Analog Implementation of Gotthans-Petrzela Oscillator with Virtual Equilibria

Jiri PETRZELA, Tomas GOTTHANS, Zdenek HRUBOS

Department of Radio Electronics, Brno University of Technology, Purky ova 118, 612 00 Brno, Czech Republic

petrzelj@feec.vutbr.cz, xgotth00@stud.feec.vutbr.cz, xhrubo00@stud.feec.vutbr.cz

Abstract. This paper presents a novel analog oscillator reported recently by the first two authors. It came from the mathematical description of the so-called labyrinth chaos but it is much more suitable for fully analog circuitry implementation. The necessity of realization of the three independent channels performing goniometrical transfer functions is replaced by its rough approximation, namely the periodical sign functions. The brief mathematical background is completed by a number of the laboratory experiments. Finally some future perspectives, possible modifications and practical usability are discussed.

Keywords Labyrinth chaos, Brownian motion, analog oscillator, dynamical system.

1. Introduction Chaos can be roughly considered as global long-time

unpredictable behavior of the nonlinear dynamical system with at least three degrees of freedom. From definition it is impossible to obtain the closed form analytic solution of the potentially chaotic system. Thus the analysis is restricted to the numerical integration process and derived procedures, like bifurcation diagram visualization or estimation of the Ljapunov exponents.

The circuitry implementation of the potentially chaotic dynamical systems attracts increasing interest in the circuit designer community. It is because it is a straightforward approach to the modeling real dynamical systems as long as the describing differential equations are known. It has been discovered and verified that the set of the differential equations known as Thomas dynamical system can produce a complex state space attractor, labyrinth chaos (LCH). The deep study of this unique dynamics published in [1], [2] reveals the possibility to utilize this system as a simulator of the Brownian motion, i.e. microscopical interaction between particles. Any state trajectory is wandering the phase space volume around the pattern of the fixed points.

Volume which is necessary to fully cover some attractor (capacity dimension) is given by dissipation represented by one internal system parameter.

2. Numerical Analysis Thomas system belongs to the class of the systems

with cyclically symmetrical vector fields and smooth, in fact goniometrical, nonlinearity. From the viewpoint of practical verification it is very difficult to realize such oscillator using commercially available devices. It has been discovered that the nature of global behavior is preserved if the rough approximations of the goniometrical functions are used, in detail the sign function equivalents. To be more specific the new dynamical system under inspection (GP) is

bxazzbxazzbzayybzayybyaxxbyaxx

cossignsinsigncossignsinsigncossignsinsign

(1)

where dots denote time derivatives of the state variables. Each of these systems has multiple equilibria, i.e. points in the state space where associated vector field disappears. Note that dissipation factors a are bounded to a certain state variable and indirectly denote the size of the state space attractors in the corresponding axis.

Fig. 1. Numerical integration of GP with the dissipation factors

ax=2, ay=2, az=0.01 and bx=by=bz=100.

978-1-61284-324-7/11/$26.00 ©2011 IEEE

Fig. 2. Numerical integration of GP with the dissipation factors

ax=2, ay=0.01, az=0.01 and bx=by=bz=100.

Fig. 3. Numerical integration of GP with the dissipation factors

ax=0.01, ay=0.01, az=0.01 and bx=by=bz=100.

Fig. 4. Numerical integration of GP using nonlinear component

value bx=10, by=10, bz=10 and ax=ay=az=1.

This fact is documented in Fig. 1, Fig. 2 and Fig. 3 in which dissipation constants associated with the state variables are marked as ax, ay and az. For the numerical integration process Mathcad and build-in fourth-order Runge-Kutta method has been utilized with initial conditions [0,1 0 0]T, final time 500 and time step 0,05. Similar rewriting of state equations (1) can be used also in the case of parameter b, which defines a distance between the virtual fixed points. The influence of this constant on the global attractor is visible in Fig. 4, Fig. 5 and Fig. 6.

Fig. 5. Numerical integration of GP using nonlinear component

value bx=100, by=10, bz=10 and ax=ay=az=1.

Fig. 6. Numerical integration of GP using nonlinear component

value bx=100, by=100, bz=10 and ax=ay=az=1.

3. Circuitry Implementation As mentioned above the main advantage of (1) is its

relative easy implementation as analog electronic circuit. The desired transfer functions, i.e. variation of positive and negative output voltage with the change of input variable can be done using a parallel connection of the comparators, as it is shown in Fig. 7. For comparators and inverting integrators the integrated circuit TL084 has been utilized. As current summer and current-to-voltage converter second generation positive current conveyor with output voltage follower AD844 is employed. This four-port building block is ideally described by the equations

ZOXZYXY VVIIVVI 0 . (2)

The breakpoints of the nonlinear transfer function are uniquely determined by voltage dividers with threshold levels given by the resistors Rd1=Rd12=100k , Rd2=Rd3= Rd4=Rd5=Rd6=Rd7=Rd8=Rd9=Rd10=Rd11=2k and symmetrical voltage supply Vcc=-Vee=15V. The output current of the individual comparator is set by the resistors Rai=15k to 1mA. The dissipation factors ax, ay and az can be adjusted independently by the resistors Rk changing from 10k up to 100k for non-trivial state space attractor.

Fig. 7. Circuitry implementation of GP oscillator.

Fig. 8. Pspice circuit simulation using dissipation ax=ay=az=2

and nonlinear constant bx=by=bz=10, plane projections.

Fig. 9. Pspice circuit simulation using dissipation ax=ay=az=0.1

and nonlinear constant bx=by=bz=10, plane projections.

Fig. 10. Mathcad and Pspice verification using ax=ay=az=2

and nonlinear constant bx=by=bz=10, time domain.

4. Experimental Verification Due to the complexity of the final oscillator structure

it has been verified by means of the Orcad Pspice circuit simulator, see Fig. 8 and Fig. 9. It turns out that oscillator, especially its time domain simulation setup, is extremely sensitive to hysteresis effect of the comparators. These building blocks work more ideally in the lower frequency range. Thus the integration constant was chosen as R=10k and C=100nF. The initial conditions should be put into circuit via IC1 component, final time is 100ms.

5. Modifications and Perspectives The main drawback of the proposed oscillator is in

the simultaneous change of the individual parameters b. In the analog circuit this can be done be changing of the resistors in divider, it means redefining the threshold levels for each comparator. To remove this, the nonlinear two-port with digital signal processing is indeed a good idea. A pre-study of such possibility has been performed using STM32F107 microprocessor, fast 10 bits analog-to-digital TLC2574 and digital-to-analog DAC8734 converters communicating via SPI bus. The KEIL uVision V3.90 has been used to create final program. Another interesting modification can be the automatic control of the nonlinear constants b using digital potentiometers instead of the fixed resistors Rd1 and Rd12. Problem associated with this upgrade is the maximum allowed voltage across pins of such potentiometer.

From the viewpoint of future research topics the still unanswered question is modeling of the noise in the circuis using some deterministic dynamical system. The authors believe that if this is possible the describing equations will be similar to GP oscillator. The non-existence of stochastic processes is the next inevitable step leading to the huge scientific breakthrough.

Much more general details about modeling nonlinear dynamics using electronic circuit can be found in paper [3], detailed approach to integrator-based realization is in [4] and interesting view on chaos evolution in the real circuits is provided in [5].

6. Conclusion In this contribution the novel analog oscillator

suitable for simulation random-like processes in nature is presented and numerically examined. The particular gallery of the chaotic attractors generated by synthesized circuit is also provided. A very good correspondence between theoretical expectations and circuit simulations has been achieved, see Fig. 10 for visual comparison of the state variable x(t) and y(t) in time domain.

Acknowledgements The research is a part of the COST action IC 0803,

which is financially supported by the Czech Ministry of Education under grant no. OC09016. The authors would also like to thank Grant Agency of the Czech Republic for their support through project number 102/09/P217 and number 102/08/H027. The research leading to these results has received funding from the European Community Seventh Framework Programme (FP7/2007-2013) under grant agreement number 230126. Research described in the paper was also supported by the Czech Ministry of Education under research program MSM 0021630513.

References [1] SPROTT, J. C., CHLOUVERAKIS, K. E. Labyrinth chaos.

International Journal of Bifurcation and Chaos, 2007, vol. 17, no. 6, p. 2097 – 2108.

[2] SPROTT, J. C. Elegant chaos: algebraically simple chaotic flows. World Scientific Publishing, 2010, 285 pages. ISBN 978-981-283-881-0.

[3] ITOH, M. Synthesis of electronic circuits for simulating nonlinear dynamics. International Journal of Bifurcation and Chaos, 2001, vol. 11, no. 3, p. 605 – 653.

[4] PETRZELA, J. Modeling of the Strange Behavior in the Selected Nonlinear Dynamical Systems, Part I: Oscillators. Brno: Vutium Press, 2008. 34 pages. ISBN 978-80-214-3787-6.

[5] ŠPÁNY, V., GALAJDA, P., GUZAN, M., PIVKA, L., OLEJÁR, M. Chua´s singularities: great miracle in circuit theory. International Journal of Bifurcation and Chaos, 2010, vol. 20, no. 10, p. 2993 – 3006.