Lorenz-Like Chaotic System on a Chip

LORENZ-LIKE CHAOTIC SYSTEM ON A CHIP Sachin Jambovane, Hoon Suk Rho, and Jong Wook Hong* Materials Research and Education Center, Department of Mechanical Engineering, Auburn University, Auburn, AL, USA

ABSTRACT The paper reports the development of Lorenz-like chaotic system in an all-fluidic, nanoliter scale, single-phase microfluidic platform. In our system, we have generated the Lorenz-like chaos inside the microchannel using the dynamic flow instability. The flow instability was introduced into an otherwise linear low-Reynolds number microchannel flow through the actuation of ensemble of peristaltic valves. The simultaneous interaction of valves and rounded flow channel creates dynamic flow instability inside the microchannel. In the field of microfluidics, the generation of Lorenz-like chaos is orig- inal and this could envision new understanding and applications of chaotic and peristaltic flows.

KEYWORDS: Lorenz, Microfluidics, Chaotic flow, Chaos, Flow instability, Peristaltic valves

INTRODUCTION Chaos is an integral part of natural and many man-made systems. It has been found that, its presence could be ex- tremely beneficial or it might be a complete disadvantage. Many dynamical systems, such as hydrodynamic and oscillatory chemical systems, have been deterministic in nature considering their dynamics—a phenomenon known as macroscopic chaos[1]. As a crack to the conventional wisdom, Lorenz mathematically demonstrated that simple but low dimensional systems could display chaotic behavior[2]. Moreover, since the origin of chaos, two approaches developed separately. First, it was the investigation of the properties and computer simulations of the chaotic systems. Second, the construction of working models to demonstrate chaos experimentally. In most cases, the simplicity of implementation encouraged many researchers to take advantage of electronic circuits, to realize the response of many chaotic oscillators and systems. In an effort to confirm the Lorenz model experimentally, the first mechanical analog was developed, in the form of waterwheel, by Malcos, Howard, and Krishnamurti[3]. Later Strogatz simplified this model to build its tabletop version [4]. In the case of Lorenz systems, in spite of its roots in fluid mechanics, it has remained quite unexplored except for a few attempt to build closed loop toroidal thermosyphon[5] to investigate the thermal instability. Though our device is ex- ceedingly simple, it is sufficiently rich not only to explain the existence of a Lorenz-like chaos, but also to reveal new un- anticipated behaviors of peristaltic flow or Lorenz-like chaotic system.. To the best of our knowledge, this could be the first exhaustive experimental development of the Lorenz-like model in all fluidic system.

EXPERIMENTAL The flows through the microchannels have many interesting and unexplored characteristics[6]. With this inspiration, we designed a device to generate Lorenz-like chaos. The device is fabricated using multilayer soft lithography, well known method in the field of microfluidic devices[7, 8]. The device consists of long microchannel and the integrated with number of peristaltic valves operated by control channels (Figure 1a). To provide the required stimulus to the microchannel flow, the valves were actuated by multiple digital valve operational sequences. The operational sequences changes for each of the for valve numbers, 3 to 6. For example, in the case of three valves, the sequence is 100-110-010-011-001-101. To better understand the working principle of the device, we conceptualized an equivalent circuit diagram as shown in figure 1b. As shown in the circuit diagram, the valve could be modeled as a capacitor, which works in the charging and discharging mode. Previously, we have successfully modeled the valve as capacitor[9]. Due to the nonlinear deflection behavior of PDMS membrane, the valve could be considered as nonlinear element. For maximum flow velocity, the valves are subjected to square type pressure signal. However, we hypothesized that the synchronized charging and discharging of the valves is a nonlinear phenomenon, leading to nonlinear flow behavior, flow instability, inside the microchannel. A range of methods discussed in next sections characterizes our hypothesis. In the device, we operated the fluid flowing through the microchannel only by operating the peristaltic valves. This condition of operating valves, allowed us to transfer the nonlinearity of the valve membrane, during charging and discharging of the of the valve, to the flow inside the microchannel. The general flow dynamics of the flow in the microchannel is characterized by determining the flow velocity by varying actuation of different valve numbers. Flow velocity is calculated based on the time required for the fluid front to travel a particular distance, d1 and d2 as shown in Figure 2b, and measuring the time of the travel. The procedure to cal- culate the flow velocity is vdtt111o , vdtt222 o and thus vvvmean 212 . To investigate the spatio-temporal flow characteristics of our device, we measured the intensity variation along the section of the microchannel in the captured video as shown in Figure 3b, by fixing a region of interest (ROI) of size100 100 μm2 . The average intensity variation inside the ROI of dyed fluid moving through the microchannel was plotted in the time domain (shown in Figure 3b).

Region of interest 0.5 mm

b. c. 51 40 Ensemble of valves 20 1cm 50 0 -20 Power (dB)

Intensity (A.U.) 49 -40 b. -60 120 130 140 150 01234 Pvalves(t) Time (Sec) Frequency (Hz)

Ircv Figure 3: a) The region of interest (ROI) is fixed along the Pi Qrcv Po microchannel as shown in the figure. b) Typical time do- Rrcv main response of the average intensity c) Continuous power Crcv spectra for the time series response, estimated by FFT. Ii Ri Ro Io

Qi Qrv Qo a. c. 240 Figure 1: (a) Layout of the fluidic platform designed for ge- 50 nerating systematic flow instability leading to a Lorenz-like 160 40 30 chaos; blue lines and red lines represent the flow channels 20 80 and the valve control channels respectively. Inset- Peristal- 10 Frequency Time (A.U.) tic valves assembly, (b) Equivalent fluidic circuit model of 0 the platform. 49 49.8 50.6 0 100 200 Intensity,x(t) (A.U.) Time (A.U.) b. a. b. t=t0 2100 1.0 1800 1mm 1500 3valve 4valve t=t1 1200 5valve 0.0 900 6valve dx(t)/dt 600 d1 300 t=t2 Flow velocity (µm/sec) 0 0481216 -1.0 Frequency (Hz) d2 49 50 51 Intensity,x(t) (A.U.) Figure 2 (a) flow velocity characteristics with changing frequency of operation of the peristaltic valves for different combination of valve numbers. (b) The flow velocity was de- Figure 4 Lorenz-like chaotic system. (a) Recurrence plot. termined by calculating the time (from shown pictures) fluid (b) Butterfly shaped attractor in a phase space plot. moves for unit distance.

RESULTS AND DISCUSSION Figure 1a shows the variation of flow velocity with change in the operational number of the valves. It is observed that, the velocity increases with increasing the number of valves. Considering all the combinations of valve numbers it can be observed that upto 1 Hz of actuation frequency, the mean fluidic velocity is almost linear. However, beyond the frequency of 1 Hz, the velocity increase is nonlinear. In the region of 3 to 5 Hz, the velocity was minimum for all valve combinations. This could be attributed to poor response of the valve, could be due to resonance of membrane. However, it needs further analysis for confirmation. In addition, as observed in Figure 2a, no flow was observed ahead of particular operating frequency, we can call this as a stall frequency. In addition, the stall frequency is different for different valve number combinations. During the spatio-temporal analysis, the average intensity variation inside the ROI of dyed fluid moving through the microchannel was plotted in the time domain (shown in Figure 3b). The close observation of the time signal revealed two zones of intensity levels. The first zone shows transient zone considering its lower intensity level. The intensity vs. time signal was converted to frequency domain through fast Fourier transform(FFT) in order to obtain asymptotic dynamics of

1140 the system . Considering the low amplitude of intensity in the frequency domain. The frequency spectrum was plotted as the power spectrum. Figure 3b shows continuous frequency spectrum where the lowest frequencies carry the highest power showing a nonperiodic chaotic behavior of the microflow[8]. Comparing the power spectrum with the one from Farmer et al.[9], it can be confirmed that the flow inside the microchannel could be Lorenz-like chaotic flow. However, we confirmed this with phase space plot and recurrence plot as described below. The phase-space xx ' obtained from the time series is shown in Figure 4a. This “butterfly” shaped attractor in the phase plot provides enough reasons to the authors to strongly believe that the proposed system has flow dynamics, inside the microchannel, similar to the Lorenz-like chaotic systems. To the best of our limited knowledge, this could be a unique result in the area of fluidics to show the butterfly shaped attractor in phase plot by using dynamic flow instability based turbulence inside the simple microchannel. The peculiar nature of the chaos would be perceived in the two wings of butterfly. In the left wing, we observe additional embedded butterfly type structure evolved due to zone 1 of the time signal. This also shows an advanced form of quasiperiodicity. In the right wing, we observe very chaotic response, probably due to zone 3 of the time signal. However, to characterize the system completely the phase space may not be sufficient to quantify the quasiperiodicity. To describe the behavior of phase portrait quantitatively, we plotted the histogram of intensity variation. Here, it can be observed that until intensity value of 50, the motion appears quasi-periodic while later it be- comes completely chaotic. To characterize the type of chaos further and for the quantification of the recurrence of states in the phase plot, we used a very powerful tool known as recurrence plot. Recurrence is the fundamental property of any deterministic dynamical system, which includes nonlinear and chaotic systems. The Poincaré recurrence theorem[10] states that all the trajectories of the dynamical return to a state to some neighborhood of the initial state, after a sufficiently long time in a phase space. However, the theorem provides qualitative treatment. Recurrence plot were initiated as tool to visualize hidden dynamical patterns and the recurrences of trajectories in the phase space[11]. The recurrence plot is the graphical representa- 4 tion of recurrence matrix RvvijNij, i j, , 1..... , where ɂ is the predefined threshold, is the Heaviside func- tion, 4 is a norm defining the distance between two points, and N is the length of trajectory[12]. According to the explanation of recurrence plot for Lorenz system provided in Eckmann et al.[11], we concluded that our recurrence plot, shown in Figure 4a, strongly expresses recurrence similar to Lorenz-like system.

CONCLUSION To conclude, we observed chaotic behavior in a simple device premeditatedly designed to produce Lorenz-like chaos on a microfluidic platform. By actuating appropriate valves and its sequence, dynamic instability based turbulence could promote quasi-periodic and/or chaotic regimes giving rise to interesting chaos. Lorenz-like chaos with its impressive flow instability based dynamics has never been demonstrated so far in an all-fluidic system, miniaturized or not. The chaotic behavior we observed here can be qualitatively extended to better understand the peristaltic flows in many systems such as the gastro-intestinal and reproductive systems. It could also open a new window for mathematical modeling and explora- tion due to its new chaotic attractor.

ACKNOWLEDGEMENTS The authors would like to acknowledge the help of Mr. Ryan Khodadadi in the fabrication of microfluidic device.

REFERENCES [1] P. Gaspard et al., Nature 394, 865 (1998). [2] E. N. Lorenz, Journal of the atmospheric sciences 20, 130 (1963). [3] E. N. Lorenz, The essence of chaos (Univ of Washington Pr, 1995). [4] S. H. Strogatz, Nonlinear dynamics and chaos: With applications to physics, biology, chemistry, and engineering (Perseus Books, 2001). [5] M. Sen, E. Ramos, and C. Treviño, Int. J. Heat Mass Transfer 28, 219 (1985). [6] G. M. Whitesides, Nature- 442, 368 (2006). [7] M. A. Unger et al., Science 288, 113 (2000). [8] S. Jambovane et al., Analytical chemistry 81, 3239 (2009). [9] W. S. Lee et al., Microfluidics and Nanofluidics 7, 431 (2009). [10] A. Katok, B. Hasselblatt, and L. Mendoza, Introduction to the modern theory of dynamical systems (Cambridge Univ Pr, 1997). [11] J. P. Eckmann, S. O. Kamphorst, and D. Ruelle, Turbulence, Strange Attractors, and Chaos 4, 441 (1995). [12] N. Marwan et al., Physics Reports 438, 237 (2007).

CONTACT *J.W. Hong, tel: +1-334-8447385; [email protected]

1141