aa r X i v : . [ n li n . C D ] S e p Boolean Chaos
Rui Zhang ∗ , † Hugo L. D. de S. Cavalcante ∗ , ‡ ZhengGao, Daniel J. Gauthier, and Joshua E. S. Socolar
Duke University, Department of Physics and Center forNonlinear and Complex Systems, Durham, North Carolina 27708
Matthew M. Adams and Daniel P. Lathrop
Department of Physics, IPST and IREAP,University of Maryland, College Park, Maryland 20742 (Dated: October 15, 2018)
Abstract
We observe deterministic chaos in a simple network of electronic logic gates that are not regulatedby a clocking signal. The resulting power spectrum is ultra-wide-band, extending from dc to beyond2 GHz. The observed behavior is reproduced qualitatively using an autonomously updating Booleanmodel with signal propagation times that depend on the recent history of the gates and filteringof pulses of short duration, whose presence is confirmed experimentally. Electronic Boolean chaosmay find application as an ultra-wide-band source of radio waves.
PACS numbers: 89.75.Hc, 89.70.Hj, 02.30.Ks, 05.45.-a ∗ RZ and HLDSC contributed equally to this article
1e show here that a very simple digital electronic device displays a form of deterministicchaos, a dynamical state characterized by a broadband spectrum and rapid divergence ofnearby trajectories. We also show that a modeling framework based on Boolean state tran-sitions with update times determined by signal propagation explains the origin of this novelbehavior, which we term “Boolean chaos.” Our device may be used as a building block insecure spread-spectrum communication systems [1], an inexpensive ultra-wide-band sensoror beacon, or a basis for engineering high-speed random number generators [2]. It can alsobe used to address fundamental aspects of the behavior of complex networks.Our network consists of three nodes realized with commercially-available, high-speedelectronic logic gates. The temporal evolution of the voltage at any given point in the circuithas a non-repeating pattern with clear Boolean-like state transitions, displays exponentialsensitivity to initial conditions, and has a broad power spectrum extending from dc to beyond2 GHz (see Fig. 1). Because the circuit includes feedback loops with incommensurate timedelays, it spontaneously evolves to dynamical states with the shortest possible pulse widths,a regime in which time-delay variations generate chaos. We conjecture that similar behaviorwill occur in a wide class of systems described by autonomous Boolean networks.Boolean networks have been studied extensively in a variety of contexts. For systems thatdisplay switch-like behavior, such as logic circuits and gene regulatory networks, it is oftenuseful to assume the system variables take only two values ( e. g. , “high” and “low”) that areupdated according to specified Boolean functions [3, 4, 5, 6]. Deterministic Boolean modelsoften include an external process such as a clock that synchronizes all the updates or a devicethat selects a particular order of individual gate updates. The state space of such modelsis discrete and finite, and can therefore have only periodic attractors. On the other hand,in many physical or biological systems, information propagates between logic elements withtime delays that can be different for each link [7, 8, 9]. In such systems, the future behavioris determined by specification of the precise times at which transitions occurred in the past,which makes the state space continuous. The mathematics describing these autonomousBoolean systems is much less developed, though it is known that they can display aperiodicpatterns if the logic elements have instantaneous response times [10, 11, 12].Ghil and collaborators [10, 11, 12] introduced Boolean delay equations (BDEs) to studyBoolean networks of ideal logic elements. They study the dynamics of state transitions,(called events here) in the networks under the hypothesis that the logic gates can process2 .0 0.5 1.0 1.5 2.0 2.5frequency (GHz)-50-45-40-35-30 PS D ( d B m ) BW -10dB = 1.3 GHz τ τ τ τ τ τ s i gna l ( V ) (a)(b) 12 3(c) FIG. 1: (Color online) (a) Topology of the chaotic Boolean network and truth table for logicoperation performed by the nodes 1, 2 ( xor ), and 3 ( xnor ) on their respective inputs. (b)Temporal evolution and (c) power spectral density (PSD) of the chaotic network for V CC = 2 .
75 Vwith a measurement bandwidth of 1 MHz. input signals arbitrarily fast. They consider the behavior to be complex when the event rateper unit time for the whole system grows as a power-law, and predict it can happen for awide class of Boolean networks.The complex behavior identified by Ghil leads to an ultraviolet catastrophe that can neveroccur in an experiment because real logic gates cannot process arbitrarily short pulses. Wefind that the predicted complex behavior is replaced by deterministic chaos in our exper-imental systems and numerical simulations that take into account the non-ideal behaviorsdescribed below. Given the presence of complex behavior in a large class of ideal BDEs, andgiven our observation of deterministic chaos in a simple experimental example with threenodes, we conjecture that a large class of experimental Boolean networks will display chaos.The topology of our autonomous Boolean network is shown in Fig. 1(a). It consists ofthree nodes that each have two inputs and one output that propagates to two different nodes.The time it takes a signal to propagate to node j from node i is denoted by τ ji ( i, j = 1 , , or ( xor ) logic operation, while node 3 executes the xnor (see truth tables in the Fig. 1). The three-node network has no stable fixed point and3lways leads to oscillations. Each time delay comes about from a combination of an intrinsicdelay associated with each gate and the signal propagation time along the connecting link,which we augment by incorporating an even number of not gates or Schmitt triggers wiredin series, either of which act effectively as a time-delay buffer. We stress that there is noclock in the system; the logic elements process input signals whenever they arrive, to theextent that they are able.We observe the dynamics of our network using a high-impedance active probe and an8-GHz-analog-bandwidth 40-GS/s oscilloscope. Figure 1(b) shows the typical observed be-havior when the probe is placed at the output of node 2. The temporal evolution of thevoltage is complex and non-repeating and has clearly defined high and low values, indicatingBoolean-like behavior. The rise time of the measured voltage is ∼ ∼ V CC of the logic gates,which we consider as a bifurcation parameter. Our hypothesis is that the observed dynamicschanges with supply voltage because the different characteristic times of the logic elements,such as the transition time delay, rise and fall times, etc., all depend smoothly on the supplyvoltage. To map out a bifurcation diagram for the network, we collect a 1- µ s-long timeseries of the voltage at node 2 for a fixed value of V CC and transform it into a time series ofa Boolean variable x ( t ) ∈ { , } by comparison to a threshold: x ( t ) = 0 , for V ( t ) < V CC / x ( t ) = 1 , for V ( t ) ≥ V CC / V CC by 5 mV and repeat, startingat V CC = 0 . V CC > .
40 V, where the logic gates are4iased to operate at maximum speed. CC (V)05101520 t i m e be t w een r i s e s ( n s ) FIG. 2: Bifurcation diagram of the Boolean network. The arrow indicates the value of V CC givingthe complex behavior shown in Fig. 1(b). A signature of chaos is exponential divergence of trajectories with nearly identical initialconditions, which is indicated by a positive Lyapunov exponent. We propose a method toestimate the largest Lyapunov exponent as follows. We acquire a long time series of thevoltage and transform it to a Boolean variable x ( t ). Given any two segments of x ( t ) startingat times t a and t b , we define a Boolean distance [11] between them by d ( s ) = 1 T Z s + Ts x ( t ′ + t a ) ⊕ x ( t ′ + t b ) dt ′ , (1)where T = 10 ns is a fixed parameter, ⊕ is the xor operation, and the Boolean distance d ( s ) evolves as a function of the time s . We then search in x ( t ) for all the pairs t a and t b corresponding to the earliest times in each interval T over which d (0) < .
01 (ln d (0) < − . µ s-long time series. We thencompute h ln d ( s ) i , where h i denotes an average over all matching ( t a , t b ) pairs.Figure 3(a) shows two typical segments for the voltages V ( s + t a ) and V ( s + t b ), and Fig.3(b) shows the associated Boolean variables x ( s + t a ) and x ( s + t b ). A visual inspection ofthe time series on a much finer scale (not shown) reveals that there exist small differencesin the timing of events between the two trajectories. On the scale of the figure, trajectorydivergence is noticeable around 20 ns and the two trajectories appear to be uncorrelatedafter approximately 30 ns.To quantify these observations, we determine the largest Lyapunov exponent of the at-tractor. The solid black curve in Fig. 3(c) shows the time evolution of h ln d i . It displays anapproximately constant slope for times shorter than ∼
20 ns and, finally, saturates at a max-imum value of ln 0 . ≈ − .
69, corresponding to uncorrelated x ( s + T + t a ) and x ( s + T + t b ).5 V ( V ) x 〈 l n ( d ) 〉 (a)(b)(c) FIG. 3: (Color online) (a) Typical segments of similar voltages for V CC = 2 .
75 V. (b) The resultingBoolean variables obtained from the voltages in (a). (c) Logarithm of the Boolean distance as afunction of time, averaged over the network phase-space attractor for experimental data (black)and simulations (red online).
To estimate the value of the maximum Lyapunov exponent, we assume that, in the re-gion of constant slope, the divergence of the initially similar segments is exponential, i.e. ,ln d ( s ) = ln d + λ ab s , where λ ab is the local Lyapunov exponent. The average of λ ab over allpairs of similar segments is our estimate of the largest Lyapunov exponent λ of the system.We find λ = 0 .
16 ns − ( ± .
02 ns − ), which demonstrates that the network is chaotic. Ourmethod is based on neighbor searching in the time series of a single element, as describedin Ref. [13], except that we use the Boolean distance instead of delay-coordinates.To test our analysis method, we set V CC to place the system in a nearby periodic window(2.35 V) and repeat our analysis. We find that the Boolean distance stays small ( h ln d i < − x ( t ) = x ( t − τ ) ⊕ x ( t − τ ) ,x ( t ) = x ( t − τ ) ⊕ x ( t − τ ) ,x ( t ) = x ( t − τ ) ⊕ x ( t − τ ) ⊕ , (2)where x i is the Boolean state of the i th node and the term ⊕ not oper-6tion. The values of τ ji are given in the last line of Table I. Using initial conditions( x ( t ) , x ( t ) , x ( t )) = (0 , ,
0) for t <
0, we find that the average event rate for x ( t ) (orany of the variables) grows as a power law with exponent ∼
2, indicating complex networkbehavior as defined by Ghil et al. [11].This increasingly fast event rate is prevented in the experimental system by the finiteresponse time of the real logic gates. We find that the dominant contribution to the non-idealbehavior of the network components is due to the series of gates used to generate the delaysin the network links; the non-ideal behavior of the xor and xnor nodes is much smaller andcan be modeled by a constant delay after an ideal gate. To quantify the non-ideal behaviorof the links, we measure simultaneously the voltage at the input and output of each linkand determine the propagation delay times. The data display the three non-ideal behaviors:(1) short-pulse rejection, also known as pulse filtering, which prevents pulses shorter thana minimum duration from passing through the gate [7, 8]; (2) asymmetry between the logicstates, which makes the propagation delay time through the gate depend on whether thetransition is a fall or a rise [8]; and (3) a degradation effect that leads to a change in thepropagation delay time of events when they happen in rapid succession [8, 14]. We note thatthese non-ideal behaviors have been proposed for Boolean idealizations of electrical [14] andbiological networks [7, 8], suggesting that studies of these effects may have wide application.The non-ideal behaviors are accounted for in our model as follows. First, followingRef. [14], we introduce a new variable to describe the degradation effect of a link on signalpropagation. Let t n be the time that event n occurs at the beginning of a link and let t ′ n bethe time that the corresponding event is observed at the end of the link. Note that t n doesnot involve the degradation associated with the link, but t ′ n does. We define P n ≡ t n + τ kji − t ′ n − , (3)where τ kji is the nominal time delay on the link for a rising ( k = r ) or falling ( k = f ) event.Typical behavior of the propagation delay τ f ,n ≡ t ′ n − t n for falling events as a function of P n for link 33 is shown in Fig. 4. We fit the experimental data for all links to τ kji,n = τ kji + Ae − BP n cos(Ω P n + φ ) (4)where τ kji , A , B , Ω, and φ are fit parameters, and τ ji,n is the delay of the n th event as itpropagates through link ji . The minimum interval P min is determined from the data based7 i
12 13 21 22 31 33 τ rji (ns) 3.13 4.30 3.201 2.47 3.08 3.62 τ fji (ns) 2.92 4.09 2.97 2.27 2.85 3.42TABLE I: Experimentally measured delay times τ kji . n (ns)2.22.42.62.83.0 τ f , n ( n s ) FIG. 4: (Color online) Experimentally measured time delay for a transition as it propagates throughthe delay line 33 (black dots) as a function of P n . Pulses are affected by the degradation effect.The measured values are fit to an empirical expression (solid line) discussed in the text. on the shortest value for which events are observed. The only parameter that dependsstrongly on the specific link and event sign is τ kji (see Table I). Based on our fit to thedata in Fig. 4 (solid line), we find that the remaining parameters take on values A = 1 . B = 1 . − , Ω = 4 . φ = 0 .
062 rad, and P min = 0 .
48 ns, which we assumeapplies to all links in our network. The next step in our simulation procedure is to solve theideal Boolean delay equations (2) with τ ji replaced by τ kji as appropriate. For each event,we evaluate P n . If P n < P min , both the new event and the previous one are eliminated.Otherwise, we adjust the newly generated transition time using Eq. (4).Using the simulated time series data, we calculate h ln d i (Fig. 3(c)) using the initialvalue of the Boolean distance of d (0) < .
001 (ln d (0) < − .
9) for choosing pairs. Wefind that λ = 0 .
10 ns − ( ± .
02 ns − ), which demonstrates that the model, modified totake into account the non-ideal behaviors of the logic gates, displays deterministic chaos.Furthermore, the Lyapunov exponents obtained in both the experiment and simulations arevery similar, demonstrating that our model captures the essential features of our electronicnetwork. A systematic study of the effect of each individual non-ideal behavior is beyondthe scope of this Letter.In summary, we observe that an autonomous Boolean network displays deterministicchaos in its sequence of switching times. This behavior is very different from that observed8n Boolean networks with periodic or clocked updating, where only periodic behavior ispredicted. Our research may have important implications for understanding other networksobserved in nature. We note, for example, that chaos was observed in a system of differentialequations of a form relevant to the modeling of genetic regulatory networks [8], thoughthe source of chaos was not identified. To make the connection to other natural systemsprecise, measurements of non-ideal logic elements are needed. We believe that the threeeffects identified here are likely to be generic, though they may be difficult to study directly.Further theoretical study is also needed to determine the extent to which modified Booleandelay equations can serve as a guide for designing and understanding real network behavior.RZ, HLDSC, ZG, DJG, and DPL gratefully acknowledge the financial support of theOffice of Naval Research, grant Nos. N00014-07-1-0734 and N00014-08-1-0871, and the adviceof John Rodgers. JESS gratefully acknowledges the support of the NSF under grant PHY-0417372. HLDSC and RZ thank Steve Callender for tips on soldering techniques. † Electronic address: [email protected] ‡ Electronic address: [email protected][1] A. R. Volkovskii, L. S. Tsimring, N. F. Rulkov, and I. Langmore, Chaos , 033101 (2005).[2] I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, Phys. Rev. Lett. , 024102 (2009).[3] F. Jacob and J. Monod, J. Mol. Biol. , 318 (1961).[4] F. Jacob and J. Monod, Cold Spring Harb. Symp. Quant. Biol. , 193 (1961).[5] E. H. Davidson, The Regulatory Genome: Gene Regulatory Networks in Development andEvolution (Academic Press, San Diego, California, 2006).[6] A. Pomerance, E. Ott, M. Girvan, and W. Losert, Proc. Natl. Acad. Sci. , 8209 (2009).[7] K. Klemm and S. Bornholdt, Phys. Rev. E , 055101(R) (2005).[8] J. Norrell, B. Samuelsson, and J. E. S. Socolar, Phys. Rev. E , 046122 (2007).[9] L. Glass, T. J. Perkins, J. Mason, H. T. Siegelmann, and R. Edwards, J. Stat. Phys. , 969(2005).[10] D. Dee and M. Ghil, SIAM J. Appl. Math. , 111 (1984).[11] M. Ghil and A. Mullhaupt, J. Stat. Phys. , 125 (1985).[12] M. Ghil, I. Zaliapin, and B. Coluzzi, Physica D , 2967 (2008).
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