y1 ∂y2 ∂y 21 2 ∗ ∂ vi + w22 (i = 1, 2). (20) ∂y 22 By construction, the Jacobian matrix {Lij } of the vector field (v1 , v2 ) ∂vi Lij = ∂yj possesses a zero eigenvalue at criticality. Let u1 u= u2 be the associated eigenvector. We define the order parameter x associated to the problem by the relation
u1 w1 = x(t) . (21) w2 u2 Multiplying both sides of Eq. (20) by the null eigenvector
+ u1 u+ = u+ 2
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of the transposed Jacobian matrix {Lji } we arrive then after same straightforward calculations at the following (closed) equation for x(t) in the vicinity of the criticality: dx = λ(t) + bx2 dt
(22a)
λ(t) = a(φ(t) − φ∗ ) = a(φ0 − φ∗ ) + at.
u+ 1
As an example, for η = 0.01 one obtains u1 = 0.711, u2 = −0.706 and the following explicit form of Eq. (22a): dx = −0.112(φ(t) − φ∗ ) + 0.163x2 dt = −0.112(φ0 − φ∗ ) − 0.00112t + 0.163x2 . (24) To complete the bifurcation diagram we need the third branch of solutions, which exists on both sides of the limit point φ∗ . Proceeding as in Sec. 2.2 we obtain the diagram of Fig. 6, where full and dashed lines denote stable and unstable solutions, respectively. For reference, the dots denote values of the solutions of the full system. The agreement is quite satisfactory, extending over a sizable range of values of φ. Notice that in the terminology of Sec. 3 the diagram in Fig. 6 is to be qualified as “adiabatic”, since the branches drawn are given by the quasi steady-state solutions of Eqs. (22) or (24) in which t and φ are merely related through Eq. (16). Interestingly, the effective control parameter λ(t) appearing in the normal form depends on time in a way similar to the dependence of the control parameter φ, except for a system-dependent scaling factor a multiplying the original ramp parameter . This reflects, once again — albeit in a milder form compared to the case of pitchfork bifurcation — the fact that the explicit form of λ(t) depends on the various transformations carried out in the derivation of the normal form.
(22b)
The coefficients a and b in Eq. (22), which is nothing but the normal form of a limit point bifurcation, are given by the following relations ∗
∂v1 ∗ ∂v2 + u+ 2 ∂φ ∂φ a= + + (u1 u1 + u2 u2 ) 2 2 ∗ 2 ∗ ∗ ∂ v1 ∂ v1 ∂ v1 + 2 u1 + 2 u1 u2 + u22 u1 2 ∂y1 ∂y2 ∂y 1 ∂y 22 b= + (u1 u+ 1 + u2 u2 ) 2 2 ∗ 2 ∗ ∗ ∂ v2 ∂ v2 ∂ v2 + 2 u1 + 2 u1 u2 + u22 u2 2 ∂y1 ∂y2 ∂y 1 ∂y 22 . + + (u1 u+ 1 + u2 u2 )
where the effective control parameter λ(t) is given by [cf. also Eq. (16)]
(23a)
(23b)
We now turn to the time-dependent behavior beyond the adiabatic limit. The objective is to identify the new effects induced by the time dependence of the control parameter. To get some insight on this point we start with an initial value of the control parameter close, but prior to the criticality
Fig. 6. Bifurcation diagrams as obtained from the full system [Eq. (18)] in the adiabatic approximation (dots) and from the normal form equation [Eq. (22)] for η = = 0.01. Full and dashed lines denote stable and unstable branches, respectively.
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and with an initial condition for x(t) close to, and slightly below the limit point. At the level of Eqs. (22) or (24) this implies that φ0 is close to φ∗ , λ0 is a small positive number and x(0) is a negative number of small absolute value (we anticipate that a < 0, b > 0). Linearizing Eq. (22a) around this state we obtain the following equation for the short time behavior of the deviation δx = x − x(0): dδx = λ0 + at + 2bx(0)δx. dt
(25)
By construction, for λ fixed and equal to λ0 , x(0) would evolve toward the stable branch of solutions which lies at a finite distance from the limit point (Figs. 2 and 7). On the other hand, for sufficiently small , and given that λ0 and x(0) are also small (see above), the initial stages of the evolution of δx as given by Eq. (25) will be very slow. It is thus possible that when the ramp is switched on the value attained by the control parameter after some lapse of time t, λ(t) = λ0 + at, will bring the system into the range of attraction of the lower (stable) branch of the bifurcation diagram, before the escape to the upper branch takes place. In other words, the time variation of the control parameter could in principle contribute to the stabilization of the less favorable state for a certain range of values of λ, and x(0) [Davies & Krishna, 1996; Nicolis & Nicolis, 2014]. To substantiate this conjecture, the full equations (18) are integrated for a range of initial conditions distributed in a narrow band around a state of equipartition (y1 ≈ y2 ) corresponding to two different φ0 values, one much smaller than and one close to, but slightly less than the critical value φ∗. Figure 7 summarizes the behavior of the mean fraction of y1 within the population versus the instantaneous value of φ, for two different values of the parameter . In both cases, the majority of trajectories are eventually attracted toward the more favorable option. Still, while the dominance of state 1 is quite clearcut for = 0.01, it is less so for = 0.1. This reflects the fact that for this latter value of there exists a time interval during which a non-negligible fraction of trajectories is attracted by the less favorable option, in agreement with the predictions drawn earlier from Eq. (25). This trend becomes more pronounced for initial conditions such that the mean ratio is less than 0.5, which tend to eventually evolve toward the attraction basin (in the sense of the adiabatic approximation) of the less favorable state.
Fig. 7. Mean fraction of y1 as a function of time covering a range of φ values from 0 to 5, for = 0.01 (full line) and = 0.1 (dashed line) and for initial φ much smaller than the critical value φ∗ (left part) and slightly less than φ∗ (right part). In both cases initial conditions are in a narrow band around a state of equipartition. Number of initial conditions as in Fig. 3.
5. Conclusions In this paper, we outlined the derivation of the normal form of a dynamical system giving rise to pitchfork and to limit point bifurcations associated to the choice between different options in a population of group-living organisms, in the presence of time-dependent control parameters. We have shown that the general structure of the normal forms is not affected by the time dependence of the parameters. On the other hand the effective control parameter appearing in the normal form depends on time in a system-specific fashion, typically different from the one in the original equations, reflecting the specific transformations that lead from these equations to the normal form. A second conclusion emerging from our study is that, far from being driven passively by the variations in the control parameter, the system responds in many cases in a highly nontrivial manner. A first example is provided by the transient stabilization of the unstable state and the associated delay in the occurrence of the transition to the new states in the case of the pitchfork bifurcation.
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A second example is provided by the important role of the ramp parameter in determining the relative attractivities of the states associated to the different options in the case of the limit point bifurcation. Depending on the magnitude of this parameter the choice of the more favorable option may be further optimized or, on the contrary, the system may be attracted transiently toward the less favorable option and remain trapped in this state thereafter. In nature, among the mechanisms at the origin of systematic variations of control parameters on a slow time scale is climatic variability. The flexible responses identified in the present work provide some clues on how a population subjected to cooperative interactions may cope with changes of this kind. In this respect it would be interesting to extend our study to account for the variability of parameters other than the flux φ present in the original equations, such as the rate ν [Eq. (4)] at which an option may be abandoned. In this latter case, the competition between the adverse actions of the two parameters may bring out some unsuspected effects. Finally, it would be interesting to extend the analysis to an arbitrary number of options, as in this case some of the bifurcations occurring will be associated to eigenvalues of increasing degeneracy [Nicolis & Deneubourg, 1999].
Acknowledgment S. C. Nicolis acknowledges support from the European Research Council Grant (Ref. IDCAB 220/ 104702003).
References Beckers, R., Deneubourg, J.-L. & Goss, S. [1992] “Trails and u-turns in the selection of a path by the ant Lasius-niger,” J. Theoret. Biol. 159, 397–415. Benoˆıt, E. [1991] Dynamic Bifurcations (Springer, Berlin). Camazine, S., Deneubourg, J.-L., Franks, N. R., Sneyd, J., Theraulaz, G. & Bonabeau, E. [2001] Self-Organization in Biological Systems (Princeton University Press). Davies, H. G. & Krishna, R. [1996] “Nonstationary response near generic bifurcations,” Nonlin. Dyn. 10, 235–250. Guckenheimer, J. & Holmes, P. [1983] Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, NY). Nicolis, G. [1995] Introduction to Nonlinear Science (Cambridge University Press). Nicolis, S. C. & Deneubourg, J.-L. [1999] “Emerging patterns and food recruitment in ants: An analytical study,” J. Theoret. Biol. 198, 575–592. Nicolis, S. C. & Dussutour, A. [2008] “Self-organization, collective decision making and resource exploitation strategies in social insects,” Eur. Phys. J. B-Cond. Matt. Compl. Syst. 65, 379–385. Nicolis, S. C., Zabzina, N., Latty, T. & Sumpter, D. J. T. [2011] “Collective irrationality and positive feedback,” PloS One 6, e18901. Nicolis, C. & Nicolis, G. [2014] “Dynamical responses to time-dependent control parameters in the presence of noise: A normal form approach,” Phys. Rev. E 89, 022903.
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