Statistical multimoment bifurcations in random-delay coupled swarms

Abstract

We study the effects of discrete, randomly distributed time delays on the dynamics of a coupled system of self-propelling particles. Bifurcation analysis on a mean field approximation of the system reveals that the system possesses patterns with certain universal characteristics that depend on distinguished moments of the time delay distribution. Specifically, we show both theoretically and numerically that although bifurcations of simple patterns, such as translations, change stability only as a function of the first moment of the time delay distribution, more complex patterns arising from Hopf bifurcations depend on all of the moments.

I. INTRODUCTION AND MODEL

Recently, much attention has been given to the study of interacting multi-agent, particle or swarming systems in various natural and engineering fields. Interestingly, these multi-agent swarms can self-organize and form complex spatiotemporal patterns, even when the coupling between agents is weak. Many of these investigations have been motivated by a multitude of biological systems, such as schooling fish, swarming locusts, flocking birds, bacterial colonies, ant movement, etc. [1–7], and have also been applied to the design of systems of autonomous, communicating robots or agents [8–10] and mobile sensor networks [11].

A topic of intense ongoing research in interacting particle systems, and in particular in the dynamics of swarms, is the effect of time delays. It is well known that time delays can have profound dynamical consequences, such as destabilization and synchronization [12,13], and delays have been effectively used for purposes of control [14]. Initially, such studies focused on the case of one or a few discrete time delays. More recently, however, the complex situation of several and random time delays has been researched [15–17]. An additional important case is that of distributed time delays, when the dynamics of the system depends on a continuous interval in its past instead of on a discrete instant [18–20].

There exists a complex interplay between the attractive coupling, time delay, and noise intensity that produces transitions between different spatiotemporal patterns [21,22] in the case of a single, discrete delay. Here, we consider a more general swarming model where coupling information between particles occurs with randomly distributed time delays. We perform a bifurcation analysis of a mean field approximation and reveal the patterns that are possible at different values of the coupling strength and parameters of the time delay distribution.

We model the dynamics of a 2D system of N identical self-propelling agents that are attracted to each other in a symmetric manner. We consider the effects of finite communication speeds and information-processing times so that the attraction between agents occurs in a time delayed fashion. The time delays are nonuniform τ_ij, for particles i and j, as well as constant in time. The dynamics of the particles is described by the following governing dimensionless equations:

$${\ddot{\mathbf{r}}}_i=(1-\mid {\stackrel{.}{\mathbf{r}}}_i{\mid }^2){\stackrel{.}{\mathbf{r}}}_i-\frac{a}{N}\sum _{\begin{array}{l}j=1\\ i\ne j\end{array}}^N[{\mathbf{r}}_i(t)-{\mathbf{r}}_j(t-{\tau }_{ij})],$$ (1)

for i = 1, 2 …, N. The vector r_i denotes the position of the ith agent at time t. The term (1 − |ṙ_i|²)ṙ_i represents self-propulsion and frictional drag forces that act on each agent. The coupling constant a measures the strength of the attraction between agents, and the time delay between particles i and j is given by τ_ij. The form of our model is based on the normal form for particles near a supercritical bifurcation corresponding to the onset of collective motion [23]. Similar models have been applied to the study of swarms in the past [21–26]. In addition, the functional form of the attractive terms may be thought of as representing the first term in a Taylor series around a stable equilibrium point of a more general time-delayed potential. The N(N − 1)/2 different time delays 0 < τ_ij are drawn from a distribution ρ(τ) whose mean and standard deviation are denoted by μ_τ and σ_τ, respectively. For simplicity, we assume symmetric time delays among pairs of agents τ_ij(=τ_ji); we leave the treatment of the more general case for future work.

II. BIFURCATIONS OF THE MEAN-FIELD APPROXIMATION

We obtain a mean field approximation of the swarming system by measuring the particle’s coordinates relative to the center of mass r_i = R + δr_i, for i = 1, 2 …, N, where $\mathbf{R}(t)=\frac{1}{N}{\sum }_{i=1}^N{\mathbf{r}}_i(t)$. Following the approximations from [27], we obtain a mean field description of the swarm:

$$\ddot{\mathbf{R}}=(1-\mid \stackrel{.}{\mathbf{R}}{\mid }^2)\stackrel{.}{\mathbf{R}}-a\left[\mathbf{R}(t)-{\int }_0^{\infty }\mathbf{R}(t-\tau )\rho (\tau )d\tau \right].$$ (2)

The approximations necessary to obtain Eq. (2) require that N be sufficiently large so that $\frac{1}{N(N-1)}{\sum }_{i=1}^N{\sum }_{j=1_{i\ne j}}^N\mathbf{R}(t-{\tau }_{ij})\approx {\int }_0^{\infty }\mathbf{R}(t-\tau )\rho (\tau )d\tau$ and that the swarm particles remain close together. Since there is no parameter in the equations that directly controls how close the particles are, we use simulations of finite swarm populations to compare our theoretical mean field predictions with the behavior of the full swarm model.

Equations (2) admit a uniformly translating solution R(t) = R₀ + V₀t (R₀ and V₀ are any constant 2D vectors), where the speed |V₀| satisfies

$$\mid {\mathbf{V}}_0{\mid }^2=1-a{\int }_0^{\infty }\tau \rho (\tau )d\tau =1-a{\mu }_{\tau },$$ (3)

which shows that this solution is possible as long as the system parameters lie below the hyperbola aμ_τ = 1 in the (a, μ_τ) plane. Remarkably, the speed of the translating state depends exclusively on the mean of the distribution ρ(τ) and not on any of the higher moments.

The linear stability of the translating state is examined by taking $X(t)=\sqrt{1+a{\mu }_{\tau }}t+\delta X(t)$ and Y(t) = δY(t). The two linearized equations decouple and the stability of motions parallel and perpendicular to the translating direction are determined by the characteristic equations (λ) and (λ), respectively:

$${\mathcal{D}}_{\Vert }(\lambda )=\mathcal{F}(\lambda )-(3a{\mu }_{\tau }-2)\lambda ,{\mathcal{D}}_{\perp }(\lambda )=\mathcal{F}(\lambda )-a{\mu }_{\tau }\lambda ,$$

where (λ) = a(1 − 〈e^−λτ〉) + λ². The function 〈e^−λτ〉 is the moment generating function of ρ(τ) since the nth moment is equal to $\langle {\tau }^n\rangle ={(-1)}^n\frac{d^n}{d{\lambda }^n}\langle e^{-\lambda \tau }\rangle {\mid }_{\lambda =0}$. Regardless of the choice of a and ρ(τ), the characteristic functions and have a zero eigenvalue arising from the translation invariance of Eq. (2) [28]. There is a fold bifurcation as an eigenvalue of crosses the origin when aμ_τ = 1, which marks the disappearance of the translating state as seen from Eq. (3). Numerical analysis [29] reveals an additional curve on the (a, μ_τ) plane (below the curve aμ_τ = 1), along which perturbations parallel to the translation undergo a Hopf bifurcation as a complex pair of eigenvalues of cross the imaginary axis.

As for perturbations perpendicular to the translational motion, there is another fold bifurcation as an eigenvalue of crosses the origin along the curve a〈τ²〉 = 2, which represents a bifurcation in which the translating state merges with a circularly rotating state of infinite radius, as discussed below.

Considering a fixed σ_τ, the overall stability picture of the translating state of Eq. (2) is as follows (see Fig. 1). For values of (a, μ_τ) below the curves a〈τ〉 = 1 and a〈τ²〉 = 2 (region A) the translating state is linearly stable. These two curves may cross at a point that we call the “zero frequency Hopf point” (ZFH). The transverse direction of the translating state becomes unstable along the curve a〈τ²〉 = 2, where this state merges with the circularly rotating state (along the mentioned curve the rotating state has an infinite radius); transverse perturbations of the translating state will thus produce a transition to the rotating state in regions B and C. From the ZFH point, there emanates a Hopf bifurcation curve where the parallel component of the translating state becomes unstable so that in region C there is a transition from the translating state to oscillations along a straight line. Finally, the translating state ceases to exist along the curve a〈τ〉 = 1, where there is a pitchfork type bifurcation with the stationary steady-state solution. The possible behaviors in region D are discussed below.

FIG. 1.

(Color online) Bifurcation structure of the translating state of the mean field Eq. (2) for the exponential distribution described in the text (σ_τ = 0.2). The translating state merges with: (i) the stationary state along the continuous red curve aμ_τ = 1; and (ii) the circularly rotating state along the dashed black curve a〈τ²〉 = 2. The component of the translating state parallel to the motion undergoes a Hopf bifurcation along the green, dotted curve.

We compare the mean field bifurcation results with the full swarm system via numerical simulations. Here, we make use of two different time delay distributions with mean μ_τ and standard deviation σ_τ to test our findings. The first is an exponential distribution $\rho (\tau )=e^{\frac{\tau -{\mu }_{\tau }+{\sigma }_{\tau }}{{\sigma }_{\tau }}}/{\sigma }_{\tau }$ for τ ≥ μ_τ − σ_τ and zero otherwise; we require σ_τ ≤ μ_τ for proper normalization. The second distribution is a uniform $\rho (\tau )=\frac{1}{2\sqrt{3}·{\sigma }_{\tau }}$ for ${\mu }_{\tau }-\sqrt{3}{\sigma }_{\tau }\le \tau \le {\mu }_{\tau }+\sqrt{3}{\sigma }_{\tau }$ and zero otherwise; here, we require $\sqrt{3}{\sigma }_{\tau }\le {\mu }_{\tau }$. Moreover, when carrying out simulations for different values of μ_τ, we consider two cases: the “sliding” case (μ_τ varies, σ_τ = const.) and the “widening” case (both μ_τ and σ_τ vary, μ_τ ∝ σ_τ). Note that for any single simulation μ_τ and σ_τ are fixed in time.

Figure 2(a) compares the speed of the swarm obtained from Eq. (3) with the time-averaged speed of the center of mass obtained from simulations (after the decay of transients). The swarm particles are all located at the origin at time zero and move with the speed obtained from Eq. (3) along the x axis. In these simulations, we use both “sliding” (σ_τ fixed) and “widening” (both μ_τ and σ_τ vary) versions of the exponential and uniform distributions described above. Figure 2(a) shows that the swarm converges to the translating state (seen from the full swarm time-series) up to a certain value of μ_τ. Beyond this critical value, the swarm converges to a state in which it oscillates back and forth along a line with a speed that has a small average value over a time much longer than the oscillation period. The behavior in Fig. 2(b) is similar, except that at the higher values of μ_τ the system converges to the stationary state (see below) instead of oscillating on a line. The mean-field equations do predict the observed departures from the translating state; however, this departure occurs earlier for the full swarm than for the mean-field. The full simulation results show that in this deviation from the mean field, the swarm particles become spread out too far apart and render the approximations leading to Eq. (2) invalid.

FIG. 2.

(Color online) Speed of the translating state of Eqs. (1) and (2) as a function of μ_τ ; here, N = 150, a = 0.2 (a), and a = 1.2 (b). Parameter values chosen so as to cross from region A to B to D (a) and from A to D (b) of Fig. 1. The red line represents the mean field result from Eq. (3); the continuous segment marks where the translating state is linearly stable and the dashed segment where it is unstable. The symbols represent numerical simulations of Eq. (1) for different time delay distributions: sliding exponential and sliding uniform σ_τ = 0.5 (a), σ_τ = 0.05 (b); widening exponential μ_τ = σ_τ (a) and (b); widening uniform ${\mu }_{\tau }=\sqrt{3}{\sigma }_{\tau }$ (a) and (b).

In addition to the translating state, Eq. (2) always possess a stationary state solution R(t) = R₀ = const. In the full system, Eq. (1), the stationary state for the center of mass manifests itself in a swarm “ring state,” where some particles rotate clockwise and others counterclockwise on a circle around a static center of mass. The characteristic equation that governs the linear stability of the stationary state has the form [ (λ)]² = 0, where (λ) = (λ) − λ. Once more there is a zero eigenvalue for all choices of a and ρ(τ) that arises from the translation invariance of Eq. (2). Also, since (0) = 0, (0) = aμ_τ − 1, and lim_λ→∞ (λ) = ∞, the condition aμ_τ − 1 < 0 guarantees the existence of at least one real and positive eigenvalue, which renders the stationary state linearly unstable. Thus, aμ_τ = 1 is a bifurcation curve on the (a,μ_τ) plane along which the uniformly translating state bifurcates with the stationary state.

The stationary state undergoes Hopf bifurcations when the equation (iω) = a(1 − 〈e^−iωτ〉) − ω² − iω = 0 for ω ≠ 0 is satisfied. The function 〈e^−iωτ〉 is called the characteristic function of ρ(τ) and is related to the moment generating function of the distribution; its Taylor series contains all of the moments of ρ(τ). This shows that the location of the Hopf bifurcations depends on the values of all moments of the time delay distribution. This is in contrast to the region where the translating state exists aμ_τ < 1, which involves the first moment only.

We see that circular orbits bifurcate from the stationary state along the Hopf bifurcation curves by writing Eq. (2) in polar coordinates and arguing as follows. The radius R and frequency ω of such orbits are given by

$${\omega }^2=a(1-\langle cos\omega \tau \rangle ),R=\frac{1}{\omega }\sqrt{1-\frac{a}{\omega }\langle sin\omega \tau \rangle },$$ (4)

the first of Eqs. (4) and the condition R = 0 are precisely the real and imaginary parts of the Hopf bifurcation conditions (iω) = 0.

Generically, the Hopf conditions for the stationary state (iω) = 0 yield a family of curves in the (a, μ_τ) plane (Fig. 3). The first member of the Hopf family emanates from the crossing of the curves aμ_τ = 1 and a〈τ²〉 = 2; the former curve is where the translating state bifurcates from the stationary state in a pitchfork-like bifurcation. Hence the name “Zero Frequency Hopf” for the Hopf-fold point [Fig. 3(a)]. The first Hopf curve is supercritical and gives rise to a circularly rotating orbit with radius and frequency given by the first solution of Eqs. (4). Below this first Hopf curve and aμ_τ = 1, in region A, the stationary state is stable. From Eqs. (4) it follows that this circularly rotating orbit collides with the translating state along the curve a〈τ²〉 = 2, where its radius tends to infinity and its speed to that of the translating state, $\sqrt{1-a{\mu }_{\tau }}$. Thus, in regions B and C the system converges to the circularly rotating orbit. The different regions change shape for the other panels of Fig. 3, but the dynamics remain as described above.

FIG. 3.

(Color online) Bifurcation curves of the mean field Eq. (2) at fixed σ_τ for the two time delay distributions ρ(τ) described in the text: exponential (top) and uniform (bottom). The translational state disappearance curve aμ_τ = 1 (red), bifurcation of the translational state with circularly rotating state curve a〈τ²〉 = 2 (black). The first four members of the stationary state Hopf bifurcation curves are also shown (blue, dashed green, dotted-dashed cyan, and dotted magenta). In each panel, σ_τ has the value (a) 0.2, (b) 0.95, (c) 0.2, and (d) 0.3, respectively.

We compare the mean field prediction for the location of the birth of the circularly rotating state (first Hopf curve) with the full system. Figure 4 shows the results from numerical simulations of Eq. (1) at a fixed value of σ_τ for increasing μ_τ. We plot the speed of the center of mass averaged over a long time interval, after the decay of transients. In that plot, the near-zero values of the mean speed (in the interval 0 ≲ μ_τ ≲ 1.4) indicate that the particles have converged to the ring state, while for all higher values of μ_τ the swarm converges to the rotating state (seen from the full swarm time-series). The transition to the rotating state is predicted by the mean-field, although its location does not match the one observed with the full swarm equations. Remarkably, for the highest values of μ_τ, the center of mass of the swarm moves faster than unit speed, the asymptotic speed of uncoupled particles. The reason is that while in its rotating orbit, the mean time-delayed position of the swarm is actually “ahead” of the center of mass at the current time, causing the particles to accelerate forward along the circular orbit.

FIG. 4.

(Color online) Center of mass speed as a function of μ_τ for the “sliding” exponential time delay distribution (a) and the “sliding” uniform distribution (b). Here, N = 150, a = 2, and σ_τ = 0.3. Parameter values chosen so as to cross from region A to B of Figs. 3(a) and 3(b). The continuous red curve represents the mean field result from Eq. (4), symbols represent the results from numerical simulations of Eq. (1).

III. DISCUSSION

In summary, we have considered a randomly delay distributed coupled swarm model and analyzed the mean-field bifurcations of various patterns as a function of delay characteristics and coupling strength. In particular, we have shown that the location and shape of the Hopf bifurcation curves is strongly dependent on all the moments of ρ(τ). This dependence, in addition to the fact that the succeeding Hopf curves in Fig. 3 exhibit higher frequencies of rotation, makes the higher-order patterns equally sensitive to all the moments of the delay distribution. In the single delay case with distribution ρ = δ(τ − τ₀), where δ(τ) is a Dirac delta function, all of the succeeding Hopf bifurcations are all subcritical and continuous. In contrast, when all moments are present, the bifurcations may not even be continuous, presenting their structure as isolated closed curves bounded by fold bifurcations, as seen in Fig. 3(d) for a uniform distribution. Numerical simulations show that the pattern transitions of the full swarm equations do occur at approximately the parameter values predicted by the mean-field. Our analysis implies that for wider delay distributions (Fig. 3): (i) the regions with stable translating and stationary states, respectively, diminish greatly in size, eventually making these states unattainable in practice; and (ii) the higher order Hopf bifurcation curves occur at much lower and more accessible values of a. Thus, the result of having wide delay distributions enhances some patterns at the expense of others. Finally, we expect that once distributed delays are accounted for, other coupled oscillator systems such as lasers, neurons, ecological populations, etc., with multiple synchronous states and other complex global behavior [19,30–33], will generically display behaviors involving bifurcations that include all moments of the coupling delay distribution.