A nonlinear switched-capacitor circuit that generates chaotic signals is reported. The circuit is described by a first-order piecewise-linear discrete equation that exhibits a chaotic dynamics. Experimental results illustrating the circuit performance and its use as a noise generator are included.Comisión Interministerial de Ciencia y Tecnología 3467-8

Huertas Díaz, José Luis

Pérez Verdú, Belén

Rodríguez Vázquez, Ángel Benito

Rueda Rueda, Adoración

English

idUS. Depósito de Investigación Universidad de Sevilla

respect to the principal (cardinal) axes, the contour graphsare shown for one quadrant in Q-<j) space.The array of the double square loops set in a square latticeforms an FSS which is stationary at any 90 degree rotation,and has principal axes at 45 degree intervals. Therefore, theincidence in the plane at </> = 45 degrees does not produce anydepolarisation (Fig. 3), and peak values of the orthogonal pol-arised component appear at nearly </> = 22-5 degrees from thisplane.Conclusions: The proposed method overcomes two basic diffi-culties in FSS applications: it reduces the computation time tocalculate the FSS parameters and also requires less FSS datastorage space. This is achieved by relating the values of thescattering coefficients for any polarisation and direction of theincident wave to the principal-plane values. An obviousadvantage of this method is that only the principal-planevalues of the scattering coefficients need be known before theantenna analysis program is invoked. When the FSS informa-tion is supplied in the form of measured data, the storagerequirement is considerably reduced, since the storage space isonly required for the principal planes. For the FSS with asquare lattice of elements which are stationary under a 90degree rotation, such as rings or square loops, the values ofthe scattering coefficients need be specified only on one prin-cipal plane. When the FSS is to be represented by a mathe-matical model, the model can be simplified considerably forthe case of principal-plane incidence, and hence save computa-tion time. For a type of FSS with elements composed ofstraight-line segments, the proposed method offers an addi-tional advantage. This type of FSS includes rectangular(square) loops, double square loops, a rectangular (square)patch array, a looped patch array, their gridded versions andtheir complements. They can be represented by theirtransmission-line equivalent-circuit models using strip gratingformulas. The computation time by the equivalent-circuitmethod is much less than required by a moment method solu-tion of comparable accuracy.M. J. ARCHER 4th March 1987British AerospaceSpace & Communications DivisionStevenage SGI 2AS, United KingdomReferences1 ANDERSON, i.: 'On the theory of self-resonant grids', Bell Syst.Tech. J., 1975, 54, pp. 1725-17312 MARCUVITZ, N.: 'Waveguide handbook' (McGraw-Hill, 1951), pp.280-2843 ARCHER, M. J.: 'Periodic multielement strip gratings', Electron.Lett., 1982,18, pp. 958-9594 ARCHER, M. j . : 'Inductive twin-strip gratings in incident TE waves',ibid., 1983,19, pp. 118-1205 ARCHER, M. J.: 'Wave reactance of thin planar strip gratings', Int. J.Electron., 1985, 58, pp. 187-230CHAOS VIA A PIECEWISE-LINEARSWITCH ED-CAPACITOR CIRCUITIndexing terms: Oscillators, Switched-capacitor circuitsA nonlinear switched-capacitor circuit that generates chaoticsignals is reported. The circuit is described by a first-orderpiecewise-linear discrete equation that exhibits a chaoticdynamics. Experimental results illustrating the circuit per-formance and its use as a noise generator are included.Introduction: The study of chaotic behaviour appearing innonlinear circuits has attracted great attention in the past fewyears.1"811 In particular, several circuit structures have beenreported that can be used as electronic chaos generators.Together with their circuit-theoretic properties, these circuits662are also interesting because they provide simple practical toolsfor experimenting on chaotic phenomena.To date, most of the proposed chaos generators arecontinuous-time circuits described by systems of ordinary dif-ferential equations that exhibit chaotic solutions.3"7 However,chaos has also been observed in numerous dynamical systemsmodelled by nonlinear finite-difference equations, henceforthcalled discrete maps. Moreover, it has been shown elsewherethat discrete maps are the simplest mathematical models forexplaining chaos, the most extended way to deal with thisphenomenon being to simulate discrete maps using a digitalcomputer.910 Nevertheless, in Reference 8 a switched-capacitor circuit described by a parabolic discrete map isgiven. It provides us with an empirical tool that can substitutedigital computers for the study of chaotic phenomena arisingin different areas of science and technology.In this letter we report a novel switched-capacitor circuitwhich is modelled by a piecewise-linear discrete map. The newcircuit is suitable for generating chaos while using fewer com-ponents than the circuit in Reference 8; in particular, theinconvenience of using analogue multipliers is avoided. In thissense, we can say the circuit to be described below is thesimplest reported circuit modelling a discrete map.Mathematical model and circuit structure: The circuit we willpropose herein is intended to implement the following discretemap:A + BxnA - f i x .x n < 0x..>0 (1)The chaotic dynamics of this map have been extensivelystudied.9"10 Its asymptotic behaviour comes from the valuesof both A and B parameters. For values of B less than 1 astable point is reached for any value of A. For values of Bgreater than 2 all solutions are attracted to — oo. On the otherhand, for values of B e (1, 2) all solutions are aperiodic. This iscalled the chaotic region. Inside this region, the values of sub-sequent iterates of x are randomly distributed on one or moresubintervals of the real axis, the number of subintervals andtheir corresponding length depending on the value of A.Let us now consider the switched-capacitor circuit shown inFig. la. Switches labelled V (respectively V) turn ON in syn-chronisation with the even (respectively odd) phase of theclock signal shown in Fig. \b. On the other hand, switchesz X I Y comparatorJL-r-  p ifFig. 1a Piecewise-linear switched-capacitor circuitb Timing diagram for switcheslabelled Z (respectively Z) are controlled by the binary outputsignal (respectively, by the logical complement) of the compa-rator.The circuit operates as follows: during the (n + l/2)th oddphase capacitor Cx charges atU+l/2 ~ Xn+l/2 ~ A (2)ELECTRONICS LETTERS 4th June 1987 Vol. 23 No. 12Authorized licensed use limited to: Universidad de Sevilla. Downloaded on March 31,2020 at 15:47:04 UTC from IEEE Xplore.  Restrictions apply. while the charge in capacitor C2 will depend on the sign ofxn + 1 / 2 as below,- x n+l/2Xn+l/2Xn+l/2 (3)Moreover, the charge in C3 does not change during the oddphase, which meansXn+l/2 ~ Xn (4)During the next (n + l)th even phase both capacitors Ct andC2 deliver all the previously stored charge to C3, thus fromthe charge conservation principle,C 3(x n + 1-x n)= - (5)Combining eqns. 2 to 4 with eqn. 5 we obtain eqn. 1, A beingthe value of the external battery shown in Fig. la and B thecapacitor ratio C2/C3.Experimental results: We have built the circuit in Fig. la usingoff-the-shelf components (a //A741 opamp, a /iA709 compara-tor, MC14066 analogue switches, and 20% tolerance capac-itors ranging between 1 nF and 10 nF). We have used a clockfrequency fc = 20 kHz and made many experiments by varyingboth A and B parameters. Fig. 2 shows a set of measuredfrequency spectra for different values of B comprised in thechaotic region, namely for B = 101 (Fig. 2a), B = 1-2 (Fig.2b), B = 1-33 (Fig. 2c) and B = 1-83 (Fig. Id). For each case•I i ii it i^rthr^ft I M in ia [IT972]Fig. 2 Frequency spectra for some aperiodic output signals for circuit ofFig. laa A = 3,B= 101b A = 3,B=l-2c A = 3,B= 1-33d A = 3,B = 1 83we used A = 3. The peakings appearing at/c/4,/c/2 and 3fJ4in Figs. 2a-b are a consequence of the accumulation of iteratesinside separate subintervals of the real axis, either 4 (Figs. 2a,b) or 2 (Fig. 2c). Notice that, as B increases, the spectrumbecomes approximately flat for most of the frequencies, beingflat along the whole frequency range going from DC to^ forB = 1-83 (Fig. 2d). Since white noise is known to be character-ised by a flat frequency spectra, we can appreciate the poten-tial use of the new circuit as a white-noise source.The time-domain evolution of the chaotic signal providedby the circuit is illustrated in Figs. 3a and b. There we showtime, 400(js/div time, 400 ps /divfn9/3]Fig. 3 Oscilloscope displays measured from circuit in Fig. la for A = 3,B = 1-83 by single-shooting at two different arbitrary time instancesELECTRONICS LETTERS 4th June 1987 Vol. 23 No. 12two displays measured from the circuit for A = 3, B = 1-83and obtained by single shooting the oscilloscope at two differ-ent time instances. As corresponds to a chaotic regime, noperiodicity can be observed. Moreover, a careful observationshows us that apparently coincident values of xn result inquite different values after a few iterations, this kind of behav-iour relying on the very foundations of any chaotic regime.910Discussion of results: We have reported a new circuit andillustrated its performance as a generator of chaos. It can beuseful either for the simulation of chaotic phenomena or as asource of white noise. The circuit is also of circuit-theoreticinterest because its dynamic equation is equivalent to a wellknown discrete map. Going beyond the purpose of this letter,the obtained results reveal the existence of a close connectionbetween nonlinear switched-capacitor circuits and discretemaps: chaos can occur in a one-integrator loop switched-capacitor circuit owing to the presence of nonlinearities. Thisconnection is worth considering in future research becauseswitched-capacitor circuits are important in VLSI tech-nology.11Acknowledgment: This work was supported by CAICYTResearch Contract 3467-83.A. RODRIGUEZ-VAZQUEZ 20th March 1987A. RUEDAB. PEREZ-VERDUJ. L. HUERTASDto. de Electricidad y ElectrbnicaFacultad de FisicaUniversidad de Sevilla, 41012 Sevilla, SpainReferences1 CHUA, L. o., KOMURO, M., and MATSUMOTO, T.: 'The double scrollfamily', IEEE Trans., 1986, CAS-33, pp. 1073-11182 TANG, Y. s., MEES, A. i., and CHUA, L. o.: 'Synchronization andchaos', ibid., 1983, CAS-30, pp. 620-6263 MATSUMOTO, T., CHUA, L. o., and TANAKA, s.: 'Simplest chaotic non-autonomous circuit', Phys. Rev. A, 1984, 30, pp. 1155-11574 FREIRE, E., GARCIA-FRANQUELO, L., and ARACIL, J.: 'Periodicity andchaos in an autonomous electronic system', IEEE Trans., 1984,CAS-31, pp. 237-2475 AZZOUZ, A., DUHR, R., and HASLER, M.: Transition to chaos in asimple nonlinear circuit driven by a sinusoidal voltage source',ibid., 1983, CAS-30, pp. 913-9146 NEWCOMB, R. w., and SATHYAN, s.: 'An RC op amp chaos gener-ator', ibid., 1983, CAS-30, pp. 54-567 MATSUMOTO, T.: 'A chaotic attractor from Chua's circuits', ibid.,1984, CAS-31, pp. 1055-10588 RODRIGUEZ-VAZQUEZ, A., HUERTAS, j . L., and CHUA, L. o.: 'Chaos ina switched-capacitor circuit', ibid., 1985, CAS-32, pp. 1083-10859 MAY, R. M.: 'Simple mathematical models with very complicateddynamics', Nature, 1976, 261, pp. 459-46710 SCHUSTER, H. G.: 'Deterministic chaos: an introduction' (Physik-Verlag, Weinheim, W. Germany, 1984)11 RODRiGUEZ-VAZQUEZ, A., HUERTAS, J. L., RUEDA, A., PEREZ-VERDU, B.,and CHUA, L. O.: 'Chaos from switched-capacitors circuits: discretemaps', Proc. IEEE, to be publishedERRATUMFORSBERG, G. s.: 'Optical receiver with optical feedback', Elec-tron. Lett., 1987, 23, (9), pp. 478^80In the seventh line on p. 479, T should be defined as the 'timedelay due to delay in optical source'Eqn. 1 should readUt) = RPJit)Eqn. 3 should readA(s)e-*T = P0(s)/V,<s)663Authorized licensed use limited to: Universidad de Sevilla. Downloaded on March 31,2020 at 15:47:04 UTC from IEEE Xplore.  Restrictions apply. 

Chaos via a piecewise-linear switch ed-capacitor circuit

https://idus.us.es/bitstream/handle/11441/94757/CHAOS%20VIA%20A%20PIECEWISE-LINEAR.pdf?sequence=1&isAllowed=y

Chaos via a piecewise-linear switch ed-capacitor circuit

Abstract

Similar works

Full text

Available Versions

idUS. Depósito de Investigación Universidad de Sevilla