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Autonomous Duffing-Holmes type chaotic oscillator/Autonominis Duffingo-Holmeso chaotinis generatorius.

Introduction

The Duffing-Holmes nonautonomous oscillator is a classical example of a nonlinear dynamical system exhibiting complex also chaotic behaviour [1-3]. It is given by the second order differential equation with an external periodic drive term:

[d.sup.2] x/[dt.sup.2] + b dx/dt - x + [x.sup.3] = a sin [omega]t. (1)

Three different techniques are used to solve the Duffing-Holmes equation and to process its solutions electronically. The first approach is a hybrid one making use of integration the equation in a digital processor and of the digital-to-analogue conversion of the digital output for its further analogue processing, analysis and display [4, 5]. The second method employs purely analogue hardware based on analogue computer design [6-9]. For example, analogue computer has been used to simulate Eq. (1) and to demonstrate the effect of scrambling chaotic signals in linear feedback shift registers [6-8]. Later analogue computer has been suggested for demonstration of chaos from Eq. (1) for the undergraduate students [9]. The third technique is based on building some specific analogue electrical circuit imitating dynamical behaviour of Eq. (1). The Young-Silva oscillator [10] and used to demonstrate the effect of resonant perturbations for inducing chaos [11] is an example. Recently the Young-Silva circuit has been essentially modified and used to test the control methods for unstable periodic orbits [12, 13] and unstable steady states [14] of dynamical systems. The modified version of the Young-Silva oscillator has been characterized both numerically and experimentally in [15].

Evidently 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 restriction the "intrinsic" electrical circuits have an attractive advantage due to their extreme simplicity and cheapness.

One may think of such analogue electrical circuits as of analogue computers. This is true from a mathematical and physical point of view in the sense that the underlying equations are either exactly the same or very similar also that the dynamical variables 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 rather different. Any analogue computer is a standard collection of the following main processing blocks: inverting RC integrators, inverting adders, inverting and non-inverting amplifiers, multipliers, and piecewise linear nonlinear units. Meanwhile the specific 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).

In this paper, we introduce, as an alternative for the nonautonomous Eq. (1), an autonomous version of the Duffing type oscillator given by

[d.sup.2] x/[dt.sup.2] - b dx/dt - x + [x.sup.3] + kz = 0,

[d.sup.2] x/[dt.sup.2] = [[omega].sub.f] (dx/dt - z) (2)

or equivalently by a set of three first order equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2a)

Here z is the third independent dynamical variable, [[omega].sub.f] is its characteristic rate, and k is the feedback coefficient. We emphasize in Eq. (2) an opposite sign of the damping term, compared to Eq. (1). The negative damping, -bdx/dt in Eq. (2), or + by in Eq. (2a) yields additional spiral instability. Also we propose a specific electrical circuit imitating solutions of Eq. (2a).

Electronic circuitry

The novel autonomous circuit is presented in Fig. 1.

[FIGURE 1 OMITTED]

The stage containing the operational amplifier OA1, the diodes, the resistors R1, R2 and R3 is exactly the same as in the nonautonomous oscillator [15]. The autonomous oscillator lacks the external periodic drive source, but includes two additional linear feedback loops. The circuit composed of the OA2 based stage and the resistor R5 introduce the positive feedback loop, specifically negative damping 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 positive damping term kz in Eq. (2).

Simulation results

The oscillator in Fig. 1 has been simulated using the ELECTRONICS WORKBENCH package (SPICE based software) and the results are shown in Figs. 2-4. The following element values have been used in the simulation: L = 19 mH, C = 470 nF, C1 = 20 nF, R = 20 [ohm], R1 = 30 k[ohm], R2 = 10 k[ohm], R3 = 30 k[ohm], R4 = 820 [ohm], R5 = 75 k[ohm], R6 = R7 = 10 k[ohm], R8 = 20 k[ohm]. The OA1 to OA3 are the LM741 type or similar operational amplifiers, the diodes are the 1N4148 type or similar general-purpose devices.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

Hardware Experiments

The autonomous oscillator has been built using the elements described in the previous section. Typical experimental results are presented in Fig. 5-7.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

Concluding remarks

We have designed and built a novel Duffing type autonomous 3rd-order chaotic oscillator. In comparison with the common nonautonomous Duffing-Holmes type oscillator [15] 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, 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 (Fig. 3-7). Some discrepancy (about 10%) between the model and the hardware prototype, namely R5 = 75 k[ohm] in the model (Fig. 3-5) and R5 = 68 k[ohm] in the experimental circuit (Fig. 6-7) can be explained in the following way. The inductive element in the model is an ideal device in the sense that its L = const. Meanwhile the inductance of a real inductor, e.g. a coil wound on a ferrite toroidal core has a slight dependence on the current through it: L = L(I).

We note that the structure of the proposed oscillator is rather different in comparison to many other 3rd-order autonomous chaotic oscillators described so far. The basic unit of the RC Wien-bridge [16-18] and LC tank [19-21] based oscillators is the 2nd-order linear unstable resonator. An additional degree of freedom required for chaos is introduced by supplementing the resonator with the 1st-order inertial nonlinear damping loop [16-21]. The same approach of building chaotic oscillators is used in higher order circuits [22-24] (some of the design principles are overviewed in a book chapter [25]). In contrast, the oscillator described in this paper contains a nonlinear unstable resonator and an inertial linear damping loop.

Received 2009 02 06

References

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[9.] Jones B. K., Trefan G. The Duffing Oscillator: A Precise Electronic Analog Chaos Demonstrator for the Undergraduate Laboratory // American Journal of Physics.--2001.--Vol. 69.--P. 464-469.

[10.] Silva C. P., Young A. M. High Frequency Anharmonic Oscillator for the Generation of Broadband Deterministic Noise.--U.S. Patent No. 6,127,899.--October 3, 2000.

[11.] Kandangath A., Krishnamoorthy S., Lai Y.-C., Gaudet J. A. Inducing Chaos in Electronic Circuits by Resonant Perturbations // IEEE Transactions on Circuits and Systems--I: Fundamental Theory and Applications.--2007.--Vol. 54.--P. 1109-1119.

[12.] Tamasevicius A., Tamaseviciute E., Mykolaitis G., Bumeliene S. Stabilisation of Unstable Periodic Orbit in Chaotic Duffing-Holmes Oscillator by Second Order Resonant Negative Feedback // Lithuanian Journal of Physics.--2007.--Vol. 47.--P. 235-239.

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[14.] Tamasevicius A., Tamaseviciute E., Mykolaitis G., Bumeliene S. Switching from Stable to Unknown Unstable Steady States of Dynamical Systems // Physical Review E.--2008.--Vol. 78, No. 2.--P. 026205.

[15.] Tamaseviciute E.,Tamasevicius A., Mykolaitis G., Bumeliene S. Lindberg E. Analogue Electrical Circuit for Simulation of the Duffing-Holmes Eequation // Nonlinear Analysis: Modelling and Control.--2008.--Vol. 13.--P. 241-252.

[16.] Namajunas A., Tamasevicius A. Modified Wien-Bridge Oscillator for Chaos // Electronics Letters.--1995.--Vol. 31, No 5.--P. 335-336.

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[23.] Tamasevicius A., Namajunas A., Cenys A. Simple 4D Chaotic Oscillator // Electronics Letters.--1996.--Vol. 32, No. 11.--P. 957-958.

[24.] Tamasevicius A., Cenys A., Mykolaitis G., Namajunas A., Lindberg E. Hyperchaotic Oscillator with Gyrators // Electronics Letters.--1997.--Vol. 33, No. 7.--P. 542-544.

[25.] Lindberg E., Murali K., Tamasevicius A.. Chaotic Oscillators--Design Principles. Chapter 1 of Chaos in Circuits and Systems.--Eds. G. Chen and T. Ueta.--World Scientific Pb.--2002.--P. 1-21.

A. Tamasevicius, S. Bumeliene

Plasma Phenomena and Chaos Laboratory, Semiconductor Physics Institute, A. Gostauto str. 11, LT-01108 Vilnius, Lithuania, phone: +370 5 2619589; e-mail: tamasev@pfi.lt

R. Kirvaitis

Department of Electronic Systems, Faculty of Electronics, Vilnius Gediminas Technical University, Naugarduko str. 41, LT-03227 Vilnius, Lithuania, phone: +370 5 2745005; e-mail: raimundas.kirvaitis@adm.vgtu.lt

G. Mykolaitis

Department of Physics, Faculty of Fundamental Sciences, Vilnius Gediminas Technical University, Sauletekio av. 11, LT-10223 Vilnius, Lithuania, phone: +370 5 2619589; e-mail: gytis@pfi.lt

E. Tamaseviciute

Department of General Physics and Spectroscopy, Faculty of Physics, Vilnius University, Sauletekio av. 9, Vilnius, LT-10222, Lithuania, phone: +370 629 19788; e-mail: elena.tamaseviciute@ff.vu.lt

E. Lindberg

DTU Elektro Department, Electronics & Signalprocessing, 348, Technical University of Denmark, DK-2800 Lyngby, Denmark, phone: +45 4525 3650; e-mail: el@elektro.dtu.dk
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Title Annotation:ELECTRONICS/ELEKTRONIKA
Author:Tamasevicius, A.; Bumeliene, S.; Kirvaitis, R.; Mykolaitis, G.; Tamaseviciute, E.; Lindberg, E.
Publication:Elektronika ir Elektrotechnika
Article Type:Report
Geographic Code:4EXLT
Date:Jun 1, 2009
Words:1894
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