Experimental and Simulated Chaotic RLD Circuit Analysis with the Use of Lorenz Maps

Transcription

1 Experimental and Simulated Chaotic RLD Circuit Analysis with the Use of Lorenz Maps N.A. Gerodimos, P.A. Daltzis, M.P. Hanias, H.E. Nistazakis, and G.S. Tombras Abstract In this work, the method Ed. Lorenz used to analyze the dynamics of his strange attractor is applied in both an experimental and a simulation setup of a weakly-chaotic resistor-inductor diode (RLD) circuit. For that, the time-series generated by the simulation and the time-series captured from a digital oscilloscope connected to the real circuit implementation, are first investigated for the presence of chaos and then analyzed in order to collect the local maxima using TISEAN software. The present analysis shows that one-dimensional maps can be generated in both cases being also unimodal, verifying thus the period-doubling route to chaos of the RLD circuit as it has been reported in a previous work. 1 Introduction Lorenz Map is a method of obtaining a one-dimensional map of the form z nc1 D f.z n /, like the logistic map, from a time-series of a dynamical system in chaotic state, by collecting the successive local maxima of this time time-series, label them as z n where z n is the n-th local maximum and plot z nc1 versus z n for various n. By constructing the Lorenz map of a chaotic system, one can visualize not only its chaotic state, but also local maxima predictability. Hence, since a one-dimensional map of the system is constructed by using only one time-series generated by that system, discarding the rest, the selected time-series must be able to depict the dynamics of that system. For chaotic systems, this suggests that the systems strange attractor must be slightly more than two-dimensional or in other words, no matter how many degrees of freedom the system has, only two or three should be really active, [1]. N.A. Gerodimos ( ) P.A. Daltzis M.P. Hanias H.E. Nistazakis G.S. Tombras Faculty of Physics, Department of Electronics, Computers, Telecommunications and Control, National and Kapodistrian University of Athens, Athens GR 15784, Greece gerodn@phys.uoa.gr S.G. Stavrinides et al. (eds.), Chaos and Complex Systems, DOI 1.17/ , Springer-Verlag Berlin Heidelberg

2 44 N.A. Gerodimos et al. From Feigenbaum s work [2], it is known that quadratic maps like the logistic map and even quadratic-like maps like the sine map, show a period-doubling route to chaos following Feigenbaum constant. Since in [3, 4], it has been verified that resistor-inductor diode (RLD) circuits can have near two-dimensional strange attractors and that chaotic state is reached through period-doubling bifurcations following Feigenbaum constant, in this work Lorenz Map method is applied to two RLD circuits, in order to construct their one-dimensional maps and investigate if RLD circuits can also have a quadratic-like one-dimensional map. For that, an experimental and a simulation setup of a weakly-chaotic RLD circuit are used and the time-series generated from the simulation and the one captured from a digital oscilloscope connected to the real circuit implementation, are at first investigated for the presence of chaos and then analyzed in order to collect the local maxima using the TISEAN software [5]. 2 Multisim and Experimental RLD Circuit Implementations The schematic diagrams of both the RLD circuits used are shown in Fig. 1. Both consist of a series connection of an ac-voltage source, a linear resistor R, a linear inductor L and a diode D, which is the only nonlinear circuit element. The input signal in both circuits is a sinusoidal voltage u s with frequency f s near to half the resonance frequency f r,[4], of each circuit respectively as a rule of thumb, while the voltage u R across resistor R is considered as the circuit output signal. As it is explained in [6], chaotic operation in an RLD circuit may eventually result due to the un-recombined electrons and holes that cross the forward-biased pn junction, which as the diode changes state-of-bias, diffuse back to their origin. Since input signal amplitude controls the amount of un-recombined electrons and holes that exists when the diode changes state-of-bias, thus the reverse recovery time t r, input signal amplitude is chosen to be the chaos control parameter. Element values chosen for Multisim circuit of Fig. 1 are L D 28:5 mh, R D 1 and the diode type is 1N45GP. The resonance frequency f r of the circuit, by using the diode s junction capacitance value C j taken from the SPICE model, is found to be approximately equal to f r D 21 khz, [4]. Also, for this implementation, an ideal inductor is used with no resistance or capacitance, nor energy dissipationradiation. This approach helps in controlling non-linearities that are not generated due to diode operation. For the experimental circuit, the element values of Fig. 1 are L D 9:5 mh, R D 33 and the diode type is 1N45. For this circuit, the resonance frequency f r, by using the diode s junction capacitance value C j taken from the diode datasheet, is found to be approximately equal to f r D 23 khz, [4]. Here, the real-life inductor used has all the afore-mentioned inductor non-linearities, resulting in more complicated state equations of the circuit.

3 Experimental and Simulated Chaotic RLD Circuit Analysis with the Use of Fig. 1 Multisim (left panel) and experimental (right panel) RLDcircuit Voltage (V).1 Voltage (V) Time(Sec) Time(Sec) Fig. 2 Multisim circuit output voltage for u in 4 mv pp (periodic, left panel) andv in 926 mv pp (chaotic, right panel) 2.1 Presentation of Both Circuits Periodic and Chaotic Oscillations In Figs. 2 and 3, periodic and chaotic oscillations of both circuits are presented. Chaotic operation in Multisim circuit is achieved for input signal amplitude V pp D 926 mv. Also, correlation dimension for that RLD circuit is found to be equal to D 2:3. For the real circuit implementation, chaotic operation is achieved for input signal amplitude V pp D 4:74 V. Correlation dimension this time, is found to be equal to D 2:5. Time series analysis that reveals the presence of chaos in both circuits was done using TISEAN [5] according to [3] and[7]. Both circuits input frequency f s is set to 1 khz which is, as stated above, approximately :5 f r. 3 Lorenz Map of Both RLD Circuits For chaotic time-series presented above in Figs. 2 and 3, Lorenz maps are generated and presented in Fig. 4.ForthataTISEAN[5] extrema function variation is used. Extrema does a quadratic interpolation in order to get a better estimate of the

4 46 N.A. Gerodimos et al Voltage (v) Voltage (v) Time(msec) Time(msec) Fig. 3 Real circuit output voltage for u in 2:3 V pp (periodic, left panel) andv in 4:74 V pp (chaotic, right panel) z(n+1) in Volts Mutlisim RLD circuit z(n+1) in Volts RLD circuit Fig. 4 Lorenz maps for Multisim (left panel) and real (right panel)rldcircuit extremum, but since both timeseries are chaotic it is decided to rule out quadratic interpolation functionality of extrema. Moreover in the real circuit, since input frequency is 1 khz, and sampling frequency 1 MHz thus 1 samples per cycle, local maxima which are a few samples away, are ruled out. As shown in Fig. 4, all points of both Lorenz maps lie on an almost onedimensional unimodal curve, for which an approximate relationship of the form z nc1 D f.z n / with f being unimodal can be found. In the real RLD circuit in Fig. 4, no points are near the f.x/ D x curve as in Multisim circuit. This can be due to low number of iterations or additional non-linearities, mainly due to the inductor. Nevertheless, the fact that chaotic orbits do not cross near this area, is an indication that attracting and repelling fixed points of the phase space, are aligned is such way that such crossings are not favored. Fixed points of one-dimensional maps, satisfy z D f.z /, analogously for unimodal curves of Fig. 4, fixed points can be located at the intersection with f.x/ D x. Such fixed points represent closed orbits that if exist further investigation is needed in order to reveal if the chaotic circuit strange attractor is just a stable limit cycle that will reveal after adequate number of integrations.

5 Experimental and Simulated Chaotic RLD Circuit Analysis with the Use of Mutlisim circuit.25 Real circuit z(n+2) in Volts q z(n+2) in Volts q.2 p.5 p Fig. 5 Second-iterate maps for Multisim (left panel) and real (right panel) RLDcircuit The stability of a fixed point is decided by finding the slope at z D f.z /.If jœj <1 fixed point z is stable else if jœj >1 fixed point z is unstable were œ D f.z /. Here, a rough calculation of slope in both curves shows, jœj 2 for Multisin circuit and jœj 1:43 for the real circuit implementation. The fact that both slopes are greater than 1 in magnitude, thus fixed point are unstable, indicates that even if closed orbits exist, they are unstable ruling out the possibility that RLD circuits examined here are chaotic for a while and then settle down to a stable limit cycle in other words experiencing transient chaos [1, 8]. Continuing with the application of unimodal maps properties on RLD circuit, at the critical slope f.z / D 1 one-dimensional chaotic maps undergo a flip bifurcation which is often associated with period-doubling [1]. In order for a flip bifurcation to spawn a 2-cycle, thus period doubling, in one dimensional maps, 2 points p and q should exist such that f.q/ D pandf.p/ D q, or equivalently f.f.p// D p, hence points p, q are fixed points of second iterate map z nc2 D f.z n /. For the RLD circuits examined here, the existence of such points is illustrated in second iterate Lorenz maps of Fig. 5. Generation of 4-cycle thus a second period doubling would result in 4 points such that p 1 D f.q 1 /,q 1 D f.p 2 /,p 2 D f.q 2 / and q 2 D f.p 1 /. By substituting, p 1 D f.f.f.f.p 1 //// or in other words 4 fixed points in the fourth iterate map should exist. For the RLD circuits examined here, the existence of such points is illustrated in fourth iterate Lorenz maps of Fig Lorenz Map of RLD Circuits with Higher Correlation Dimension As stated above, in order for a system to have a near one-dimensional Lorenz map, the system s strange attractor has to be near two-dimensional. For the RLD circuits studied so far this was achieved by setting the input signal frequency f s to 1 khz,

6 48 N.A. Gerodimos et al. Fig. 6 Fourth-iterate maps for Multisim (left panel) and real (right panel) RLDcircuit Fig. 7 Lorenz maps for Multisim (left panel) and real (right panel)rldcircuit about half the RLC resonance frequencies, which lead to correlation dimension in both cases close to the value 2. By re-adjusting input frequencies of RLD circuits to 21 khz for the Multisim circuit and 23 khz for the real circuit, both circuits are operated near to their resonance frequencies. This causes non-linearities that where suppressed to emerge, giving higher correlation dimensions in both cases. This time, chaotic operation is achieved for input signal amplitude V pp D 2:58 V in Multisim circuit and for V pp D 5:6 V in the real circuit implementation. Correlation dimension in both circuits was calculated using TISEAN, [5], and was found v D 2:25 for Multisim circuit and v D 2:26 for the real circuit implementation. For the chaotic time series captured for near resonance input frequencies, Lorentz maps are generated and presented in Fig. 7. As shown in Fig. 7, this time both Lorenz maps are not an almost onedimensional curve, and thus an approximate relationship of the form z nc1 D f.z n / can t be found. In this case RLD chaotic circuit complexity can be depicted by Lorenz map but n-th local maximum can t predict (n C 1)-th local maximum [1]. Since period-doubling route to chaos following Feigenbaum constant, [2], is also the case here [4], the existence of a unimodal Lorenz map is a sufficient but not necessary condition for a period doubling route to chaos.

7 Experimental and Simulated Chaotic RLD Circuit Analysis with the Use of Conclusion In this work, we have shown that the complexity of a simple RLD chaotic circuit can be captured by Lorenz map. The applicability of unimodal maps properties in a RLD circuit has been examined and a plausible explanation has been given to why stable limit cycles do not occur for the RLD circuits studied. Acknowledgements This research was partially funded by the University of Athens Special Account of Research Grants no References 1. Strogatz, S.H.: Nonlinear Dynamics and Chaos. Westview Press, New York (21) 2. Feigenbaum, M.J.: Quantitative universality for a class of non-linear transformations. Ann. N. Y. Acad. Sci. 357, (198) 3. Hanias, M.P., Giannaris, G., Spyridakis, A., Rigas, A.: Time series analysis in chaotic diode resonator circuit. Chaos Solitons Fract. 27, (26) 4. Hanias, M.P., Avgerinos, Z., Tombras, G.S.: Period doubling, Feigenbaum constant and time series prediction in an experimental chaotic RLD circuit. Chaos Solitons Fract. 4, (29) 5. Hegger, R., Kantz, H., Schreiber, T.: Practical implementation of nonlinear time series methods: the TISEAN package. CHAOS 9, 413 (1999) 6. Mariz de Moraes, R., Anlage, S.M.: Unified model, and novel reverse recovery nonlinearities, of the driven diode resonator. Phys. Rev. E 68, art. no (23) 7. Kantz, H., Schreiber, T.: Non Linear Timeseries Analysis, 2nd edn. Cambridge University Press, London (24) 8. Lai, Y.-C., Tel, T.: Transient Chaos. Springer, Berlin (211)