regular and chaotic behaviors of plasma oscillations...

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Available at: http://www.ictp.it/~pub- off IC/2005/041 United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS REGULAR AND CHAOTIC BEHAVIORS OF PLASMA OSCILLATIONS MODELED BY A MODIFIED DUFFING EQUATION H.G. Enjieu Kadji, J.B. Chabi Orou 1 Institut de Math´ ematiques et de Sciences Physiques, B.P. 613, Porto-Novo, B´ enin and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy and P. Woafo 2 Laboratoire de M´ ecanique, Facult´ e des Sciences, Universit´ e de Yaound´ e I, B.P. 812, Yaound´ e, Cameroun and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. MIRAMARE – TRIESTE July 2005 1 Senior Associate of ICTP. Corresponding author: [email protected] 2 Regular Associate of ICTP.

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Page 1: REGULAR AND CHAOTIC BEHAVIORS OF PLASMA OSCILLATIONS …users.ictp.it/~pub_off/preprints-sources/2005/IC2005041P.pdf · 2005-11-07 · Abstract The regular and chaotic behavior of

Available at: http://www.ictp.it/~pub−off IC/2005/041

United Nations Educational Scientific and Cultural Organizationand

International Atomic Energy Agency

THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

REGULAR AND CHAOTIC BEHAVIORS OF PLASMA OSCILLATIONS

MODELED BY A MODIFIED DUFFING EQUATION

H.G. Enjieu Kadji, J.B. Chabi Orou1

Institut de Mathematiques et de Sciences Physiques,

B.P. 613, Porto-Novo, Benin

and

The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy

and

P. Woafo2

Laboratoire de Mecanique, Faculte des Sciences,

Universite de Yaounde I, B.P. 812, Yaounde, Cameroun

and

The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.

MIRAMARE – TRIESTE

July 2005

1Senior Associate of ICTP. Corresponding author: [email protected] Associate of ICTP.

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Abstract

The regular and chaotic behavior of plasma oscillations governed by a modified Duffing

equation is studied. The plasma oscillations are described by a nonlinear differential equation

of the form x + ω20x + βx2 + αx3 = 0 which is similar to a Duffing equation. By focusing on

the quadratic term, which is mainly the term modifying the Duffing equation, the harmonic

balance method and the fourth order Runge-Kutta algorithm are used to derive regular and

chaotic motions respectively. A strong chaotic behavior exhibited by the system in that event

when the system is subjected to an external periodic forcing oscillation is reported as β varies.

1

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1 Introduction

In the past decades, a few works had been performed to study the nonlinear plasma instability

effects. In some of these papers published in mid 70, the approach was to point out the frequency

shifts, the asymmetry of the amplitude, the resonance response spines and the effect of an

external forcing oscillation near the fundamental or its sub harmonics. In this paper, we intend

to tackle the regular and chaotic behaviors of the system submitted to an external forcing

oscillation with chosen frequencies. The nonlinear differential equation of the form

x + ω20x + βx2 + αx3 = E cos Ωt (1)

comes from a suggestion which many authors confirmed later by experimental studies [1]. In

that equation, ω0 and Ω are respectively the internal and the external frequencies. E stands for

the amplitude of the external excitation while α and β are the quadradic and cubic nonlinearities

parameters. It should be noticed that α and β are called the nonlinear saturation coefficient

in ion-sound instability in a plasma. The above equation has been derived from a standard ion

equation of continuity in which the source term consists of an ion source that can be caused

by the presence of large amplitude fluctuations in the plasma [1]. Equation (1) describes for

example in plasma physic the electron beam surfaces, the Tonks-Datter resonances of mercury

vapor and low frequency ion sound waves [2]. Keen and Fletcher in the early seventies [1] and

Hsuan [3] from thermodynamics arguments had shown that the source term can be expressed

such as the last three terms in the above nonlinear differential equation. The cubic nonlinearity

(so-called Duffing nonlinearity) has been studied in several occasions [4-9] and it will be worth

to look at the contribution of the quadratic term in these oscillations of the plasma which is

indeed essentially a nonlinear medium. Interests according to plasma oscillations are due to

their potential applications. Indeed, radio-wave propagation in the ionosphere was an early

stimulus for the development of the theory of plasma. Nowadays, plasma processing is viewed

as a critical technology in a large number of industries, and semiconductor device fabrication

for computers is may be the best known. It is also important in other sectors such as bio-

medicine, automobiles, defence, aerospace, optics, solar energy, telecommunications, textiles,

papers, polymers and waste managment [10]. The organization of the paper is as follows: In

the next section, we deal with the harmonic response of the system using the harmonic balance

technique [11] to explore the influence of the quadratic term on the amplitude of oscillations.

Section 3 addresses the phase portraits, the largest Lyapunov exponent and the bifurcation

diagrams from which a concluding remark can be made in connection to the tendency of the

system to have a quasi-periodic evolution and a chaotic one according to the choice of initial

conditions which match with the basin of attraction found. The last section devoted to the

conclusion will point out the contribution of the quadratic term in both regular and chaotic

behavior of plasma oscillations.

2

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2 Forced harmonic oscillatory states

We use the harmonic balance method [11] to derive the amplitude of the forced harmonic

oscillatory states of equation (1) by assuming that the fundamental component of the solution

has the period of the external excitation. Thus, we consider

x(t) = A cos Ωt + η, (2)

where A stands for the amplitude of the oscillations and η a constant. Inserting the solution

x(t) in equation (1) and equating the cosine and constant terms separately, we have

(ω20 − Ω2)A +

4A3 + 2αη = E,

(3α

2A2 + ω2

0)η +β

2A2 = 0. (3)

After some algebraic manipulations, we have that the amplitude satisfies the following nonlinear

algebraic equation

9α2

4A5 + (

2ω2

0 − 3αΩ2− 2β)A3

− 3αEA2 + 2ω20(ω

20 − Ω2)A − 2ω2

0E = 0. (4)

We investigate the effect of the quadratic term on the behavior of the amplitude of oscillations A

by solving equation (4) using the Newton-Raphson algorithm. The obtained results are reported

in figure 1 where a jump phenomena is found. For a given value of β taken within the domain

of jump phenomenon, we have represented the variation of the amplitude versus the frequency

of the external excitation for fixed values of the parameters α and ω0 with different values of

the external force E (see figure 2). One can notice that as E increases, the jump phenomenon

disappears. For some values of the parameters, the model can display periodic oscillations

through limit cycles whose topology is well influenced by the quadratic term of nonlinearity

as shown in figure 3. In that figure, the phase portraits are plotted for different values of β.

One notices that as β increases, the shape of the limit cycle changes from a standard aspect

to a quasi-periodic state signature. In between, there are non smooth boundaries limit cycles

indicating the eventual richness of the behavior the system can show. It should be emphasized

that these periodic oscillations can switch to quasi-periodic oscillations and chaotic regime also

without changing the physical parameters of the model. Only the choice of the set of initial

conditions has a significant effect on that transition.

3 Bifurcation and transition to Chaos

Our aim in this section is to investigate the way under which chaotic motions arise in the

model described by equation (1) for resonant states since they are of interest in plasma oscilla-

tions. For this purpose, we solve numerically that equation using the fourth-order Runge Kutta

algorithm [12] and draw the resulting bifurcation diagram and the variation of the correspond-

ing largest Lyapunov exponent as the amplitude E, the parameters of nonlinearity α and β are

3

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varied.

The largest Lyapunov exponent which is used here as the instrument to measure the rate of

chaos in the system is defined as

Lya = limt→∞

ln(√

dx2 + dv2x)

t, (5)

where dx and dvx are the variations of x and x respectively.

The time period of the periodic stroboscopic bifurcation diagram used to map the transition is

T = 2π

Ω. For the set of parameters α = 1.5, β = 2.05, ω0 = 1,Ω = 3 and (xo, x0) = (1, 1), it

is found that the model can switch from periodic to quasi-periodic oscillations and finally to

chaotic states when the amplitude of the external excitation is varied as shown in the bifurcation

diagram and its corresponding Lyapunov exponent (see figure 4). In order to illustrate such

situations, we have represented the phase portraits using the parameters of the bifurcation

diagram for which periodic and quasi-periodic oscillations are observed respectively in figures 5

and 6. As concerning chaotic states, one can find many ranges of the parameter E for which such

motions occur. Among these domains, we can have the following: [2.128; 2.836], [4.009; 4.367],

[5.396; 5.481], [6.134; 6.553]. For E = 6.33, chaotic motions predicted by the Lyapunov exponent

are found as represented in figure 7. Since the model is highly sentitive to the initial conditions,

it can leave a quasi-periodic state for a chaotic state without changing the physical parameters.

Therefore, its basin of attraction has been plotted (see figure 8) in order to situate some regions

of the initial conditions for which chaotic oscillations are observed. In that figure, the black zone

stands for the area where the choice of the initial conditions lead to a chaotic motion while the

white area is the domain of periodic or quasi-periodic oscillations. To show how the parameters of

nonlinearity can influence the chaotic motion in the model, the Lyapunov exponent has also been

plotted versus β and α with the parameters of figure 7 and the following results are observed:

As β increases, the rate of chaos in the model becomes more impotant and remains unchanged

as shown in figure 9 (a) while on the other hand, as α increases, windows of chaotic and regular

motions decreases (see figure 9 (b). Consequently a set of physical parameters of the model,

α and β can be used to increase or dismiss the rate of chaotic motions in the model. Phase

portraits in the subharmonic and harmonic resonance have also been represented respectively

in figures 10 and 11 and the following comments are made:

• Both phase portraits have been drawn using the same initial conditions as in the superharmonic

case one (see figure 7)

• The harmonic case (see figure 11) shows a chaotic state

• The portrait phase in the sub-harmonic case seems to be like a torus. One might expect to see

a quasi-periodic oscillation since the overall torus resembles the one observed in case of solenoidal

attractor (the so-called quasi-periodic attractor). Indeed by looking at its corresponding time

series, one can observe a quasi-periodic state as shown in figure 12.

4

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4 Conclusion

From the source term, which is represented by the three last terms on the left-hand side in

equation 1, one can see that the potential energy is not symmetric as stated by Mahaffey [2],

hence both quadratic and cubic terms are needed. This equation has of course the features of

the Duffing equation. But coefficient β of the quadratic term plays a non negligeable role. This

paper has shown a good quantitative agreement between theory and numerical investigations.

The largest Lyapunov exponent, as obtained here, matches very well the regime of oscillations

expected. Also the frequency of the applied forcing term dictates the evolution of the system in

accordance with α and β. One must take into account the influence of the quadratic term when

it comes to use both nonlinearities. Some interesting features development will come later in

managing the chaotic states noticed.

Acknowledgments: This paper has been written during a visit at ICTP in the summer

2005. This work was done within the Associateship Scheme of the Abdus Salam International

Centre for Theoretical Physics, Trieste, Italy. The authors express special thanks to the Swedish

International Development Cooperation Agency (SIDA) for financial support.

5

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References

[1] B. E. Keen and W. H. Fletcher, Nonlinear plasma instability effects for subharmonic

and harmonic forcing oscillations, Journal of Physics A: Gen. Phys. Vol 5, (1972).

[2] R. A. Mahafhey, Anharmonic oscillator description of plasma oscillations, The Physics

of Fluids, Vol 19, 9 (1976).

[3] H. C. S. Hsuan, Phys. Rev. 172, 137 (1968).

[4] A. Venkatesan and M. Lakshmanan, Phys. Rev. E 63, 6321 (1997).

[5] L. Jianguo, Communication in Nonlinear Sciences and Numerical Simulation 4, 132 (1999).

[6] L. C. J. Albert and H. P. S. Ray, Journal of the Franklin Institute 334, 447 (1997).

[7] H. Jihuan, Communication in Nonlinear Sciences and Numerical Simulation 4 78 (1999).

[8] R. Yamapi, Dynamics and Synchronization of Electromechanical Devices with a Duffing

Nonlinearity, PhD thesis (2003).

[9] R. Yamapi and P. Woafo, Journal of Sound and Vibration 285, 1151 (2005).

[10] J. Proud and al., Plasma Processing of Materials: Scientific Opportunities and Technolo-

gies Challenges. National Academy Press, Washington D.C. (1991).

[11] A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations, (Wiley, New York, 1979).

[12] N. Piskunov, Calcul Differentiel et integral, Tome II, 9e edition, MIR, Moscou (1980).

6

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