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DYNAMICS OF A NONLINEAR ELECTRICAL OSCILLATOR

DESCRIBED BY DUFFINGS EQUATION

Ioannis M. KyprianidisPhysics Department, Aristotle University of Thessaloniki

Thessaloniki, Greecekyprianidis@physics.auth.gr

Abstract: We study the dynamics of a nonlinear electrical oscillator described by

Duffings equation. Forward and reverse period-doubling sequences are observed

following the Feigenbaums scenario.

The electrical oscillator under consideration is a nonlinear electric circuit driven

by a sinusoidal voltage source (Fig.1). The nonlinear element is a nonlinear inductor.The nonlinear inductor is an inductor with a ferromagnetic core, which can be

modeled, if an abstraction of the hysteresis phenomenon is made, by an i nonlinear

characteristic. i is the current and the magnetic flux through the inductor. This

characteristic is approximated by a constitutive relation of the form

i = a1 + a33 (1)

where a1and a3 are constants peculiar to the inductor [1].

Fig.1. The electrical nonlinear oscillator.

The nonlinear differential equation, which describes the circuit is

233 o1

2

aa Ed 1 d cost

dt RC dt C C R + + + = (2)

If we define 3 O1a Ea1

= , a = , b = and B =RC C C RC

, we take the following

Duffing equation

23

2

dxd x ax bx = Bcost

dt dt+ + + (3)

The state equation of the circuit is

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3

dxy

dt

dyy ax bx Bcosz

dtdz

dt

=

= +

=

(4)

where x = , y =L and z = t.

Y. Ueda studied chaotic dynamics for the case a = 0, b=1 and = 1 [2], while

Parlitz & Lauterborn [3,4] studied the nonlinear resonances of Eq.(3) for the

parameter values a = 1, b = 1 and = 0.2.

1. The Dynamics of the System

1a. Bifurcation Diagrams and Poincarmaps.Depending on the values of the parameters, this nonlinear circuit periodic or non-

periodic behavior. The later is characterized by extreme sensitivity to initial

conditions, so it is a chaotic behavior. Chaotic behavior has been verified by the

calculation of Lyapunov exponents [5].

Fig. 2a. Bifurcation diagram for = 0.20, a = 1.0, b = 1.0, = 0.75, in the range

40.0 B 100.0.

The bifurcation diagrams shown in Figs.2-4 is a verification of the rich nonlinear

dynamics of the system.

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Fig. 2b. Bifurcation diagram for = 0.20, a = 1.0, b = 1.0, = 0.75, in the range

94.0 B 124.0.

Fig. 3. Bifurcation diagram for = 0.20, a = 1.0, b = 1.0, = 1.0.

The transition from periodic to chaotic behavior proceeds the period-doubling

route (Feigenbaums scenario [6]). In the bifurcation diagram of Fig.4, a reverse

period-doubling sequence is observed, for B 29.0. This phenomenon is known as

antimonotonicity [7].

In the bifurcation diagram of Fig.4, we can observe three chaotic regions. The

Poincarmaps for each chaotic region are shown in the following figures 5-7.

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Fig. 4. Bifurcation diagram for = 0.180, a = 1.0, b = 1.0, and = 0.8.

Fig. 5. Poincarmap for = 0.180, a = 1.0, b = 1.0, = 0.8, and B = 23.5.

Fig. 6. Poincarmap for = 0.180, a = 1.0, b = 1.0, = 0.8, and B = 26.7.

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Fig. 7. Poincarmap for = 0.180, a = 1.0, b = 1.0, = 0.8, and B = 28.5.

1b. Phase Portraits for = 0.180, a = 1.0, b = 1.0, = 0.8.

The phase portraits for = 0.180, a = 1.0, b = 1.0, and = 0.8 are shown in the

following figures 8-16 covering the range of the bifurcation diagram of Fig.4.

Fig. 8. Phase portrait for = 0.180, a = 1.0, b = 1.0, = 0.8, and = 19.0 (period1).

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Fig. 9. Phase portrait for = 0.180, a = 1.0, b = 1.0, = 0.8, and = 21.2 (period2).

Fig. 10. Phase portrait for = 0.180, a = 1.0, b = 1.0, = 0.8, and = 22.0 (period-4).

Fig. 11. Phase portrait for = 0.180, a = 1.0, b = 1.0, = 0.8, and = 22.5 (chaos).

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Fig. 11. Phase portrait for = 0.180, a = 1.0, b = 1.0, = 0.8, and = 23.5 (chaos).

Fig. 12. Phase portrait for = 0.180, a = 1.0, b = 1.0, = 0.8, and = 26.7 (chaos).

Fig. 13. Phase portrait for = 0.180, a = 1.0, b = 1.0, = 0.8, and = 28.5 (chaos).

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Fig. 14. Phase portrait for = 0.180, a = 1.0, b = 1.0, = 0.8, and = 28.9 (chaos).

Fig. 15. Phase portrait for = 0.180, a = 1.0, b = 1.0, = 0.8, and = 29.3 (period-2).

Fig. 16. Phase portrait for = 0.180, a = 1.0, b = 1.0, = 0.8, and = 30.0 (period-1).

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1c. Waveforms for = 0.180, a = 1.0, b = 1.0, = 0.8.

The phase portraits for B = 19.0 and B = 30.0 althougth they correspond to a

period-1 steady state, they are not the familiar to us limit cycles. This happens,

because their corresponding waveforms are not sinusoidal-like, as we can see in

Figs.17 and 18.

Fig. 17. Waveform for = 0.180, a = 1.0, b = 1.0, = 0.8, and = 19.0

Fig. 18. Waveform for = 0.180, a = 1.0, b = 1.0, = 0.8, and = 30.0

The period-1 steady state of Duffing electrical oscillator is present in a wide range

of values of the amplitude of the driving voltage signal for = 0.80 and = 0.18, as

we can see in the bifurcation diagram of Fig.19.

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Fig. 19. Bifurcation diagram for = 0.180, a = 1.0, b = 1.0, and = 0.8.

In the following figures we can see the change of the form of the period-1 phase

portraits, as B is increased from 0.5. For low values of B, the phase portraits have the

well-known shape of the limit cycle, and the waveforms are almost sinusoidal

(Fig.20). For higher values of B, the shape of the period-1 phase portraits is changed,

becoming more complex (Figs.22 and 23).

Fig. 20. Phase portrait and waveform for = 0.180, a = 1.0, b = 1.0, = 0.8 and B =

0.5. The waveform is almost sinusoidal (T = /2 is the period of the voltage source

signal).

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Fig. 21. Phase portrait and waveform for = 0.180, a = 1.0, b = 1.0, = 0.8 and B =

1.0.

(a) (b)

Fig. 22. Phase portraits for = 0.180, a = 1.0, b = 1.0, = 0.8. (a) B = 2.5 and (b) B =

5.0.

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(a) (b)

Fig. 23. Phase portraits for = 0.180, a = 1.0, b = 1.0, = 0.8. (a) B = 7.5 and (b) B =

12.5.

2. Antimonotonicity

For a =1.0, b=1.0 and w=0.8 antimonotonicity is observed as the parameter takes

values in the range 0.130 <

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Fig. 25. Bifurcation diagram for = 0.245. We can observe the primary bubble, which

follows the sequence p-1 p-2 p-1.

Fig. 26. Bifurcation diagram for = 0.230. The primary bubble is present, but the

region of the p-2 mode has been increased.

Fig. 27. Bifurcation diagram for = 0.225. We can observe the sequence p-1 p-2 p-4 p-2 p-1 .

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Fig. 28. Bifurcation diagram for = 0.220. We can observe the sequence p-1 p-2

p-4 p-8 p-16 p-8 p-4 p-2 p-1 .

Fig. 29. Bifurcation diagram for = 0.219. We can observe the chaotic bubble.

Fig. 30. Bifurcation diagram for = 0.218. We can observe the evolution of the

chaotic bubble.

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Fig. 31. Bifurcation diagram for = 0.215.

Fig. 32. Bifurcation diagram for = 0.210.

Fig. 33. Bifurcation diagram for = 0.200. The central chaotic region has beenbroken.

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Fig. 34. Bifurcation diagram for = 0.190. The evolution of the central chaotic

region. A period-3 window has been formed.

Fig. 35. Bifurcation diagram for = 0.160. Antimonotonicity is still present.

Fig. 36. Bifurcation diagram for = 0.150. Antimonotonicity has been destroyed.

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3. Coexisting Attractors

There is a case in nonlinear systems, when the same system can be oscillated in

different steady states, when the initial state of the system is changed. This is the case

of coexisting attractors.For = 0.8 , = 0.20 and B = 25.0, the circuit can be appeared in two different

dynamic states depending on the initial conditions. For initial conditions x0 = 2.0 and

y0 = 1.0, the circuit is in a chaotic state, while for initial conditions x0 = 1.0 and y0 =

2.0, the circuit is in a period-3 state (Fig.37).

(a) (b)

Fig. 37. Phase portraits for = 0.8 , = 0.20 and B = 25.0.

(a) For initial conditions x0 = 2.0 and y0 = 1.0, the circuit is in a chaotic state.

(b) For initial conditions x0 = 1.0 and y0 = 2.0, the circuit is in a period-3 state.

References

[1] M. Hasler and J. Neirynck, "Nonlinear Circuits", Artech House, 1986[2] Y. Ueda, "Random Phenomena Resulting from Nonlinearity in the System

Described by Duffings Equation", Int. J. Non-Linear Mechanics, vol. 20, pp. 481-

491, 1985.

[3] U. Parlitz and W. Lauterborn, "Superstructure in the Bifurcation Set of the

Duffing Equation 3 cos( )x dx x x f t + + + =!! ! ", Phys. Lett., vol. 107A, pp. 351-

355, 1985.

[4] U. Parlitz, "Common Dynamical Features of Periodically Driven Strictly

Dissipative Oscillators", in Complexity and Chaos, Eds. N. B. Abraham, A. M.

Albano, A. Passamante, P. E. Rapp and R. Gilmore (World Scientific, 1993), pp.

219-231.

[5] A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, "Determining LyapunovExponentsfrom a Time Series", Physica D, vol. 16, pp. 285-317, 1985.

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[6] M. J. Feigenbaum, "Universal Behavior in Nonlinear Systems", Los Alamos

Science, vol. 1, pp. 4-27, 1980. (Also in the volume of collected papers,

Universality of Chaos, Ed. P. Cvitanovic, Adam Hilger, 1984) [7]

[7] I. . yprianidis, "The Phenomenon of Antimonotonicity", in Order and Chaos

in Nonlinear Dynamical Systems, vol.7 ,Eds. . Bountis, D. Hellinas and .

Gryspolakis, pp.. 135-147, G. A. Pnevmatikos editions, Athens 2002, (in Greek).