A Chaotic RC Oscillator

Transcription

1 A haotic Oscillato Konstantine Giannakopoulos and Theodoe eliyannis epatment of Physics Univesity of Patas 6500 Patas, Geece Astact An autonomous chaotic oscillato is descied. It is using the eliyannis single amplifie iquad, a negative impedance convete, a simulated esonant cicuit and one diode employed as the nonlinea device. The chaotic oscillato is descied y a set of fou diffeential equations. The chaotic oscillations have een chaacteized using the Lyapunov exponents and the Lyapunov dimension. esults fom numeical simulation of the poposed oscillato fully coincide with expeimental esults. I. INTOUTION uing the last two decades, a lage nume of electonic cicuits exhiiting chaotic ehaviou have een poposed in the liteatue [-3]. A nume of chaotic oscillatos ae ased on the Wien-idge oscillato and ae thid-ode systems [4-6]. Seveal othe oscillatos ae ased on two esonant cicuits coupled y means of a nonlinea device such as a diode [7]. Such systems ae fouth-ode systems and can exhiit hypechaotic ehaviou chaacteized y two positive Lyapunov exponents. The inteest in the chaotic and hypechaotic oscillatos is inceasing ecause of thei possile applications in secue communications such as analogue cyptogaphy [,3,8-0]. In this pape we pesent the esults of studying a chaotic oscillato [-3] using the eliyannis single amplifie iquad (SAB) [4,5] and a L esonant cicuit. The inducto of the esonant cicuit is ealized actively with the well-known Antoniou gyato cicuit [6]. The cicuit (Fig. ) consists of a L esonant cicuit and the SAB coupled y means of a diode. The negative esistance, implemented using a negative impedance convete, intoduces negative losses in ode to peseve the oscillations. The tansfe function of the SAB (Fig. ) is the following: V V in whee k= a /. s + + k + s k s + out = () In ode to use this SAB as an oscillato we have to set the input teminal to gound and select such a value fo k that the tem in paenthesis in the denominato of () ecomes nonpositive. This can e easily achieved y selecting a suitale value of a. The inducto L has een ealized in the final cicuit (Fig.3) y means of the Antoniou gyato cicuit and its value is given y L = () L L II. IUIT ESIPTION Figue. The chaotic oscillato cicuit. Figue. eliyannis SAB.

2 Then, the aove set of diffeential equations (3) takes the fom of the following fouth-ode system: x = αx z c( x y ) H ( x y ) g y = khy e h + g + w khc( x y ) H ( x y ) k z = x w = khy e( h + g) w khc( x y ) H ( x y ) (6) whee H(u) is the Heaviside function, that is Figue 3. The final cicuit. The negative esisto has een ealized y a negative impedance convete cicuit [5]. III. NONLINEA ANALYSIS The chaotic oscillato can e descied y a set of fou diffeential equations. These equations deived easily fom the cicuit (Fig. 3) ae as follows: dil dv V V = L + il + + i = 0 dt dt V dv dv3 dv dvo i = + + dt dt dt dt a dv dv3 V3 V V3 = Vo = + dt dt a whee i is the cuent in the diode. Assuming a piecewise linea model fo the diode we have o (3) 0, V V < 0 i = V V V (4), V V 0 whee and V ae the diode fowad iased esistance and the diode voltage dop espectively. In ode to modify the aove equations in a way convenient fo numeical analysis we intoduce the following notation: V x = t θ = τ c = V y = du u = dθ e = il z = L = g = V3 w = α = h = τ = L = a k = (5) IV. 0, u < 0 H ( u) = (7), u 0 ONFIMATION OF HAOS The staility popeties of the aove continuous autonomous system have een studied y calculating the Lyapunov exponents (LEs) and the Lyapunov dimension [3,7-9]. The LEs ae defined y λi = lim lnσ i ( t) (8) t t } n i ( t i= if the limit exists. { σ ) ae the eigenvalues of the system. Let λ λ n e the LEs of a dynamic system, and j e the lagest intege such that λ + +λ j 0. The Lyapunov dimension L, as defined y Kaplan and Yoke [0], is as follows: L j+ j λi λ λ j i= = j + = j + λ λ The LEs measue the ate at which neay tajectoies convege o divege. Thee ae as many LEs as thee ae dimensions of the system in the state space, ut the lagest exponent is the most impotant. If the lagest LE is negative, then the tajectoies convege in time and the system is insensitive to initial conditions. If the lagest LE is positive, then the distance etween neay tajectoies gows exponentially in time and the system exhiits dependence on initial conditions. Fo an attacto, contaction must outweigh expansion so the sum of all LEs has to e negative. Fo each tajectoy, which does not teminate at a fixed point, at least one LE vanishes []. If the attacto is an equiliium point, all the LEs ae negative and L =0. If the attacto is a limit cycle, thee exists one zeo LE and the othe LEs ae negative, while L =. In case thee is a two-tous, it has anothe LE zeo while the othe LEs ae negative and L =. If it is a K- tous, it has K zeo LEs and the othe LEs ae negative and L =K, whee K is an intege. In case it is chaotic, it has one j+ (9)

3 positive LE and the Lyapunov dimension is nonintege. If it has two positive LEs, it is hypechaotic. The LEs and Lyapunov dimension of the set of equations (6) have een calculated fo vaious values of the paamete α and fo a lage nume of iteations y means of Lyapunov Exponents Toolox (LET) [], which uns on MATLAB. They ae shown in Figs. 4 to 7. Figue 7. Lyapunov dimension. It can e seen that the system exhiits a chaotic ehaviou in the ange of α fom 0.5 to 0.5 and especially fo α=0.0. The fouth LE (Fig. 6) is deeply negative. Figue 4. Lagest Lyapunov exponent. V. SIMULATION AN EXPEIMENTAL ESULTS We have simulated the cicuit in Fig. 3 as well as uilt it in the laoatoy. Fo the L esonant cicuit we have selected a fequency of.5khz implemented with =6.9nF, L =80.8nF and L =kω (coesponding to L=80.8mH). The opamps UA74 and the diode N448 with V =0.65V and =5Ω at 5mA wee used. The negative esistance was implemented with =5.6kΩ and a POT of 0kΩ. Fo the design of the SAB [4,5] we assume = and = /. Then we take = = πf o πf o k = a (0) Figue 5. Thee lage Lyapunov exponents. Optimizing the aove SAB [5] we assume = f o. Selecting = =0nF and a fequency f o =800Hz we take =.99kΩ and =99kΩ. Fo the value of k we select k=0.0 implemented with =00.kΩ and a y means of a POT of 5kΩ. Figue 6. Fouth Lyapunov exponent. Figue 8. Phase potait (i L against V ).

4 Figue 9. Phase potait (V 3 against V ). Figue. Phase potait (V 3 against i L). Thee is vey good ageement etween the PSPIE and MATLAB simulation and the expeimental esults. Figue 0. Phase potait (V 3 against V ). Figue 3. Time wavefoms of chaotic vaiales V (t) and V (t). Figue. Phase potait (i L against V ). The oscillato has een simulated y means of MATLAB setting k=0.0, α=0.0, =0.86, c=68.4, e= and g=h=6.. The coectness of ou simulations has een confimed also y means of PSPIE and ynamics Solve [3]. esults ae depicted in figues 8 to 6. Figue 4. Time wavefoms of chaotic vaiales i L(t) and V 3(t).

5 Figue 5. 3 Phase potait (i L against V and V ). Figue 6. 3 Phase potait (i L against V and V 3). VI. ONLUSIONS An autonomous chaotic oscillato using the eliyannis SAB has een pesented. It has een shown that it exhiits chaotic ehaviou chaacteized y a positive Lyapunov exponent and lage nonintege Lyapunov dimension. The oscillato has een implemented in the laoatoy as an active cicuit and was simulated y means of PSPIE and MATLAB. EFEENES [] Special issue on chaos in nonlinea electonic cicuits - pat A: tutoials and eviews, IEEE Tans. icuits Syst.-I, vol. 40, pp , 993. [] Special issue on chaos in nonlinea electonic cicuits - pat B: ifucations and chaos, IEEE Tans. icuits Syst.-I, vol. 40, pp , 993. [3] Special issue on chaos in nonlinea electonic cicuits - pat : applications, IEEE Tans. icuits Syst.-II, vol. 40, pp , 993. [4] A. Namajūnas and A. Tamaševičius, Modified Wien-idge oscillato fo chaos, Electon. Lett., vol. 3, pp , 995. [5] A. S. Elwakil and A. M. Soliman, A family of Wien-type oscillatos modified fo chaos, Int. J. ic. Theo. Appl., vol. 5, pp , 997. [6] A. Tamaševičius, G. Mykolaitis and A. Čenys, Wien-idge chaotic cicuit with compaato, Electon. Lett., vol. 34, pp , 998. [7] A. Tamaševičius, A. Čenys, G. Mykolaitis, A. Namajūnas and E. Lindeg, Hypechaotic oscillato with gyatos, Electon. Lett., vol. 33, pp , 997. [8] M. Hasle, Synchonization pinciples and applications, IEEE icuits and Systems Tutoials, pp , 994. [9] Special issue on chaos synchonization, contol, and applications, IEEE Tans. icuits Syst.-I, vol. 44, pp , 997. [0] Special issue on communications, infomation pocessing and contol using chaos, Int. J. Theo. Appl., vol. 7, pp , 999. [] K. Giannakopoulos, T. eliyannis and J. Hadjidemetiou, Means fo detecting chaos and hypechaos in nonlinea electonic cicuits, Poc. 4 th Int. onf. SP, Santoini, Geece, pp , 00. [] K. Giannakopoulos, Study and implementation of nonlinea electonic cicuits: chaotic oscillatos, applications, jump phenomenon, Ph Thesis, Univ. of Patas, 003. [3] K. Giannakopoulos and T. eliyannis, A compaison of five methods fo studying a hypechaotic cicuit, Int. J. Electon., vol. 9, pp , 005. [4] T. eliyannis, High-Q facto cicuit with educed sensitivity, Electon. Lett., vol. 4, pp , 968. [5] T. eliyannis, Y. Sun and J. K. Fidle, ontinuous-time Active Filte esign, Pess LL, 999. [6] A. Antoniou, ealization of gyatos using opeational amplifies and thei use in -active-netwok synthesis, Poc. IEE, vol. 6, pp , 969. [7] T. S. Pake and L. O. hua, haos: a tutoial fo enginees, Poc. IEEE, vol. 75, pp , 987. [8] A. A. Tsonis, haos Fom Theoy to Applications, Plenum Pess, New Yok, 99. [9].. Hilon, haos and Nonlinea ynamics: An Intoduction fo Scientists and Enginees, Oxfod Univ. Pess, New Yok, 000. [0] P. Fedeickson, J. L. Kaplan, E.. Yoke and J. A. Yoke, The Liapunov dimension of stange attactos, J. iff. Eq., vol. 49, pp , 983. [] H. Haken, At least one Lyapunov exponent vanishes if the tajectoy of an attacto does not contain a fixed point, Phys. Lett., vol. 94A, pp. 7-7, 983. [] S. W. K. Siu, Lyapunov Exponents Toolox, ity Univ. of Hong Kong, 998. [3] J. M. Aguiegaiia, ynamics Solve ve..74, Univ of the Basque ounty, 008.