Mobility induces global synchronization of oscillators in periodic extended systems

We study synchronization of locally coupled noisy phase oscillators which move diffusively in a one-dimensional ring. Together with the disordered and the globally synchronized states, the system also exhibits several wave-like states which display local order. We use a statistical description valid for a large number of oscillators to show that for any finite system there is a critical spatial diffusion above which all wave-like solutions become unstable. Through Langevin simulations, we show that the transition to global synchronization is mediated by the relative size of attractor basins associated to wave-like states. Spatial diffusion disrupts these states and paves the way for the system to attain global synchronization.

Synchronization of oscillators is a widespread phenomenon in nature [1][2][3]. In biology, synchronization can occur at scales that range from groups of single cells to ensembles of complex organisms [4]. When oscillators hold fixed positions in space and the interaction that drives synchronization is short ranged, spatial and temporal patterns can self-organize. Such is the case in cardiac tissue, where cells generate spiral patterns that shape the heartbeats [5].
Also in central pattern generators, the oscillating neural network self-organizes to produce coordinated movements of the body [6].
A different situation arises when the oscillators are not fixed in space but are able to move around. The problem of synchronization of moving oscillators has many applications in the domain of chemistry [7], biology [8], and technology [9]. Small porous particles loaded with the catalyst of the Belousov-Zhabotinsky reaction behave as individual chemical oscillators, undergoing a density-dependent synchronization transition as the stirring rate is increased [7]. The same particles support wave propagation in the form of dynamic target and spiral patterns when the particles are not moving [10]. This phenomenon illustrates a wider scenario: mobility and mixing remove local defects and patterns, enabling global order. This effect has far reaching consequences in finite systems. For example, in ecosystems of competing populations with cyclic interactions, biodiversity can be sustained if dispersal is local, but it is lost when dispersal occurs over large length scales [11]. The dynamics of such cyclic competition was described by a complex Ginzburg-Landau equation near a Hopf bifurcation, displaying complex oscillatory patterns indicative of biodiversity for low mobility, while in the case of high mobility diversity is wiped out [12,13].
In this paper we study the effects of mobility -spatial diffusion-on the macroscopic collective dynamics of locally coupled, moving phase oscillators subjected to noise, in a onedimensional ring. When oscillators are fixed in space, these systems can exhibit a series of steady states where local order is present [14][15][16]. Such states have been called m-twist solutions [16], Fig. 1. Here we show that mobility can destabilize all m-twist solutions, enhancing the stability of the synchronized solution. We find that in finite systems there is a critical mobility above which either the synchronized or the disordered state is stable.

I. DIFFUSING PHASE OSCILLATORS
We consider an ensemble of N identical phase oscillators that diffuse on a ring of perimeter L. Oscillators are coupled to other oscillators in their local neighborhood, within an interaction range r. The dynamics of phase and position is described bẏ where i = 1, . . . , N is the oscillator label, θ i (t) and x i (t) are the phase and position of the i-th oscillator at time t, ω is the autonomous frequency, and γ is the coupling strength -whose inverse characterizes the typical relaxation time of the interaction. Each oscillator interacts with its n i neighbors in the range r through the coupling function in brackets, which defines an attractive interaction towards the local average of the phase. In steady state the spatial density is uniform and the number of neighbors is on average constant, n = n i = N r/L. With this definition of the coupling the thermodynamic limit is well defined, and the system reduces to the noisy Kuramoto model for r = L/2 [17]. The fluctuation terms ξ θ,i and ξ x,i represent two uncorrelated Gaussian noises such that ξ θ,i (t) = ξ x,i (t) = 0, . The strength of angular fluctuations is determined by the angular diffusion coefficient C, while the spatial diffusion coefficient D

III. STATISTICAL DESCRIPTION
The role of twisted states can be studied using a statistical description that is valid when the number of oscillators is large. Given that the oscillators have identical autonomous frequencies ω, it is convenient to make the transformation θ → θ − ωt to a rotating reference frame. We coarse grain the microscopic model and describe the system in terms of ρ(x, θ, t), the density of oscillators at position x with phase θ, which obeys the Fokker-Planck equation where g(x−x ′ ) is a kernel accounting for the range and relative strength of local interactions, while denotes the effective number of oscillators in this range. In this paper we choose g(x−x ′ ) = 1 Since the movement of the oscillators is purely diffusive, see Eq. (2), the spatial density of oscillators is uniform in steady state, 2π 0 dθρ(x, θ, t) = N/L ≡ ρ 0 , and n(x) = 2rρ 0 . For small N, fluctuations in the spatial density can induce the formation of gaps in which the nearest oscillator is beyond the range of interaction. In this paper we consider large densities such that the lifetime of these gaps is much shorter than other typical time-scales.

A. Local order parameter
The statistical description (3) can be cast in a more transparent form introducing a local mean field. Local order can be characterized by a local order parameter [14,15] where R(x, t) is a measure of local order and ψ(x, t) is the local average of the phase. Eq. (3) can be expressed in terms of this local order parameter as reflecting the fact that ψ(x, t) acts as a local mean field and R(x, t) is a local modulation to the coupling strength.

IV. TRANSITION FROM DISORDER TO LOCAL ORDER
Eq.
(3) has a trivial steady state ρ(x, θ, t) = ρ 0 /2π ≡ ρ d which corresponds to the disordered state of the system. We study the stability of ρ d by inserting ρ(x, θ, t) = ρ d + ǫf (x, t) cos(ℓθ) in Eq. (3) and keeping terms of order O(ǫ) [20]. Linear stability analysis reveals that the disordered solution ρ d becomes unstable for ℓ = 1 when C < C * , with This threshold is independent of ρ 0 and D, and determines the value of C below which local order sets in. The critical C * given by Eq. (6) coincides with the existence [17] and stability [20] threshold displayed by globally coupled noisy oscillators.

V. LOCAL ORDER SOLUTIONS
Once local order has set in, the system also supports twisted solutions. We specifically look for steady state solutions to Eq. (5) of the form ρ s (x, θ) = f (θ − ψ(x)). Such wavelike solutions describe densities in which the angular distribution has the same shape, but is centered at position dependent phases ψ(x). Setting ∂ t ρ = 0 we obtain an ordinary differential equation for f that we can solve together with periodic boundary conditions in phase and space to determine the arbitrary constant C(x) and phases ψ(x). Periodicity of the phase is consistent with solutions that fulfill ψ ′′ (x) = 0 and C(x) = 0, and periodicity of space sets the wave numbers k = 2πm/L with m integer. We obtain the m-twist steady state solutions where N is a normalization constant such that Fig. 3(a). We have introduced the effective diffusion coefficient which is a combination of angular diffusion C, and mobility D scaled by the square of the wave number k. Effective diffusion competes with the local coupling γR and controls the width of the angular distribution, which has a mean ϕ = kx and variance see Fig. 3(b,c). A trivial solution to Eq. (10) is R m = 0. Apart from this, an expansion of Eq. (10) for R m ≪ 1 reveals that non-vanishing solutions  Fig. 3(b,c).

B. Existence of twisted solutions
We can unfold the effects of spatial and angular fluctuations by writing D eff in terms of its components D and C. Setting R m = 0 in Eq. (11) we get C + (2πm/L) 2 D = (γ/2) sinc (2πmr/L) , where we have expressed k in terms of L to stress system size dependence. For m = 0, Eq. (12) reduces to C = C * = γ/2 and corresponds to global synchronization.
Existence of global synchronization in steady state is not affected by mobility, as indicated by the dotted red line in Fig. 4(a,c). However, existence of twisted solutions in finite systems is controlled by angular diffusion C, mobility D, and range of interaction r as indicated by Eq. (12), Fig. 4. As L → ∞, the critical value of the spatial diffusion coefficient diverges for all m. Therefore, in the infinite system size limit all twisted solutions coexist with global synchronization for any finite D. These result is in agreement with [22], and indicates that identical noisy phase oscillators cannot exhibit a global synchronized state in 1D in this limit.

C. Stability of twisted solutions and states
While existence and stability thresholds coincide for the global order solution, Eqs. (6) and (12), this is not the case for m-twist solutions in finite systems, Fig. 5. We address the stability of the 1-twist solution by performing a numerical study of Eq. (3), using a finite difference scheme. To estimate the stability boundary, we continue a stable twist solution until it becomes unstable against small perturbations. We find that the instability is of modulational type. The m-twist solutions become stable only after the corresponding local order parameter R m becomes larger than a certain value, i.e. twisted solutions become stable with a finite amplitude, Fig. 5. For vanishing spatial and angular diffusion C = D = 0, we encounter the system studied by Wiley et al. [16], see purple open circle in Fig. 5(a).
The numerical solution seems to approach this point as C/γ → 0, but the numerics become

VI. ATTRACTION BASINS
Twisted solutions co-exist with global order and among themselves, Fig. 4. As mobility is increased from low values, twisted solutions become unstable one by one, e.g. Fig. 4(c) and Fig. 5(c), and global order is enhanced resulting in an increasing value of the ensemble average of the global order parameter as displayed in Fig. 2

VII. DISCUSSION
We have investigated the effects of mobility in a generic 1D model of locally coupled moving phase oscillators, and showed that oscillator mobility dramatically affects the collective behavior of finite systems. More specifically, our results show that in low dimensional systems global synchronization is compromised by the presence of multiple m-twist states exhibiting local order. At the onset of local order, the system can fall into the global synchronized state. However, the coexistence of local order m-twist states implies that the attraction basin of the global synchronization state is reduced. Strong mobility of the oscillators destabilizes these m-twist states, and thus promotes global synchronization.
In this paper we have considered a high density limit such that the connectivity of the system is never interrupted by gaps. In the dilute limit, gaps in the connectivity play a crucial role in the synchronization dynamics. This problem was studied in the context of moving neighborhood networks, and under the assumption of a fast exchange of neighbors, a mean-field condition for the existence and stability of the global synchronization state has been derived [23]. According to this study, whenever the global synchronized state is stable the system reaches global synchronization, regardless of its spatial dimensionality. In other words, the study overlooks the possibility of coexistence of multiple solutions. Our findings reveal a different role for mobility, unrelated to the existence and stability of the global synchronization state: mobility disrupts all these multiple solutions except for the global synchronized state. Extensions of the current study towards dilute systems will be the subject of further investigations.
Two-dimensional systems display a similar phenomenology, though the competing local order states can now take other forms, e.g. vortexes [24]. It has been recently reported that chaotic oscillators moving in a two-dimensional space can synchronize provided that spatial dynamics is fast enough [25]. A related albeit different scenario occurs with chaotic advection mixing in two dimensional systems, where synchronization of excitable media is enhanced by strong mixing [26,27]. These results indicate that mobility may also enhance synchronization in two-dimensional systems. We speculate that global synchronization may be achieved by destabilizing local deffects, as we show here for 1D systems. Further work is intended to clarify these issues.
The theoretical framework introduced here may provide insight into other related problems, as when movement is coupled to the oscillator phases. In this case synchronization can be interpreted as collective motion [28]. As a result of this coupling, strong spatial fluctuations and clustering effects dominate the system dynamics [29], and global order prevails even in the thermodynamic limit [30].
Finally, a compelling biological application of our framework may be found in the vertebrate segmentation clock, where global coupling is a good effective description of the system because of the high mobility of cells [31]: by precluding the appearance of local defects, mobility promotes global synchronization. Moreover, it has been recently shown that mobility decreases the relaxation times to achieve synchronization in a model of the segmentation clock that allows for flipping between neighboring cells [32]. However, this system also hosts spatial patterns [33], and mobility is not accounted for in current distributed models. It will be interesting to see how mobility affects the synchrony recovery times and pattern reorganization after perturbation in such models.