Even though nonlinear temporal dynamics is well understood1, the understanding of systems having complex dynamics in both time and space is still limited2,3,7. Spatiotemporal dynamics may underlie phenomena as varied as weather8, evolution of geophysical patterns9, the movement of stock markets6, the flux through metabolic pathways4, biomechanical processes10 and morphogenesis2,5. So-called large spatiotemporal systems—systems in which the number of effective degrees of freedom is large—demonstrate the applicability of amplitude and phase equations2,11 in the characterization of spatiotemporal patterns2,7. Small systems—systems with only a few degrees of freedom—are substantially more complicated to study because the boundaries of these systems strongly influence their dynamics and are difficult to treat analytically. Truly tractable experimental demonstrations12,13 of spatiotemporal dynamics are scarce11, and the need for them is correspondingly high2. Here we provide an example of such a tractable system that is amenable to rational design and modification.

The system comprises coupled microfluidic flow-focusing devices. A single flow-focusing device14,15 (Fig. 1a) was first implemented in a microfluidic chip by Anna et al. 16; the system has three inlet channels—two outer supplying liquid and the centre one supplying gas—that merge at a junction leading to a single outlet channel. In this geometry, the pinch-off process that generates bubbles is regulated17 by the inflow of liquid into the orifice (at a typical speed uinflow0.01–1 m s−1); this inflow is slower by two to five orders of magnitude than the relaxation rates (the typical interfacial and bulk relaxation speeds are uinterfacialγ/μ100 m s−1 and usound1,000 m s−1, respectively, where γ is the interfacial tension and μ is the viscosity of the liquid). As a result, the breakup process in the simple flow-focusing device is highly reproducible and leads to the generation of monodisperse bubbles17,18.

Figure 1: The geometry of the system and an example of the periodic behaviour.
figure 1

a,b, Schematic diagrams of the flow-focusing device with a single orifice (a) and the multi-orifice system (b) that we used in this study. We supply the gaseous phase (nitrogen) through a channel that runs along the axis of the device. Liquid (water containing 2% w/w Tween 20 surfactant) flows from both sides into the central channel through a network of five side channels; the five orifices in which bubbles are formed are numbered from 1 to 5. The width of the gas-inlet channel and the orifices is 40 μm, the channels that deliver the liquid to the orifices comprise two sections of different widths, and the width of the section adjacent to the centre-line channel is 20 μm. The spacing between the liquid-carrying inlets along the centre channel (the lengths of the orifices) and the width of the outlet channel are 200 μm. All channels have a uniform height of 44 μm (the Supplementary Information gives a detailed diagram of the experimental device). The gas–liquid interface is represented with a thick solid line. The shaded areas correspond to the elastomeric polydimethylsiloxane (PDMS) walls of the device. ck, The dynamics of the system at Q=1.66 μl s−1 and the value of the pressure applied to the gas stream p=76 kPa. The part of the system shown in these micrographs corresponds to the area marked by the dashed box in diagram b. ch are chronological micrographs of the system showing formation of the sequence of 29 bubbles produced within one period. This sequence is repeated indefinitely provided that the pressures applied to the gas and liquid inlets remain approximately constant. hk (highlighted with the grey background) illustrate the system at the same phase of the period—that is right after the last of the 29 bubbles enters the outlet channel. The intervals between the pairs of micrographs h and i, i and j, and j and k correspond to 1, 101 and 1 period(s) correspondingly. The numbers by the images shown in ck indicate the time (in ms) elapsed after the first micrograph (c) was taken.

We have incorporated this well-characterized17 device into a multi-orifice design (Fig. 1b) in which the stream of gas can pinch-off at any orifice in an array of five orifices. Each liquid inlet splits into five connected channels that feed the array of orifices. These five channels couple the dynamics at each orifice by pressure feedback loops. For an externally fixed rate of flow, Q, of the liquid, the viscous resistance to flow downstream determines the pressure at each orifice and the flow through it. As the gas fills the centre channel, it blocks certain orifices and increases the flow through a smaller number of (unblocked) inlet channels. The pressures (pi) in—and the rates of inflow (qi) into—each of the orifices depend on the length and position of the gaseous thread and of the bubbles. When the thread breaks and the bubbles move downstream, they also alter the distribution of pressure, and consequently the dynamics of the breaking of the thread.

We examined the behaviour of this system using N2 as the gas and water containing 2% of a surfactant (Tween 20) as the liquid. As the pressure applied to the stream of gas increases, the behaviour of the system becomes progressively more complex and then it re-simplifies. For a fixed Q, and for the pressure applied to the gas stream p only slightly above the minimum pressure needed to form bubbles, the thread breaks into bubbles of uniform size at the first orifice (the ‘period-1’ regime). At intermediate pressures the dynamics becomes astonishingly complex: the thread enters the array of orifices, undergoes several breakup events that generate a series of bubbles of different sizes, pauses, and then the whole process repeats. In spite of the intricacy of this periodic behaviour, it is coherent—the sequence of the sizes of the bubbles and the intervals between them are almost identical from period to period (Fig. 1h–k). We have observed stable, coherent sequences of order as high as 40 bubbles in each period. At high values of p, the thread progresses all the way downstream and breaks only in the last orifice, again in a period-1 mode. We can adjust the periodicity almost continuously from order 1 (Q=1.66 μl s−1,p=29.75 kPa) to 40 (Q=1.66 μl s−1,p=80.80 kPa), see Fig. 2. Each cyclic trajectory is stable within a finite range of pressures: the period-29 cycle is stable for p in the range between 74.2 and 76 kPa.

Figure 2: Periodicity as a function of the pressure applied to the stream of gas.
figure 2

a, The number (N ) of bubbles in a period (or burst) as a function of p, for Q=1.66 μl s−1. The regions in which the cyclic trajectories are stable (white) are separated by regions of irregular dynamics (shaded grey). For each pressure we counted the number of bubbles in five consecutive bursts. If this number was always the same (suggesting a stable cyclic orbit) we have verified that the behaviour does not change over longer intervals (100 periods). b, An enlarged plot of a section of the diagram (marked with the dashed lines in a). The numbers in b denote the order of the limit cycle that is stable within the white (not shaded) band.

To exemplify the complex periodic dynamics we describe the behaviour of the system at Q=1.66 μl s−1 and p=76 kPa. For these values, there are 29 breakup events and 29 bubbles formed in each cycle. Figure 1c–h shows six chronological micrographs of the system taken within one period. Micrographs 1h–k depict the system at the same phase in the cycle—when the last of the 29 bubbles emerges from the fifth orifice. This periodic behaviour is stable over long times (103 s, corresponding to 105 periods), and we were able to recover the same behaviour of the system in multiple copies of the device.

To quantify the coherence of this behaviour, we construct a function ρ(t), where t is time, that can assume two values: ρ(t)=1 if the outlet of the last (fifth) orifice is occupied by gas, and ρ(t)=−1 if it is occupied by water. Figure 3a demonstrates four fragments of ρ(t) extracted from a video tracing the behaviour of the system over 4.3 s with a temporal resolution of 6.25 μs. Each change of ρ(t) from −1 to 1 signifies an approach of the gas phase to the end of the last orifice; a release of a bubble into the outlet channel changes ρ(t) from 1 to −1. Figure 3b shows the autocorrelation function gt)=〈ρ(t)ρ(tt)〉, where 〈〉 denotes an average over the 690,000 frames (4.31 s), with peaks at the multiples of Δt29=5.02 ms—an interval corresponding to a sequence of 29 bubbles.

Figure 3: Quantitative analysis of the cyclic behaviour of the system at Q=1.66 μl s−1 and p=76 kPa.
figure 3

a, Four consecutive traces of the function ρ(t), each representing one period; the second, third and fourth trace are delayed by Δt=5.037,10.112 and 15.100 ms, respectively. One can see that a small deviation in the intervals between bubbles (third trace) does not amplify, but decays to yield nearly ideal behaviour in the following period (fourth trace). b, The autocorrelation function gt ) calculated for the full trace of ρ(t ) (interval of 4.3215 s). The peaks occur at multiples of Δt29=5.020 ms. c,d, The Poincaré maps of the intervals τn(τn−1) between bubbles plotted for one period (c), and for the whole series of data of 25,257 bubbles (d). e,f, The Poincaré maps for τn(τn−29), again, plotted for a single period (e) and for the whole series of data (f). As 29 is a prime number, the concentration of points on the diagonal cannot be explained by a resonance with a lower-order period. The size of the spots on the Poincaré maps d and f code the number of counts for each point. We have normalized the counts by dividing them by the largest count (ci j=ci j/cmax, where ci j is the number of counts for each point, and cmax is the maximum number of counts). In order to bring out the intermediate values of counts we used a nonlinear scaling of the diameter (d ) of the dots: d proportional to 1+(3/4)log(ci j). This scale is shown to the right of panel f.

We verify the stability of these periodic dynamics by constructing Poincaré maps1 on the intervals τn between adjacent bubbles. A release of the nth bubble is signified by the nth instance of a change of ρ(t) from 1 to −1 at ttn. We define the nth interval as τn=tntn−1. Figure 3c shows a map of τn versus τn−1 plotted for 29 consecutive intervals; Fig. 3d shows the same map for all the 25,257 intervals extracted from the 4.31 s recording. Strong clustering of the points signifies periodic dynamics, and the positions of the clusters match the locations of the points on the map for a single period. We also plot a map of τn versus τn−29 in Fig. 3f. If a system showed ideally periodic behaviour, all of the points on the plot would lie on the line τn=τn−29. As 29 is a prime number, the observed concentration of points on the diagonal cannot be explained by a resonance with a lower period (as would be possible in the case for a τn versus τn−16 map of a system showing period-4 behaviour). We do not observe such clustering for any map of τn versus τnm with m<29.

There are two causes of the scatter of points in the plots shown in Fig. 3d and f. First, some bubbles flow through the last orifice very rapidly and our ability to detect them is at the limits of the temporal resolution of our camera (6.25 μs). Some of the bubbles are omitted, and these omissions introduce errors into the sequence of intervals (τn). Second, within some periods, the intervals between consecutive bubbles vary slightly (see the third trace in Fig. 3a). The observation that these variations do not propagate into irregular or chaotic behaviour indicates that the observed periodic trajectories are stable limit cycles and small deviations from them decay rather than amplify with time. This stability is in pronounced contrast with other nonlinear systems1,19 in which the multiperiodic trajectories are usually intertwined in strange attractors20, are unstable, and the observed behaviour of the system is irregular.

We believe that the stability of the long limit cycles reflects the separation of timescales between the two sets of processes that determine the behaviour of the system: (i) slow breakup of the gaseous thread, which typically proceeds at a timescale tbreakup10 μm/uinflow100 μs, with 10 μm (radius of the thread) as the characteristic length scale for the interfacial dynamics; and (ii) fast spatial coupling between different positions along the breaking thread, which proceeds at a typical timescale tcoupling1 mm/usound1 μs, where 1 mm is a typical length of the channels that transmit the feedback in pressure. This separation insulates the system against perturbations, and leads to stable periodic behaviour. The construction of our system out of modules (single flow-focusing devices) characterized by ‘linear’ dynamics17, resulting from the separation of timescales between the slow evolution of the system and fast relaxation rates, distinguishes the multi-orifice device from several other hydrodynamic systems that show archetypal examples of nonlinear dynamics, for example, the famous leaky faucet system21,22 or spatially extended dripping systems13.

Until now, interest in the field of spatiotemporal dynamics has focused more on probing the onset and properties of chaos, and less on selecting and understanding intricate periodic behaviour. An exception is the work of Ott et al. 23, who demonstrated active stabilization of an unstable periodic cycle embedded in a strange attractor. Their method 23 required continuous application of judiciously chosen perturbations to keep the system on a periodic (but unstable) trajectory. Similar strategies have stabilized periodic dynamics in simple mechanical24, electronic25 and biological systems26. Ott et al. 23 argued that the ability to choose and stabilize periodic trajectories may lead to the construction of adaptive materials and shed light on the adaptability and self-regulation shown in life10. The intricate nonlinear phenomena, especially those in the world of biology, such as, for example, morphogenesis5, dynamics of populations20,27,28 or the operation of the biological machinery regulating the distribution of enzymes in cells29, often involve intricate correlations in both space and time that are only possible in strongly nonlinear systems that operate at the edge of chaos30. The proximity to chaotic dynamics induces sensitivity to perturbations, which destroys the temporal and spatial correlations; this feature makes the experimental studies of the complex spatiotemporal patterns difficult. The example of a system that we demonstrated here—a system that shows complex yet stable periodic dynamics without active external control—suggests that the separation of timescales of the (slow) dynamics and the (fast) spatial coupling could serve as a guideline in synthesis of artificial systems showing complex nonlinear behaviour and possibly help in the studies of spatiotemporal dynamics. In a shorter perspective, modifications of the system presented here could open a way to preparation of emulsions characterized by a multimodal, space-filling distribution of sizes of the bubbles or droplets. The example of a coherent periodic behaviour in this microfluidic systems points to the potential for design of lab-on-chip devices that would reliably and reproducibly perform simple signal-processing tasks relying on nonlinear dynamics.