PHYSICS OF PLASMAS 15, 032308 共2008兲
Nonlinear dynamics of plasma oscillations modeled by an anharmonic oscillator H. G. Enjieu Kadji,1,2,a兲 B. R. Nana Nbendjo,2,b兲 J. B. Chabi Orou,1,c兲 and P. K. Talla3,d兲 1
Institut de Mathématiques et de Sciences Physiques, B.P. 613, Porto-Novo, Benin Laboratory of Modelling and Simulation in Engineering and Biological Physics, Faculty of Science, University of Yaounde I, Box 812, Yaounde, Cameroon 3 Faculty of Sciences, University of Dschang, Box 67, Dschang, Cameroon 2
共Received 28 June 2007; accepted 9 January 2008; published online 25 March 2008兲 This paper considers nonlinear dynamics of plasma oscillations modeled by an anharmonic oscillator. These plasma oscillations are described by a nonlinear differential equation of the form x¨ + 共1 + x2兲x˙ + x + x2 + ␦x3 = F cos ⍀t. The amplitudes of the forced harmonic, superharmonic, and subharmonic oscillatory states are obtained using the harmonic balance technique and the multiple time scales method. Admissible values of the amplitude of the external strength are derived. Bifurcation sequences displayed by the model for each type of oscillatory states are performed numerically through the fourth-order Runge–Kutta scheme. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2841032兴 I. INTRODUCTION
Nonlinear science is a field of growing interest to scientists due to its usefulness in such diverse fields as physics, biology, engineering, chemistry, and social sciences, among others.1–7 In particular, the nonlinear description of plasma oscillations is of interest as a result of its importance to the semiconductor industry.8,9 Experiments suggest that some plasma behavior is approximately described by anharmonic oscillations. Thus, it has been shown experimentally10 that in plasma physics, the electron beam surfaces, the Tonks-Datter resonances of mercury vapor and low frequency ion sound waves, oscillations are described by the following anharmonic equation: x¨ + 20x + x2 + ␦x3 = F cos ⍀t.
共1兲
In the equation above, the overdot denotes the time derivative, while 0 and ⍀ are, respectively, the internal and the external frequencies. F stands for the amplitude of the external excitation, and and ␦ are the quadratic and cubic nonlinearity parameters, respectively. It has also been shown that Eq. 共1兲 has many more additional features than the Duffing equation. Indeed, these features include additional shifts in the frequency of oscillation, asymmetry of the amplitude of oscillation, effects on the phase plane singularities, and changes in the resonance response curves. A discussion of each of these features has been presented and a comparison with experiments made. In order to shed more light on the dynamics of the model described by Eq. 共1兲, recently we investigated conditions under which it can display regular and chaotic motions.11 Mainly, we have derived analytically the equation above and have shown a good quantitative agreement between theory and numerical investigations. The a兲
Electronic mail:
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[email protected]. c兲 Author to whom correspondence should be addressed. Telephone: ⫹229 90015918. Fax: ⫹229 20222455. Electronic mail:
[email protected]. d兲 Electronic mail:
[email protected]. b兲
1070-664X/2008/15共3兲/032308/13/$23.00
purpose of this paper is to take into account the effect of dissipation, which has been neglected in the modeling of Eq. 共1兲 共nondissipative plasma兲, and then to investigate using analytical methods, harmonic and resonant states that can be displayed by the model in such conditions. We also aim to perform the possible bifurcation mechanisms of the model using numerical tools. The paper is organized as follows. In Sec. II, we describe the model under consideration and establish the related equations of motion. Section III deals with the amplitude of the forced harmonic oscillatory states, using the harmonic balance method.1 Section IV addresses the resonant states of the model through the multiple time scales method.2 In Sec. V, we will point out admissible values of the amplitude of the external field as a function of the dissipative coefficient and, bifurcation sequences of the model. We conclude in Sec. VI.
II. MODEL AND EQUATION OF MOTION
We consider the two-fluid model, which treats the plasma as two interpenetrating conducting fluids. This model consists of a set of fluid equations for the electrons and ions plus the complete set of Maxwell’s equations. Such a model has been source of growing interest for researchers for many years12–14 and nowadays, it remains an interesting task because of its potential applications.15–19 The two-fluid plasma system is applicable in many areas where high density plasmas interact with high frequency electromagnetic waves. A few examples include application to helicon thrusters, electron cyclotron resonance plasma sources and plasmas resulting from strong explosions in the atmosphere. In the electrostatic situation, it also has possible application in modeling discharge cathode plasma sources which are important in ion thrusters and hall thrusters. The Eulerian equations of motion in electric and magnetic fields 共E and B, respectively兲, are given as follows:20
15, 032308-1
© 2008 American Institute of Physics
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Phys. Plasmas 15, 032308 共2008兲
Enjieu Kadji et al.
dv␣ = n␣q␣共E + v␣ ⫻ B − ␣J兲 − ⵜP␣ , dt
dvi = n0e共E + vi ⫻ B0 − iJ兲 − ⵜP1 , dt
共2兲
n0 M i
n␣ + ⵜ · 共n␣v␣兲 = S,
共3兲
n1 + ⵜ . 共n0vi兲 = S,
共11兲
d 共P␣n␣−␥兲 = 0. d
共4兲
n1 P1 =␥ . n0 P0
共12兲
n␣ M ␣
S is the source term due to ionization or to large amplitude oscillations present in the plasma. The suffix “␣” stands for the species label and it will be denoted by “i” and “e,” respectively, for positive ions with charge +e and electron with charge −e. n␣ stands for the density of the species, v␣ their velocity, P␣ their pressure, ␥ the usual specific heat ratio, and ␣ the resistive collision, which is defined as
␣ =
M ␣ ␣ , n ␣e 2
共5兲
where ␣ is the collision frequency of the species ␣. The electric charge density and current J are given by
= 兺 n ␣q ␣ , ␣
共6兲
J = 兺 n ␣q ␣v ␣ . ␣
These quantities are the source terms for Maxwell’s equations. In order to deal with small amplitude waves, we consider a “background” situation representing uniform infinite plasma. The values of n␣, v␣, P␣, E, and B for this will be denoted by n0␣, etc.; however, here we shall take v␣ = E = 0 in the unperturbed state. We then have J = 0, and all of Eqs. 共2兲–共6兲 are satisfied except Eqs. 共6兲, which require = 0; hence,
兺␣ n␣q␣ = 0.
共7兲
For our simple two-species plasma, that condition of charge neutrality becomes 共8兲
n0e = n0i = n0 .
We now consider the M e = 0 ion-sound instability and introduce perturbations terms that are denoted by the suffix “1,” namely, n i = n 0 + n 1,
P i = P 0 + P 1,
B = B0 + B1 .
共9兲
Let us note that for other variables that vanish at the unperturbed state, labels “0” and “1” are not necessary. n1 is considered as a perturbed time varying function of z. We then insert the expressions 共9兲 into Eqs. 共2兲–共4兲, and after all of the second-order perturbation terms have been discarded, we obtain the following equations:
共10兲
Since in this case of the ion-sound instability under consideration, only spatial variation of the form e−jkzz 共j is a complex number兲 needs to be considered.13 Here, kz stands for the wave number in the z direction. In dealing with Eq. 共4兲 and taking each species to be a perfect gas with unperturbed temperature T 共which could be different for each species兲, we have P0 = n0kBT 共kB is Boltzmann’s constant兲 and Eq. 共10兲 can be rewritten as follows: n0 M i
dvi = n0e共E + vi ⫻ B0 − iJ兲 − ␥kBTi ⵜ n1 . dt
共13兲
To investigate the two-fluid model, we assume that E = −ⵜ 共 is the corresponding potential兲 and consider the Boltzmann distribution equation of electron given as follows: n1 e = . n 0 k BT e
共14兲
By considering dvi / dt = vi / t + 共vi · ⵜ兲vi, eliminating vi between Eqs. 共13兲 and 共11兲, we obtain after some algebraic manipulations the following equation: n 0e d 2n 1 k B 共Te + ␥Ti兲ⵜ2n1 + ⵜ · 共 v i ⫻ B 0兲 2 − d Mi Mi
冉
− i S −
冊
dn1 dS − = 0. d d
共15兲
Keen and Fletcher12,13 and Hsuan21 in the early 1970s showed that from thermodynamics argument, the source term is a function of density n1. Here, the source is taken to be of the following form: S = − n21 − n31 .
共16兲
By assuming that the model is influenced by an external sinusoidal excitation F0 cos , Eq. 共15兲 becomes kB共Te + ␥Ti兲 2 d 2n 1 2 dn1 + kជ n1 2 + 共i + 2 + 3n1兲 d d Mi + i共n21 + n31兲 − j
en0 kជ · 共vi ⫻ B0兲 = F0 cos . Mi 共17兲
If one considers the slab geometry configuration for which density varies in the x direction, and the z axis coincides with the magnetic field direction, Eq. 共16兲 takes the following expression:
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Phys. Plasmas 15, 032308 共2008兲
Nonlinear dynamics of plasma oscillations… 2
0.15
Analytical results
1.75
Numerical results
0.12
1.5 0.09
A
A
1.25
0.06
1
0.75
ε=0.05
0.03
0.5
ε=0.4 ε=2
0.25
0 0
1
0.5
Ω
2
1.5
0
3
2.5
FIG. 1. 共Color online兲 Comparison between analytical and numerical frequency-response curves with the parameters = 1, ␦ = 0.3, = 0.4, and F = 0.05.
0
1
2
Ω
3
FIG. 3. 共Color online兲 Effects of on the frequency-response curves with the parameters of Fig. 1.
x¨ + 共1 + x2兲x˙ + x + x2 + ␦x3 = F cos ⍀t.
2
d n1 2 dn1 + 20n1 + i共n21 + n31兲 2 + 共i + 2 + 3n1兲 d d = F0 cos , where
0 = k zc ,
共18兲
冋冉
c = kB
Following the rescaling t = 0 ,
i⌶ = 2 ,
n1 =
冉
Te + ␥Ti Mi
冊册
冊
i + 2 x ⬅ ⌶x, 3
⌶2 ␦= i 2 , 0
1/2
.
i + 2 , 0
=
F0 F= , ⌶20
共19兲
⍀= , 0
it results that the system is governed by the following nonlinear second-order differential anharmonic equation
共20兲
There are several physical mechanisms that could mimic the driven force F. For example, the transport of dust particles into plasma is proportional to the dust charge and as well as to the coagulation of small particles into larger ones since charged particles attract or repel each other through the coulomb potential. However, considering the fact that ultraviolet light can extract electrons from materials by photodetachment, such a light can be used as an external force to control the charge on a dust particle.22 Such forcing terms could also be mimicked through an externally applied electric field that supplies the system with an external drive.16 For the particular case where the coefficient of dissipation = 0, Eq. 共20兲 is reduced to the modified Duffing equation treated recently.11 It should be quoted that the coefficient of the dissipation term plays a key role on how a limit cycle is born and how it dies. By changing that coefficient, a Hopf bifurcation will take place in between and a homoclinic bifurcation might also show up. Additionally, nondissipative plasma that corre-
2.5
7 6
2
ε=0.05
5
1.5
ε=0.3
A
A
4
F=1.5
1
ε=0.7
3
F=0.2
2
F=0.05
0.5
1
0
0
1
Ω
2
3
FIG. 2. 共Color online兲 Effects of F on the frequency-response curves with the parameters of Fig. 1.
0
0
3
6
F
9
12
15
FIG. 4. 共Color online兲 Effects of on the amplitude-response curves with ⍀ = 0.7 and the parameters of Fig. 1.
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Phys. Plasmas 15, 032308 共2008兲
Enjieu Kadji et al.
7
1
κ=1.80 6
κ=0.50
4
0.6
F=3
A
Α
0.8
κ=1.25
5
3
0.4
F=2
2
0.2
1 0
0
5
F
10
0 -10
15
FIG. 5. 共Color online兲 Effects of on the amplitude-response curves with ⍀ = 0.7, = 0.05, and the parameters of Fig. 1.
sponds to the ideal case has been considered mainly in twofluid stationary states studies for many decades. However, since the plasma is dissipative and externally driven in realistic experimental situations, it is of interest to elaborate a formalism for investigating the driven dissipative two-fluid model in order to forecast theoretical results closer to those of the experiments. III. AMPLITUDE OF THE FORCED HARMONIC OSCILLATORY STATES
Assuming that the fundamental component of the solution and the external excitation have the same period, the amplitude of harmonic oscillations can be tackled using the harmonic balance method.1 For this purpose, we express its solutions as x = A cos共⍀ − 兲t + ,
0
5
σ
10
15
20
FIG. 7. 共Color online兲 Effects of F on the frequency-response curves of the first superharmonic resonance with ␦ = 0.3.
关共1 − ⍀2兲 + 2A + ␦ 共 43 A3 + 3A2兲兴2 + 2关A⍀共1 + 2兲 + 41 A3⍀兴2 − F2 = 0,
共22兲
共1 + 23 ␦A2兲 + 21 A2 + 2 + ␦3 = 0.
共23兲
If it is assumed that 兩兩 Ⰶ 兩A兩, i.e., that shift in x = 0 is small compared to the amplitude,8 then 2 and 3 terms in Eq. 共23兲 can be neglected and one obtains =−
A2 . 2 + 3␦A2
共24兲
Substituting Eq. 共24兲 into Eq. 共22兲 leads us to the following nonlinear algebraic equation:
关 49 ␦A5 + 共 29 ␦ − 3␦⍀2 − 22兲A3 + 2共1 − ⍀2兲A兴2 + 关2⍀2共A + 41 A3兲2 − F2兴共2 + 3␦A2兲2 = 0.
共21兲
where A represents the amplitude of the oscillations and is a constant. Inserting this solution in Eq. 共20兲 and equating the constants and the coefficients of sin ⍀t and cos ⍀t, we have
F=1 -5
共25兲
The behavior of the amplitude of plasma oscillations is investigated when the external frequency ⍀ varies and the re-
1.5
0.8 δ = 0.3
0.4
1
δ=5
A
A
0.6
δ = 20
δ=5
0.5
0.2 0 −10
δ=3.0 −5
0
σ
5
10
FIG. 6. 共Color online兲 Effects of ␦ on the frequency-response curves of the primary resonant state for F = 0.1.
0 -10
δ=0.3 -5
0
5
10
15
20
σ FIG. 8. 共Color online兲 Effects of ␦ on the frequency-response curves of the first superharmonic resonance with the parameters of Fig. 7.
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Phys. Plasmas 15, 032308 共2008兲
Nonlinear dynamics of plasma oscillations… 2
8
1.5
κ=25
A
Α
6
4
F=5
κ=10
2
1
0.5
F=4
κ=2 0 -10
0
10
20
σ
0
30
FIG. 9. 共Color online兲 Effects of on the frequency-response curves of the second superharmonic resonance with the parameters of Fig. 7.
0
3
6
σ
9
12
15
FIG. 11. 共Color online兲 Effects of F on the frequency-response curves of the second subharmonic resonance with the parameters = 1.0 and ␦ = 2.30.
IV. RESONANT STATES
sults are plotted in Fig. 1, where the comparison between analytical and numerical resonance curves A共⍀兲 of the model is shown. The agreement between analytical and numerical results is very good. The resonance obtained from Fig. 1 is also strongly related to the amplitude of the external oscillatory force as depicted in Fig. 2. Thereby, the observed resonant state obtained for a set of parameters can be destroyed according to the value taken by the amplitude of the external force. Additionally, it should be stressed that the hysteresis and jump phenomena known to be function of the cubic nonlinear coefficient can also be triggered or quenched through the dissipative coefficient 共see Figs. 3 and 4兲 and as well as via the nonlinear quadratic parameter 共see Fig. 5兲. During the hysteresis and jump phenomena processes, for any value of the frequency ⍀ and the external force F, respectively, three different amplitudes of oscillations are obtained, among which two are stable and one is unstable.
Generally, many types of oscillations can be found in a forced system additionally to the harmonic oscillatory states. Such oscillations occur when the external frequency is too close or far from the internal frequency, according also to the external excitation strength. Since these oscillations rise up at different time scales, the best tool to be used for their investigation is the multiple time scales method.2 In such a situation, an approximate solution is generally sought as follows: x共,t兲 = x0共T0,T1兲 + x1共T0,T1兲 + ¯ ,
where the fast time scale T0 and the slow time scale T1 are associated, respectively, with the unperturbed system and with the amplitude and phase modulations induced by the global first-order perturbation. The first and second time derivatives can now be rewritten as follows:
500
4
3.5
400
3
300
F
Α
共26兲
2.5
200
F=5
2
F=4
1.5
100
0
1 0
10
20
σ
30
40
FIG. 10. 共Color online兲 Effects of F on the frequency-response curves of the first subharmonic resonance for ␦ = 2.30.
0
2
4
ε
6
8
10
FIG. 12. 共Color online兲 Boundary of the admissible values of the amplitude of the external force in the 共 , F兲 plane with the parameters = 3.05, ␦ = 1.5, and ⍀ = 1.
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Enjieu Kadji et al.
FIG. 13. 共Color online兲 Bifurcation diagram 共upper frame兲 and Lyapunov exponent 共lower frame兲 vs the amplitude F with the parameters = 3.05, ␦ = 1.5, ⍀ = 1, and = 0.02.
d = D0 + D1 + ¯ , dt
d2 = D20 + 2D0D1 + ¯ , dt2
共27兲
with
Dn = n , T
Order 0: D20x0 + x0 = 0,
共28兲
Order 1: T n = nT 0,
n = 0,1,2, . . . .
D20x1 + x1 = F cos ⍀T0 − 2D1D0x0 − 共1 + x20兲D0x0 − x20 − ␦x30 .
A. Primary resonant state
In this state, the amplitude F of the external excitation and the parameters and ␦ are small 共at the order of 兲, that is F = F, = , ␦ = ␦. The closeness between both internal and external frequencies is given by ⍀ = 1 + , where is the detuning parameter. Inserting Eqs. 共26兲 and 共27兲 into Eq. 共20兲 and equating the coefficients of like powers of , we obtain
共29兲
The general solution of Eq. 共28兲 is x0 = A共T1兲exp共jT0兲 + c.c.,
共30兲
where “c.c.” represents the complex conjugate of the previous terms. A共T1兲 is a complex function to be determined from solvability or secular conditions of Eq. 共29兲. Thus, substituting the solution x0 in Eq. 共29兲 leads us to the following secular criterion:
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Nonlinear dynamics of plasma oscillations…
Phys. Plasmas 15, 032308 共2008兲
FIG. 14. 共Color online兲 Various phase portraits for several different values of F with the parameters of Fig. 13.
2jD1A + j共1 + 兩A兩2兲A + 3␦兩A兩2A −
F exp共jT1兲 = 0. 共31兲 2
In polar coordinates, the solution of Eq. 共31兲 is
A共T1兲 = 21 a共T1兲exp关j共T1兲兴,
共32兲
where a and are real quantities and stand, respectively, for the amplitude and phase of oscillations. After injecting Eq. 共32兲 into Eq. 共31兲, we separate real and imaginary terms and
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Phys. Plasmas 15, 032308 共2008兲
Enjieu Kadji et al.
FIG. 15. 共Color online兲 Various phase portraits in some intermittency areas for different values of F with the parameters of Fig. 13.
obtain the following coupled flow for the amplitude and phase: a⬘ = −
a a3 F − + sin ⌽, 2 8 2
3 F a⌽⬘ = a − ␦a2 + cos ⌽, 8 2 where the prime denotes the derivative with respect to T1 and ⌽ = T1 − . For the steady-state conditions 共a⬘ = ⌽⬘ = 0兲, the following nonlinear algebraic equation is obtained:
冊 冉
共35兲
⌽ = ⌽0 + ⌽1 ,
共33兲
冉
a = a0 + a1 ,
冊 冉
冊
9␦2 + 1 6 1 − 6␦ 4 1 + 42 2 F2 a0 + a0 + a0 − = 0, 64 4 4 8 共34兲
where a0 and ⌽0 are, respectively, the values of a and ⌽ in the steady state. This steady state is of interest only if it is stable. To study the stability of the process, we assume that each equilibrium state is submitted to a small perturbation as follows:
where a1 and ⌽1 are slight variations. Inserting Eq. 共35兲 into Eq. 共33兲 and canceling nonlinear terms enables us to obtain
a1⬘ = −
冉 冉
冊 冉 冊 冉
冊 冊
1 3 3 1 + a20 a1 − a0 − ␦a20 ⌽1 , 2 4 8 共36兲
1 1 1 9 − ␦a20 a1 − 1 + a20 ⌽1 . ⌽1⬘ = 8 2 4 a0 The stability process depends on the sign of eigenvalues ⌼ of Eqs. 共36兲, which are given through the following characteristic equation: ⌼2 + 2Q⌼ + R = 0,
共37兲
where
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Phys. Plasmas 15, 032308 共2008兲
Nonlinear dynamics of plasma oscillations…
FIG. 16. 共Color online兲 Bifurcation diagram 共upper frame兲 and Lyapunov exponent 共lower frame兲 vs the amplitude F with the parameters of Fig. 2 for = 6.
Q = 41 共a20 + 2兲, R=
1 4
共1 + 43 a20兲共1 + 41 a20兲 − 共 − 83 ␦a20兲共 − 89 ␦a20兲 .
Since Q ⬎ 0, the steady-state solutions are stable if R ⬎ 0 and unstable otherwise. Figure 6 displays the amplitudes response curves obtained from Eq. 共34兲 for different values of the parameter ␦, and one can observe that as it increases, the model goes from resonance to a hysteresis state.
Order 0: D20x0 + x0 = F cos ⍀T0 . Order : D20x1 + x1 = − 2D1D0x0 − 共1 + x20兲D0x0 − x20 − ␦x30 .
When the amplitude of the sinusoidal external force is large, other type of oscillations can be displayed by the model, namely the superharmonic and the subharmonic oscillatory states. It is now assumed that F = 0F and, therefore, one obtains the following equations at different order of :
共39兲
From Eq. 共38兲, we have x0 = A共T1兲exp共jT0兲 + exp共j⍀T0兲 + c.c.,
B. Superharmonic and subharmonic oscillations
共38兲
1
共40兲
with
=
F . 2共1 − ⍀2兲
Substituting the general solution x0 into Eq. 共39兲 yields
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032308-10
Enjieu Kadji et al.
Phys. Plasmas 15, 032308 共2008兲
FIG. 17. 共Color online兲 Bifurcation diagram 共upper frame兲 and Lyapunov exponent 共lower frame兲 vs the amplitude F with the parameters of Fig. 2 for = 0.6, = 1.8.
D20x1 + x1 = 关− 2jA⬘ − j共1 + 22 + 兩A兩2 + 3␦兩A兩2 + 6␦2兲A兴exp共jT0兲 + 关− j⍀共1 + 2兩A兩2 + 2兲 − 6␦兩A兩2 − 3␦2兴exp共j⍀T0兲 − 共j + ␦兲A3 exp共3jT0兲 − 关j共1 + 2⍀兲 + 3␦兴2A exp关j共1 + 2⍀兲T0兴 − 关j共2 + ⍀兲 + 3␦兴2A exp关j共2 + ⍀兲T0兴 − 关j共2 − ⍀兲 + 3␦兴A2 exp关j共2 − ⍀兲T0兴 − 共j⍀ + ␦兲3 exp关3j⍀T0兲 − 关j共1 − 2⍀兲 + 3␦兴2A exp关j共1 − 2⍀兲T0兴 − A2 exp关2jT0兴 − 2 exp关2j⍀T0兴 − 2A exp关j共1 + ⍀兲T0兴 − 2A exp关j共1 − ⍀兲T0兴 − 共兩A兩2 + 2兲 + c.c.. 共41兲
From Eq. 共41兲, superharmonic and subharmonic states can be found from the quadratic and cubic nonlinearities. The cases of superharmonic oscillations we consider are 3⍀ = 1 + and 2⍀ = 1 + , while the subharmonic oscillations to be
treated are ⍀ = 3 + and ⍀ = 2 + . For the first superharmonic states, i.e., 3⍀ = 1 + , the amplitude of oscillations is governed by the following nonlinear algebraic equation:
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Nonlinear dynamics of plasma oscillations…
FIG. 18. 共Color online兲 Basin of chaoticity in the primary resonant state.
冉
冊 冋 冋冉 冊
册
冉
9␦2 + 1 6 1 1 + 2 − 3␦共 − 3␦2兲 a40 a0 + 64 4 2 +
1 + 2 2
2
册
+ 共 − 3␦2兲2 a20 − 共␦2 + ⍀2兲6 = 0,
冊 冋
冉
冊
1 1 1 2 3 9␦2 + 1 4 + − ␦ − 3␦2 − 2共9␦2 a0 + 64 8 4 4 3 16
册 冋冉 冊 冉 1 + 2 2
+ 共2 − ⍀兲2兲 a20 +
2
+
− 3␦2 3
冊册 2
= 0,
共42兲 and they are stable if 1 4
and their stability is guaranteed only if
共1 + 22 + 43 a20兲共1 + 22 + 41 a20兲 + 共 − 89 ␦a20 − 3␦2兲共 − 3␦2 − 83 ␦a0兲 ⬎ 0.
共43兲
Figure 7 presents the frequency-response curves of the superharmonic resonance as a function of for different values of the external force. As the intensity of the external force increases, the resonance behavior observed is destroyed. The effects of the parameter ␦ on such superharmonic oscillations are also investigated, and results are reported in Fig. 8, showing the appearance of the hysteresis phenomenon when the nonlinear cubic parameter ␦ is increasing. On the other hand, the amplitude of the second superharmonic oscillations is given by the following equation:
冉
冊 冋 冋冉 冊
册
9␦2 + 1 6 1 1 a0 + + 2 − 3␦共 − 3␦2兲 a40 64 4 2 +
1 + 2 2
共45兲
2
册
+ 共 − 3␦2兲2 a20 − 24 = 0,
冉
1 1 + 2 + a20 2 4
冉
冊冉
冊冉
1 1 1 + 2 + a20 − − ␦2 − a20 2 8 8 9
冊
冊
9 ⫻ + ␦a20 − 9␦2 ⬍ 0. 8
共46兲
When considering the second subharmonic oscillatory states 共⍀ = 2 + 兲, we obtain that their motions are governed by the equation
冉
冊 冋 冋冉 冊 冉
冉 冊
1 1 2 3 9␦2 + 1 4 + − ␦ − 3␦2 a0 + 64 8 4 4 2 +
1 + 2 2
2
+
− 3␦2 2
2
冊册
a20
册
− 22 = 0.
共47兲
They are stable if the following criterion is fulfilled: 共44兲
and the stability criteria is the one defined through inequality 共43兲. In this case also, the influence of the quadratic parameter on such oscillations has been checked 共see Fig. 9兲. In such a state, multiple resonances obtained for small values of are reduced progressively when increasing . However, as the first subharmonic oscillations 共⍀ = 3 + 兲 are concerned, their states are described by the following equation:
冉
冊 冋冉
1 1 + 22 + a20 a20 + 6 4
冊
册
3 − 3␦2 a20 − ␦a40 ⬎ 0. 8 2 共48兲
The frequency-response curves of both types of subharmonic oscillations are plotted in Figs. 10 and 11 and the regions where such behaviors occur are obtained. From these pictures, we conclude that the range of frequency where a response can be obtained is more important in the first subharmonic state than in the other cases.
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032308-12
Phys. Plasmas 15, 032308 共2008兲
Enjieu Kadji et al.
FIG. 19. 共Color online兲 Basin of chaoticity in the first superharmonic resonant state.
V. ADMISSIBLE VALUES OF F AND BIFURCATION STRUCTURES
Since the model is submitted to the effect of an external field, it would be interesting to have a view on the admissible values of F for which possible bifurcation mechanisms are allowed to appear will be stable. Therefore, Eq. 共20兲 is solved numerically using the fourth-order Runge–Kutta algorithm. The stroboscopic time period used to map various transitions that appear in the model is T = 2 / ⍀. The boundary of the admissible values of the external strength in the 共 , F兲 plan is plotted in Fig. 12. For any fixed value of or ⍀, we find that the possible value of F is the region below the curve. For example, from Fig. 12, the bifurcation mechanisms are only possible for F 苸 兴0 , 212兴 when = 1.5 and ⍀ = 1, while for = 6 and ⍀ = 1, the choice of F is possible in the interval 兴0,38.20兴. After deriving the area of admissible values of F, it is important to find the way by which chaos appears in the model, using bifurcation diagrams and Lyapunov exponent as indicators. The Lyapunov exponent is
obtained by solving numerically Eq. 共20兲 and the corresponding variational equation. The Lyapunov exponent is defined as Lya = lim t→⬁
ln共冑dx2 + dv2x 兲 , t
共49兲
where dx and dvx are the variations of x and x˙, respectively. For the set of parameters = 3.05, ␦ = 1.5, = 0.02, and ⍀ = 1, the bifurcation diagram and its corresponding Lyapunov exponent are plotted in Fig. 13. From the bifurcation diagram, various types of motions are displayed. For instance, a period-1 orbit exists for F苸兴0,0.7兴 艛 关0.9,1.62兴艛关2.5,3.5兴艛关4.8,8兴艛关10.6,16.74兴艛关19,19.65兴 艛 关22.2,24.8兴, while quasiperiodic motions are obtained for F苸 关0.76,0.85兴艛关0.87,0.89兴 艛兴1.62,1.68兴艛关2.05,2.2关艛兴3.5,3.99兴 艛关4.6,4.8兴艛关8.0,8.30兴艛关9.7,10兴艛关16.4,17.1兴艛关17.25,18.67兴 艛关19.65,19.8兴. On the other hand, chaotic motions exist for F苸兴0.7,0.76关艛兴0.85,0.87关艛兴0.89,0.9关艛兴1.68,2.05关艛兴2.1,2.5 关艛关4.0,4.55兴艛兴8.30,9.7关艛兴10,10.6关艛兴17.1,17.25关艛兴18.67,19
FIG. 20. 共Color online兲 Basin of chaoticity in the first subharmonic resonant state.
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Phys. Plasmas 15, 032308 共2008兲
Nonlinear dynamics of plasma oscillations…
关艛关19.9,22.2兴. In order to have an idea about the system behavior as predicted by the bifurcation diagram, various phase portraits for several different values of F chosen in the abovementioned regions are plotted in Fig. 14, using the parameters of Fig. 13. It should be emphasized from Fig. 13 that there are some domains where the Lyapunov exponent does not match very well the regime of oscillations expected from the bifurcation diagram. Far from being an error that has occurred from the numerical simulation process, such a behavior corresponds to what is called the intermittency phenomenon. Therefore, within these intermittent domains, the dynamics of the model can not be predicted. For instance, some forecasted period-1 and quasiperiodic motions from the bifurcation diagram are not confirmed by the Lyapunov exponent. Indeed, in these period-1 regions, phase portraits display rather quasiperiodic or chaotic motions, while for the forecasted quasiperiodic motion, phase portraits show chaotic motions as reported in Fig. 15. The influences of both the nonlinear quadratic and dissipative parameters on the bifurcation sequences are also investigated, and the results are reported in Figs 16 and 17, respectively. By increasing the parameter , the chaotic windows as well as the ones of intermittency remain in the system. On the other hand, the parameter can be used to control the presence of chaotic oscillations and to suppress the intermittency regime in the system. The occurrence of chaotic and intermittency states has been also found in the superharmonic and subharmonic resonant states 共this is not shown here兲. Due to the high sensitivity of the model to initial conditions, basins of chaoticity 共where any choice of initial conditions that belongs to the shaded area will lead the system to chaotic states, while if the initial conditions are chosen in the nonshaded area, the system will display periodic or quasiperiodic states兲 are also checked in primary, superharmonic, and subharmonic resonant states 共see Figs. 18–20, respectively兲. From these figures, we conclude that chaos is more abundant in the subharmonic resonant states than in the superharmonic and primary resonances. This confirms what has been obtained through their bifurcation diagrams and Lyapunov exponent. VI. CONCLUSION
We have investigated the nonlinear dynamics of plasma oscillations modeled by an anharmonic oscillator. The model has been described and the corresponding equation of motion obtained. The amplitude of forced oscillatory states has been investigated as well as the different resonant states of the model by using analytical methods. For these states, several orders of amplitude oscillations have been obtained. The influences of the dissipative parameter on the resonant, hysteresis, and jump phenomena have been highlighted. Additionally, the boundaries of the admissible values of F have been constructed as a function of the dissipative parameter using
numerical simulations of the equation of motion. From these preliminary results, we can conclude that the dissipation parameters have a real impact on the dynamics of the model. Therefore, it is useful to forecast domains in which such effects could be of interest or not. The results show a way to predict admissible values of the signal amplitude for a corresponding set of parameters. This could be helpful for experimentalists who are interested in trying to stabilize such a system with external forcing. Some bifurcation structures of the model have also been investigated and domains where the intermittency phenomenon occurs are found. For practical interests, it is useful to develop tools and to find ways to control or suppress such undesirable regions. This will be also useful to control high amplitude oscillations obtained, which are generally source of instability in plasma physics. These tasks are under investigations and preliminary results have already been obtained. ACKNOWLEDGMENTS
A part of this work was done during the Victor Rothschild Fellowship visit 共April–May 2006兲 of H.G.E.K. at the African Institute for Mathematical Sciences 共AIMS兲. He would like to express special thanks to AIMS. The authors are grateful to Dr. Charlemagne Akpovo for the fruitful discussions and also to the referees for their valuable comments. C. Hayashi, Nonlinear Oscillations in Physical Systems 共McGraw Hill, New York, 1964兲, Sec. 1.5. 2 A. H. Nayfeh, Introduction to Perturbation Techniques 共John Wiley and Sons, New York, 1981兲, Sec. 4.5. 3 J. Guckenheimer and P. J. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields 共Springer-Verlag, Berlin, New York, 1984兲, Sec. 1.3. 4 S. H. Strogatz, Nonlinear Dynamics and Chaos With Applications to Physics, Chemistry, and Engineering 共Westview, Cambridge, 1994兲, Sec. 1.2. 5 J. D. Murray, Mathematical Biology, 3rd ed. 共Springer, New York, 2001兲, pp. 218–271. 6 G. V. Paeva, “Sheath phenomena in dusty plasmas,” Ph.D. thesis, Technische Universiteit Eindhoven 共2005兲. 7 S. Park, C. R. Seon, and W. Choe, Phys. Plasmas 11, 5095 共2004兲. 8 R. A. Mahaffey, Phys. Fluids 19, 1837 共1976兲. 9 K. Ostrikov, Rev. Mod. Phys. 77, 489 共2005兲. 10 K. Ostrikov and S. Xu, Plasma-Aided Nanofabrication: From Plasma Sources to Nanoassembly 共John Wiley & Sons, Weinheim, 2007兲, pp. 149– 280. 11 H. G. Enjieu Kadji, J. B. Chabi Orou, and P. Woafo, Phys. Scr. 77, 551 共2008兲. 12 B. E. Keen and W. H. Fletcher, J. Phys. A 5, 152 共1972兲. 13 B. E. Keen and W. H. Fletcher, Plasma Phys. 13, 419 共1970兲. 14 L. H. Li and M. Matsuoka, Radiophys. Quantum Electron. 99, 75 共1996兲. 15 J. Loverich and U. Shumlak, Phys. Plasmas 13, 082310 共2006兲. 16 R. Bhattacharyya and M. S. Janaki, Phys. Plasmas 13, 044508 共2006兲. 17 U. Shumlak and J. Loverich, J. Comput. Phys. 187, 620 共2003兲. 18 T. Kanki, M. Nagata, and T. Uyama, IEEE Trans. Magn. 42, 1403 共2006兲. 19 M. Arshad Mirza, T. Rafiq, and G. Murtaza, Phys. Plasmas 13, 1107 共1999兲. 20 R. Dendy, Plasma Physics: An Introductory Course 共Cambridge University Press Cambridge, 1993兲, pp. 56, 370. 21 H. C. S. Hsuan, Phys. Rev. 172, 137 共1968兲. 22 V. Land and W. J. Goedheer, IEEE Trans. Plasma Sci. 35, 280 共2007兲. 1
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