Bumeliene, Skaidra

Kirvaitis, Raimundas

Lindberg, Erik

Mykolaitis, Gytis

Tamaševicius, Arunas

Tamaševiciute, Elena

A novel Duffing-Holmes type autonomous chaotic oscillator is described. In comparison with the well-known non-autonomous Duffing-Holmes circuit it lacks the external periodic drive, but includes two extra linear feedback sub-circuits, namely a direct positive feedback loop, and an inertial negative feedback loop. SPICE simulation and hardware experimental results are presented

Online Research Database In Technology

                       General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.  • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal   If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.     Downloaded from orbit.dtu.dk on: Dec 17, 2017CHAOTIC DUFFING TYPE OSCILLATOR WITH INERTIAL DAMPINGTamaševicius, Arunas; Mykolaitis, Gytis; Kirvaitis, Raimundas; Tamaševiciute, Elena; Bumeliene,Skaidra; Lindberg, ErikPublished in:Proceedings of the 17th International Workshop on Nonlinear Dynamics of Electronic SystemsPublication date:2009Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Tamaševicius, A., Mykolaitis, G., Kirvaitis, R., Tamaševiciute, E., Bumeliene, S., & Lindberg, E. (2009).CHAOTIC DUFFING TYPE OSCILLATOR WITH INERTIAL DAMPING. In R. Stoop, & J. Joller (Eds.),Proceedings of the 17th International Workshop on Nonlinear Dynamics of Electronic Systems: NDES 09 (pp. 1-230)CHAOTIC DUFFING TYPE OSCILLATOR WITH INERTIAL DAMPINGArǌnas Tamaševiþius1, Gytis Mykolaitis1,2, Raimundas Kirvaitis3,Elena Tamaševiþiǌtơ1,4, Skaidra Bumelienơ1, and Erik Lindberg51Plasma Phenomena and Chaos Laboratory, Semiconductor Physics Institute, A. Goštauto 11, Vilnius, LT-01108, Lithuaniae-mail: tamasev@pfi.lt; gytis@pfi.lt; skaidra@pfi.ltWWW: http://www.pfi.lt/index_e/2Department of Physics, Faculty of Fundamental Sciences, Vilnius Gediminas Technical University, Saulơtekio 11, Vilnius, LT-10223, Lithuania e-mail: gytis@pfi.ltWWW: http://www.fm.vgtu.lt/en/3Department of Electronic Systems, Faculty of Electronics, Vilnius Gediminas Technical University, Naugarduko 41, Vilnius, LT-03227, Lithuania e-mail: raimundas.kirvaitis@adm.vgtu.ltWWW: http://www.el.vgtu.lt/en/4Department of General Physics and Spectroscopy, Faculty of Physics, Vilnius University, Saulơtekio 9, Vilnius, LT-10222, Lithuania e-mail: elena.tamaseviciute@ff.vu.ltWWW: http://www.vu.lt/en/5DTU Elektro Department, Electronics & Signalprocessing, 348 Technical University of Denmark, Lyngby, DK-2800, Denmark e-mail: el@elektro.dtu.dkWWW: http://www.elektro.dtu.dk/English.aspxAbstract—A novel DuffingHolmes type autono-mous chaotic oscillator is described. In comparisonwith the wellknown nonautonomous DuffingHolmes circuit it lacks the external periodic drive,but includes two extra linear feedback subcircuits,namely a direct positive feedback loop, and an inertial negative feedback loop. SPICE simulation and hardware experimental results are presented. I. INTRODUCTIONThe DuffingHolmes nonautonomous oscillator is a classical example of a nonlinear dynamical systemexhibiting complex also chaotic behaviour [13]. It is given by the secondorder differential equationwith an external periodic drive term:taxxxbx 13 sinZ   . (1)Three different techniques are used to solve theDuffingHolmes equation and/or to process itssolutions electronically. The first approach is ahybrid one making use of integration the equation ina digital processor and of the digitaltoanalogueconversion of the digital output for its further analogue processing, analysis and display [4, 5]. Thesecond method employs purely analogue hardware based on analogue computer design [69]. Forexample, analogue computer has been used to simulate Eq. (1) and to demonstrate the effect ofscrambling chaotic signals in linear feedback shift registers [68]. Later analogue computer has beensuggested for demonstration of chaos from Eq. (1)for the undergraduate students [9]. The thirdtechnique is based on building some specificanalogue electrical circuit imitating dynamicalbehaviour of Eq. (1). The YoungSilva oscillator [10] described in more details in [11] and used to demonstrate the effect of resonant perturbations for inducing chaos [12, 13] is an example. Recently the YoungSilva circuit has been essentially modifiedand used to test the control methods for unstable periodic orbits [14, 15] and unstable steady states [16] of dynamical systems. The modified version has been characterized in details both numericallyand experimentally in [17].102Evidently the first and the second techniques are rather general and can be applied to other differential equations as well. In contrast, the third approach is limited to a specific equation. Despite this restrict-ion the “intrinsic” electrical circuits havean attractive advantage due to their extremesimplicity and cheapness. One may think of such analogue electrical circuits as of analogue computers. This is true froma mathematical and physical point of view in the sense that the underlying equations are either exactlythe same or very similar also that the dynamicalvariables in the both cases are represented by real electrical voltages and/or currents. However, the circuit architecture of an analogue computer,compared to an “intrinsic” nonlinear circuit, is ratherdifferent. Any analogue computer is a standardcollection of the following main processing blocks: inverting RC integrators, inverting adders, invertingand non-inverting amplifiers, multipliers, andpiecewise linear nonlinear units. Meanwhile thespecific analogue circuits comprise only small number of electrical components: resistors, capacitors, inductors, and semiconductor diodes. In addition, they may include a single operational amplifier (in some cases several amplifiers). The differences between the “intrinsic” analogue electrical circuits, simulating behaviour of the specific dynamical systems, and the commonanalogue computers are discussed in [18].In this paper, we introduce, as an alternative for the nonautonomous Eq. (1), an autonomous versionof the Duffing type oscillator given by)(,03zxzkzxxxbxf   Z(2)or equivalently by a set of three first order differential equations ).(,,3zyzkzbyxxyyxf    Z(3)Here z is the third independent dynamical variable, Zf is its characteristic rate, and k is the feedbackcoefficient. We emphasize in Eq. (2) an opposite sign of the damping term, compared to Eq. (1). Thenegative damping,  in Eq. (2), or xb by  in Eq. (3) yields an additional spiral instability. Also we propose a specific electrical circuit imitatingsolutions of Eq. (3). II. CIRCUITRYFirst of all we recall to the nonautonomous circuitshown in Fig. 1 for comparison. The novel autonomous circuit is presented in Fig. 2.Fig. 1. Circuit implementation of the nonautonomousDuffingHolmes oscillator [17].Fig. 2. Circuit implementation of the autonomousDuffingHolmes type oscillator. Any of the nodes letteredas ‘x’, ‘y’ or ‘z’ can be taken for the output.The nonautonomous oscillator contains a single nonlinear positive feedback loop introduced by theresistor R3, two diodes, and the operational amplifier OA1. The external periodic drive is givenby asinZt.The autonomous oscillator lacks the externalperiodic drive, but includes two additional linear feedback loops. The circuit composed of the OA2based stage and the resistor R5 introduce the positive feedback loop, specifically negativedamping in Eq. (2). While the circuit including the OA2OA3 stages (note a capacitor C1 in the latter stage) and the resistor R8 compose the inertial negative feedback, specifically the inertial dampingterm kz in Eq. (2). 103III. SIMULATION RESULTSThe oscillator in Fig. 2 has been simulated using the ELECTRONICS WORKBENCH package (SPICE basedsoftware) and the results are shown in Figs. 35.Fig. 3. Snapshot of typical chaotic waveform of x(t) fromthe autonomous Duffing type oscillator. L = 19 mH, C =470 nF, C1 = 20 nF, R = 20 :, R1 = 30 k:, R2 = 10 k:,R3 = 30 k:, R4 = 820 :, R5 = 75 k:, R6 = R7 = 10 k:,R8 = 20 k:. The OA1 to OA3 are the LM741 type or similar operational amplifiers, the diodes are the 1N4148type or similar generalpurpose devices.Fig. 4. Simulated phase portraits and Poincaré section(bottom right). Element values are the same as in Fig. 3. Fig. 5. Simulated power spectrum S from the variable x(t).Circuit element values are the same as in Fig. 3. IV. HARDWARE EXPERIMENTSThe autonomous oscillator has been built using the elements described in the caption to Fig. 3. Typicalexperimental results are presented in Figs. 68.x(t)toFig. 6. Experimental snapshot of chaotic waveform x(t).Horizontal scale 2 ms/div. Vertical scale 1 V/div. Elementvalues are the same as in Fig. 3, except R5 = 68 k:.zyx xz y z=0, Ī>0y xFig. 7. Experimental phase portraits and Poincaré section(bottom right). Circuit element values are the same as inFig. 3, except R5 = 68 k:.S, dB foFig. 8. Experimental power spectrum S from the outputsignal x(t). Frequency range 0 to 5 kHz. Horizontal scale500 Hz/div., resolution 100 Hz. Vertical scale 10 dB/div.Circuit element values are the same as in Fig. 3, exceptR5 = 68 k:.104V. CONCLUDING REMARKSWe have designed and built a novel DuffingHolmes type autonomous thirdorder chaotic oscil-lator. In comparison with the common nonautonomous DuffingHolmes type oscillator the autonomous circuit has an internal positive feedback loop instead of an external periodic drive source. In addition, it is supplemented with an RC inertial damping loop providing negative feedback. The circuit has been investigated both numerically and experimentally. The main characteristics, including the time series, phase portraits, Poincaré sections, and power spectra have been calculated using the SPICE based software, also taken experimentally. Fairly good agreement between the simulation and the hardware experimental results is observed (Figs. 38). Some discrepancy (about 10%) between the model and the hardware prototype, namely R5 = 75 k: in the model (Figs. 35) and R5 = 68 k: in the experimental circuit (Figs. 68) can be explained in the following way. The inductive element in the model is an ideal device in the sense that L = const. Meanwhile the inductance of a real inductor, e.g. a coil wound on a ferrite  core has a slight dependence on the current L = L(I).Finally, we emphasize that the described autonomous oscillator is not simply a formal alter- native to the classical nonautonomous DuffingHolmes oscillator. An externally driven chaotic oscillator has a sharp and | 20 dB high peak in the power spectrum at the drive frequency f1=Z1/2S.While autonomous oscillator exhibits no peaks but essentially smoother spectra (Figs. 5, 8). This feature may have an advantage in practical applications. REFERENCES1. E. Ott. Chaos in Dynamical Systems. Cambridge: Cambridge University Press, 1993. 2. R.C. Hilborn. Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers. Oxford: Oxford University Press, 2006. 3. P. Cvitanoviü et al. Chaos: Classical and Quantum,stable version 12. Copenhagen: Niels Bohr Institute, 2008. <http://ChaosBook.org>.4. A. Namajǌnas and A. Tamaševiþius. An optoelectronic technique for estimating fractal dimensions from dynamical Poincaré maps. In IFIP Trans., v. A41. Amsterdam: Elsevier Science, 1994. pp. 289293. 5. A. Namajǌnas and A. Tamaševiþius. Simple laboratory instrumentation for measuring pointwise dimensions from chaotic time series. Rev. Sci. Instr., v. 65, pp. 30323033, 1994. 6. A. Namajǌnas, A. Tamaševiþius, A. ýenys, and A.N. Anagnostopoulos. Whitening power spectra of chaotic signals. In: Proc. NDES'99, Rønne, Denmark, 1999, pp. 137140. 7. A. Namajǌnas, A. Tamaševiþius, G. Mykolaitis, and A. ýenys. Spectra transformation of chaotic signals. Lith. J. Phys., v. 40, pp. 134139, 2000. 8. A. Namajǌnas, A. Tamaševiþius, G. Mykolaitis, and A. ýenys. Smoothing chaotic spectrum of nonautonomous oscillator. Nonlinear Phenomena in Complex Systems, v. 3, no. 2, pp. 188191, 2000. 9. B.K. Jones and G. Trefan. The Duffing oscillator: a precise electronic analog chaos demonstrator for the undergraduate laboratory. Am. J. Phys., v. 69, pp. 464469, 2001. 10. C.P. Silva and A.M. Young. High frequency anharmonic oscillator for the generation of broadband deterministic noise. U.S. Patent No. 6,127,899. October 3, 2000. 11. A.K. Kandangath. Inducing chaos in electronic circuits by resonant perturbations: Master Thesis. Tempe: Arizona State University, 2004. <http://www.ecogito.net/projects/>.12. Y. C. Lai, A. Kandangath, S. Krishnamoorthy, J.A. Gaudet, and A.P.S. de Moura. Inducing chaos by resonant perturbations: theory and experiment. Phys. Rev. Lett., v. 94, no. 21, 214101, 2005. 13. A. Kandangath, S. Krishnamoorthy, Y. C. Lai, and J.A. Gaudet. Inducing chaos in electronic circuits by resonant perturbations. IEEE Trans. Circuits and SystemsI: v. 54, pp. 11091119, 2007. 14. A. Tamaševiþius, E. Tamaševiþiǌtơ, G. Mykolaitis, and S. Bumelienơ. Stabilisation of unstable periodic orbit in chaotic DuffingHolmes oscillator by second order resonant negative feedback. Lith. J. Phys., v. 47, pp. 235239, 2007. 15. A. Tamaševiþius, G. Mykolaitis, V. Pyragas, and K. Pyragas. Delayed feedback control of periodic orbits without torsion in nonautonomous chaotic systems: theory and experiment. Phys. Rev. E, v. 76, no. 2, 026203, 2007. 16. A. Tamaševiþius, E. Tamaševiþiǌtơ, G. Mykolaitis, and S. Bumelienơ. Switching from stable to unknown unstable steady states of dynamical systems. Phys. Rev. E, v. 78, no. 2, 026205, 2008. 17. E. Tamaševiþiǌtơ, A. Tamaševiþius, G. Mykolaitis, S. Bumelienơ, and E. Lindberg. Analogue electrical circuit for simulation of the DuffingHolmes equation. Nonlinear Analysis. Modelling and Control, v. 13, pp. 241252, 2008. <http://www.lana.lt/journal/issues.php>.18. T. Matsumoto, L.O. Chua, and M. Komuro. The double scroll. IEEE Trans. Circuits and Systems,v. 32, pp. 797818, 1985. 105

CHAOTIC DUFFING TYPE OSCILLATOR WITH INERTIAL DAMPING

http://orbit.dtu.dk/ws/files/3877469/NDES2009Proceedings-102-105.pdf

CHAOTIC DUFFING TYPE OSCILLATOR WITH INERTIAL DAMPING

Abstract

Similar works

Full text

Available Versions

Online Research Database In Technology