Indian Journal of Pure & Applied Physics Vol. 47, November 2009, pp. 823-827 Experimental and numerical realization of higher order autonomous Van der Pol-Duffing oscillator V Balachandran, * & G Kandiban 2 Department of Physics, A A Government Arts College, Musiri, Tiruchirappalli 62 20 2 Department of Physics, Thanthai Hans Roever College, Perambalur 62 22 *E-mail: brsbala@rediffmail.com Received 28 April 2009; revised 7 August 2009; accepted 9 September 2009 The application of the hyperchaotic dynamics will enhance the engineering system such as chaos based security of communication in information technology. Interest in such systems has been increasing. Some interesting phenomena of higher order autonomous Van der Pol-Duffing oscillator based on fifth order hyperchaotic circuit to improve secure communication have been studied in the present paper. This circuit, which is capable of realizing the behaviour of every member of the autonomous Van der Pol-Duffing family, consists of just six linear elements (resistor, inductors and capacitors) and a smooth cubic non-linearity. In this circuit, we can confirm period doubling routes to chaos and then to hyperchaos through boundary condition, when the system parameter varies. The hyperchaotic dynamics, characterized by broad band power spectrum, is presented to confirm the hyperchaotic nature of the oscillations of the circuit. This has been investigated extensively using laboratory experiments and numerical integration of the appropriate differential equations. Keywords: Chaos, Hyperchaos, Higher order autonomous Van der Pol-Duffing oscillator, Power spectrum Introduction Hyperchaos is the non-periodic behaviour of deterministic non-linear dynamical system that is highly sensitive to initial conditions with more than one positive Lyapunov Exponents. Several fourth orders autonomous oscillatory circuits characterized by broad band power spectrum have already been reported -2. In order to obtain hyperchaotic oscillations from an electronic circuit it should be at least a fifth order one in the case of common passive non-linearity such as cubic non-linear element. The increasing interest in higher order autonomous Van der Pol-Duffing oscillator is stimulated by their possible application to secure communication 2-6. Perez and Cerderia 5 have shown that messages masked by such simple chaotic systems are not always safe. Once intercepted, there is a possibility that the message can be easily extracted using wellknown non-linear signal processing techniques 9,0. However, Pecora 6 has suggested that this problem can be over come by using higher dimensional hyperchaotic systems, which have increased randomness and higher unpredictability.higher order circuit system is used to security and high quality synchronization by giving rise to more complex time signal which is apparently not vulnerable to the unmasking procedures. Therefore, interest in hyperchaos increases due to their possible application in improving secure communication. By just including an inductor and capacitor in parallel to this third order autonomous Van der Pol- Duffing oscillator circuit, we realize a very simple fifth order non-linear dynamical system. However, an important noticeable feature is that when the parallel inductor and capacitor are included, the circuit exhibits period doubling bifurcation route to chaos and then hyperchaos through boundary condition. 2 Experimental Circuit Realization of the Higher Order Autonomous Van der Pol-Duffing Oscillator The Higher order autonomous Van der Pol Duffing oscillator circuit shown in Fig. (a), is one of the most simple fifth order autonomous electronic generators of hyperchaotic signal. The circuit contains six linear elements (three capacitors, C,C 2,C 3, two inductors, L, L 2 and one resistor, R) and one active element (cubic non-linear resistor, N R ) that can be built using off-the shift op-amps with six diodes. The chaotic behaviour of the circuit was studied numerically, confirmed mathematically and realized experimentally. Reducing the variable resistance R, while keeping the other circuit parameter at constant values, one finds that the circuit admits period doubling route to chaos. For a small range of
824 INDIAN J PURE & APPL PHYS, VOL 47, NOVEMBER 2009 resistance R, it also exhibits crisis induced hyperchaos. The important requisites for hyperchaos are (i) The minimal dimension of phase space that embeds the hyperchaotic attractor should be at least five, which requires the minimum number of coupled first order ordinary differential equations to be five, and (ii) The number of terms in the coupled equations giving raise to instability should be at least two, of which one should be non-linearity function 2. Correspondingly, for hardware implementation, (i) The number of energy storage elements (inductors or capacitors) in the circuit should be at least five. (ii) The number of active elements giving rise to instability should be one or two, of which one should necessarily be a non-linear device. Applying Kirchoff s laws, the set of five coupled first order differential equations describing the circuit is obtained as: dv C = ( V V2 ) + f ( V ) (a) dt R dv2 C2 = ( V V2 ) il il dt R 2 (b) dv3 C 3 = il dt (c) dil L = V2 V3 dt (d) dil2 L 2 = V2 dt (e) While V,V 2 and V 3 are the voltages across the capacitors C, C 2, and C 3, and i and i denote the current through the inductances L and L 2, respectively, the term f (V ) representing the characteristic of the cubic non-linear resistance can be expressed mathematically: f ( V = av + bv ) 3 2. Experimental Observations: Route to hyperchoas via period-doubling phenomenon For our present experimental study, we have chosen the following typical values of the circuit in Fig. (a): C = 0 nf, C 2 = 00 nf, C 3 = 0 nf, L = 00 mh, L 2 = 43 mh. The cubic non-linear resistance characteristics are chosen the following typical value of the circuit as shown in Fig. (b): Ga = 0.76 ms, Gb = 0.4 ms and Bp =.0 V. Here the variable resistor, R is assumed to be the control parameter. By L L 2 Fig. (a) Higher order autonomous Van der Pol-Duffing oscillator circuit; (b) Circuit realization of cubic non-linear resistor N R decreasing the value of R from 5000 Ω to 0 Ω, the circuit behaviour of Fig. (a) is found to transit from a periodic limit cycle to chaos and then to hyperchaotic attractor through boundary crisis, etc 5. The projections of the attractor on the (V V 2 ) plane and (V V 3 ) plane of cathode ray oscilloscope are shown in Fig. 2 for various values of control parameter R. The distribution of power in a signal x(t) is the most commonly quantified by means of the power density spectrum or simply power spectrum. It is the magnitude-square of the Fourier transform of the signal x(t). It can detect the presence of chaos and hyperchaos when the spectrum is broad-banded. The power spectrum corresponding to the voltage V 2 (t) waveform across the capacitor C 2 for the chaotic and hyperchaotic regimes is shown in Fig. 3 which resembles broad-band spectrum noise. 3 Numerical Simulations The hyperchaotic dynamics of circuit as shown in Fig. (a), is studied by numerical integration of the normalized differential equations. For a convenient numerical analysis of the experimental system given by Eq. (), we rescale the parameters as / 2 / 2 / 2 x = br, x = ( br) ), x = ( br) ), ( ) ) ( V 3 / 2 4 br ( il 2 ( V 2 3 ( V 3 3 / 2 5 br ( il 2 x = ( ) ), ( ) ) x =, τ = t/rc 2,
BALACHANDRAN & KANDIBAN: HIGHER ORDER AUTONOMOUS VAN DER POL-DUFFING OSCILLATOR 825 Fig. 2 Experimental phase portraits in the (V V 2 ) plane of the higher order autonomous Van der Pol-Duffing oscillator circuit of Fig.(a) period limit cycle, (b) period 2 limit cycle, (c) period 4 limit cycle, (d) one band chaos, (e) two band chaos, (f) hyper chaos, (g) boundary condition, (V V 3 ) plane of the higher order autonomous Van der Pol-Duffing oscillator circuit of fig. (h) period limit cycle, (i) period 2 limit cycle, (j) period 4 limit cycle, (k) one band chaos, (l) two band chaos, (m) hyper chaos, (n) boundary condition Fig. 3 Experimental power spectrum of the signal V 2 (t) from the circuit of (a) chaotic attractor; (b) hyper chaotic attractor
826 INDIAN J PURE & APPL PHYS, VOL 47, NOVEMBER 2009 β = C 2 R 2 /L, β 2 = C 2 R 2 /L 2, α = (+ar),υ = C 2 /C, υ 2 = C 2 /C 2, υ 3 = C 2 /C 3, and then redefine τ as t. Then the normalized equations of the higher order autonomous Van der Pol-Duffing oscillator circuit [Fig. (a)] are : 3 x = ν( αx + x x2 ) (2a) x 2 = ν 2 ( x x2 x4 x5 ) (2b) x 3 = ν 3 ( x 4 ) (2c) x = β ( x 3 ) (2d) 4 2 x x 5 = β 2 ( x 2 ) (2e) The dynamics of Eq. (2) now depends upon the parameters υ, υ 2, υ 3, β, β 2, a =.02 and b = 0.55. The experimental results have been verified by computer simulation of the normalized Eq. (2) using the standard fourth order Runge-Kutta integration routine for a specific choice of system parameters employed in the laboratory experiments. That is, in the actual experimental set-up the resistor R is varied from R = 5000 Ω downward to zero. Therefore, in the Fig. 4 Numerical Phase Portraits in the (V V 2 ) plane of the higher order autonomous van der Pol-Duffing oscillator circuit of Fig. (a) period limit cycle, (b) period 2 limit cycle, (c) period 4 limit cycle, (d) one band chaos, (e) two band chaos, (f) hyper chaos, (g) boundary condition, (V V 3 ) plane of the higher order autonomous van der Pol-Duffing oscillator circuit of fig. (h) period limit cycle, (i) period 2 limit cycle, (j) period 4 limit cycle, (k) one band chaos, (l) two band chaos, (m) hyper chaos, (n) boundary condition
BALACHANDRAN & KANDIBAN: HIGHER ORDER AUTONOMOUS VAN DER POL-DUFFING OSCILLATOR 827 numerical simulation, we study the corresponding Eq. (2) for R in the range R = (5000 Ω, 0 Ω). From our numerical investigations, we find that for the value of R above 5000 Ω, limit cycle motion is obtained. When the value of R is decreased to lower than 5000 Ω, particularly in the range R = (5000 Ω, 0 Ω), the system displays a period doubling bifurcation scenario to chaotic motion and then to hyperchaotic motion through boundary crisis. These numerical results are summarized in the phase portraits as shown in Fig. 4 in (x x 2 ) and (x x 3 ) plane. The transience of the system from periodic limit cycle to chaos and hyperchaos is shown in the projection of the attractors in the (x x 2 ) and (x x 3 ) plane in Fig. 4, which are the counter part of the attractors obtained experimentally in Fig. 2 in the (V V 2 ) and (V V 3 ) plane. 4 Conclusions A new higher order autonomous Van der Pol- Duffing oscillator based on fifth order circuit has been designed and investigated. The hyperchaotic broadband power spectrum analysis is characterized with the circuit oscillations. Due to the usage of stable oscillators coupled through a non-linear resistor, this circuit has better reproducibility, higher stability and lower sensitivity. Also, this circuit can be realized easily and can be used for synchronization and ensuring secure communication applications. References Zou Y L, Zhu J, Chen G & Luo X S, Chaos, Solitons & Fractals, 25 (2005) 245. 2 Rossler O E, Phys Lett, 7 (979) 55. 3 Matsumoto T, Chua L O & Kogayashi K, IEEE Trans Circuits and systems, 33 (986) 43. 4 Saito T, IEEE Trans Circuits and Systems-I, 37 (990) 399. 5 Tamasevicius A, Cenys A, Mykolaitis G, Namajunas A & Lindberg E, Electronics Lett, 33 (997) 542. 6 Lindberg E, Proc The European conference on circuit theory and design, Budapest, 997, p 7. 7 Tamasevicius A, Proc 5 th Int Workshop on Non-linear Dynamics of Electronic Systems, Ronne, Denmark, 997, p 97. 8 Tamasevicius A & Cenys A, Chaos, Solitons & Fractals, 9 (998) 5. 9 Murali K, Tamasevicius A & Lindberg E, Proc 7 th Int Workshop on Non-linear. 0 Murali K, Tamasevicius A, Mykolaitis G, Namajunas A & Lindberg E, Hyperchaotic circuits with unstable oscillators, Nonlinear phenomena in complex systems, 2000, p7. Tamasevicius A, Cenys A, Mykolaitis G, Namajunas A & Lindberg E, Electronics Lett, 33 (997) 2025. 2 Tamasevicius A & Cenys A, Phys Rev E, 55 (997) 297. 3 Kocarev L & Parlitz U, Phys Rev Lett, 74 (995) 5028. 4 Peng J, King E J, Ding M & Yang W, Phys Rev Lett, 76 (996) 904. 5 Perez G & Cerderia H A Phys Rev Lett, 74 (995)970. 6 Pecora L, Phys World, 9 (996) 7.