Physics Letters A 372 (2008) 5329–5335

Contents lists available at ScienceDirect

Physics Letters A

www.elsevier.com/locate/pla

On the chaotic aspects of three wave interaction in a magnetized plasma

Anirban Ray a, Dibakar Ghosh a,b, A. Roy Chowdhury a,∗a High Energy Physics Division, Department of Physics, Jadavpur University, Calcutta 32, Indiab Department of Mathematics, Dinabandhu Andrews College, Garia, Calcutta 700 084, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 19 March 2008Received in revised form 9 June 2008Accepted 17 June 2008Available online 20 June 2008Communicated by F. Porcelli

PACS:05.45.-a52.35.Mw52.35.Ra52.35.Bj

Keywords:Magnetized plasmaThree wave processGalerkin approximationDNLS equationChaosHopf bifurcation

Nonlinear processes in magnetized plasma are very much important for the proper understanding ofmany space and astrophysical events. One of the most important type of study has been done in thedomain of Alfven waves. Here we show that a Galerkin type approximation of the DNLS (DerivativeNonlinear Schrödinger) equation describing such wave propagation leads to a new type of nonlineardynamical systems, very much rich in chaotic properties. Starting with the detailed analysis of fixedpoints and stability zones we make an in depth study of the unstable periodic orbits, which span thewhole attractor. Next the birth of a Hopf bifurcation is identified and normal form, limit cycle analyzed.In the course of our study the detailed structure of the attractor is analyzed. A possibility of internal crisisis also indicated. These results will help in the choice of the plasma parameters for the actual physicalsituation.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

Of late rigorous research activities have been seen to investigatecollective plasma modes in classical and quantum plasmas [1–6].The inherent nonlinearity of such processes makes it imperativethat a detailed stability analysis is undertaken. One of the mostimportant class of problems occur in the domain of magnetizedplasma, where the analysis of Alfven waves plays a leading role.In particular it may be mentioned that the coupling of Alfven–Langmuir–Whistler waves [7–9] plays a significant role in planetarymagnetosphere. The nonlinear interaction of such waves is still avery important [10–12] one, and requires a very detailed study[13–16]. A recent paper by Brodin et al. [17] has studied the threewaves process in a cold magnetized plasma. Also it has been seenthat radio energy bursts from sun can be produced via a nonlin-ear conversion of Langmuir waves into high frequency electromag-netic cyclotron pulse, through coupling with similar low frequencywaves. On the other hand, a kinetic Alfven and Whistler wave the-ory have been proposed by Voitenko and Goossens [15] in this

* Corresponding author. Tel.: +91 33 24163708; fax: +91 33 473 4266.E-mail addresses: bubai82@rediffmail.com (A. Ray), diba.ghosh@gmail.com

(D. Ghosh), asesh_r@yahoo.com (A. Roy Chowdhury).

0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2008.06.035

respect. So it is very apparent that a sort of three wave interac-tion process plays a central role in these processes. Here in thiscommunication we have considered a Galerkin type approxima-tion [18] of the DNLS equation describing the dynamics of a largeamplitude nonlinear Alfven wave propagating along the ambientmagnetic field, to obtain a new set of nonlinear ordinary differen-tial equations (ode’s), describing the three wave process. As thisset of equations is highly nonlinear it is very much important toinvestigate those set of parameter values which lead to stable orunstable mode [19–22]. Below we show how the instability is builtup and through the formation of unstable periodic orbits span thewhole chaotic attractor. In this connection, a new branch of insta-bility is also identified leading to Hopf bifurcation [23–26], whosedetailed structure is exposed via normal form analysis. To startwith we analyze the origin and stability of various fixed pointsby the Routh–Hurwitz criterion. Here we observe the transition toa Hopf bifurcation channel later explored with the help of nor-mal form. The form of limit cycle is also obtained. Later a detailedinvestigation is undertaken to explore the population of unstableperiodic orbits of the attractor by a modified approach of Newton–Raphson method for flow [27]. Lastly we have used MATCONT [28]to study the unfolding of bifurcation, with its transition to Hopfstate which shows agreement with the detailed numerical results.

5330 A. Ray et al. / Physics Letters A 372 (2008) 5329–5335

2. Formulation

The dynamics of large amplitude Alfven waves propagatingalong the ambient magnetic field can be described by the DNLSequation,(

i∂

∂t− γ

)B + iα

∂

∂x

(B|B|2) + β

∂2 B

∂x2= 0 (1)

here γ is the linear growth or damping, α stands for the sign ofthe nonlinearity and β that of dispersion.

The three term Galerkin approximation is done by writing B as

B =3∑

i=1

Bi(t)exp{−i(ki x − ωit)

}(2)

where we assume 2k1 = k2 + k3. From the linear dispersion rela-tion we have ωi = −βk2

i (i = 1,2,3) and ω2,3 − ω1 = �′1,2 is the

phase difference. We furthermore represent each complex ampli-tude Bi(t) as

Bi(t) = Ri(t)exp{

iθi(t)}

(3)

i = 1,2,3, while (Ri, θi(t)) are real. To simplify the resulting dy-namical system equations we impose some conditions on the var-ious quantities,

R1 = R2, r1 = r2, α = β = −1,

alongwith T = γ0t and k1 ≈ k2 ≈ k3,

a21 = k1

γ0R2

1, a22 = k1

γ0R2

2, γ = −γ1

γ0,

φ = −ψ, δ = −�′

γ0(4)

where �′ = (�′1 + �′

2)/2. So that one gets

a1 = a1 + 2a1a22 sin φ,

a2 = −γ a2 − a21a2 sin φ,

φ = −2δ + 2(a2

2 − a21

) + 2(2a2

2 − a21

)cosφ (5)

where a1, etc., denote da1/dT and

ψ = 2θ1 − θ2 − θ3 − 2�′t.Eq. (5) describes the three wave process under consideration.

3. Stability analysis

To start with we observe that the fixed point of Eq. (5) is givenas

a∗0 =

√γ

D; a∗

1 = 1√2D

; θ∗ = − sin−1(D),

where D is given as

D = (γ − 1)√

4δ2 − 4γ + 3 − δ(2γ − 1)

2{δ2 + (γ − 1)2}provided 4δ2 − 4γ + 3 > 0. Furthermore we fix δ = −6 and varyγ as the bifurcation parameter. The characteristic equation corre-spondence to this is

f (λ) = λ3 + e1λ2 + e2λ + e3 = 0 (6)

where

e1 = 2(γ − 1),

e2 = 2γ

D2

(2D2 + 4 − 3ξ

),

e3 = 8γ {(γ − 1)D − 6ξ

}

D

with ξ = −12D+2γ −12(γ −1)

. By the Routh–Hurwitz criterion the real partof the roots of Eq. (6) are negative if and only if

e1 > 0, e3 > 0, e1e2 > e3

leading to the restriction 1 < γ < 1.3215. Note that all the coef-ficients of Eq. (6) are positive for γ > 1. Consequently the insta-bility arises if there is two complex conjugate zeros, say λ1 = iΩ ,λ2 = −iΩ , then λ3 = −2(γ − 1) which is on the boundary of thestability and we obtain the critical value γ = γ0 = 1.3215 andλ3 = −0.643 < 0. This set of complex eigenvalues leads to Hopfbifurcation, which we take up in a later section. But here we studythe general structure of the unstable periodic orbits (UPO’s) whichpopulate the attractor.

Before going to the actual results we briefly describe themethodology adopted for getting information about the UPO’s.Though basically it is a variant of the Newton’s method for findingthe fixed points of a Poincaré map, yet some details are given herefor the sake of completeness.

Consider a dynamical system,

x = f (x) (7)

where x = {x1, x2, . . . , xd} is a d-dimensional vector and f ={ f1(x), f2(x), . . . , fd(x)}. Let Xt(X0) be the flow of (7) for a giveninitial condition X0, that is the value of the orbit of X0 at time t .Let t = σ be the time of a trajectory started in X0 takes to com-plete one iteration of some Poincaré map, so

X1 = Xσ (X0)

is the next point in the map iteration. The fixed point is a solutionof

Xσ ( X) = X .

By Newton’s approach we set

X = X0 + �X

with initial guess X0 close to the desired point and an initial guessτ0 for the return time of X

σ = τ0 + �τ

then Taylor expansion of X yields

Xσ ( X) = Xτ0(X0) + ∂ Xτ0(X0)

∂t�τ + D X Xτ0(X0)�X,

where∂ Xτ0 (X0)

∂t = f (Xτ0(X0)). Now using Xτ ( X) = X = X0 + �X ,one gets

(I − J )�X − f (X1)�τ = X1 − X0, (8)

where I is the identity matrix, J is the Jacobian to be obtained byintegrating the variational equation

J = D X f (X0) J (9)

along with Eq. (7), with J = I as the initial condition. The solutionX1 of Eq. (7) at each iteration are not necessarily on the Poincarésection unless a stroboscopic section is used. A possible method torestrict this is to assume that Poincaré section be given as

(X1 − X0)a = 0 (10)

where a is a vector normal to the plane, X0 is on it. Adding (8) to(10) guarantees that X1 is always on the plane when we iterate(

I − J − f (X1)

a 0

)(�X�τ

)=

(X1 − X0

0

). (11)

In the present case, we have two bifurcation parameters (γ and δ).In Fig. 1(a) we have fixed δ to −6.015 and shown the bifurcation

A. Ray et al. / Physics Letters A 372 (2008) 5329–5335 5331

(a)

(b)

(c)

Fig. 1. (a) Bifurcation diagram with respect to γ and (b) δ, (c) variation of largestLyapunov exponent with change of δ.

(a)

(b)

Fig. 2. (a) Bifurcation near internal crisis and (b) variation of largest Lyapunov ex-ponent with δ within period-3 window.

scenario with respect to γ . On the other hand in Fig. 1(b), γ isset to 6.74 and a bifurcation diagram is shown with respect to δ.The corresponding nature of the dynamical state is evident fromthe computation of the largest Lyapunov exponent (LLE) displayedin Fig. 1(c) (we have used Wolf’s method for calculating Lyapunovexponents [29]), where we display its variations with respect to δ.In Fig. 2(a) we have shown the bifurcation diagram near internalcrisis i.e. between δ = −6.01 and δ = −6.004. In Fig. 2(b), we havegiven an enlarged version of the variation of largest Lyapunov ex-

ponents with δ within the period-3 window for the self-checkingby the reader. Here the dashed line represents the unstable pe-riodic orbit obtained through modified Newton–Raphson methodwhich is detailed above. We have taken the period three windowto show the internal crisis because appearance of period-3 impliesthat all other periodic orbits will be present (this is an implicationof forcing rule and Sarkovskii theorem) [30,31]. This method hasalso been used for (δ = −6.015, γ = 6.74). Note that these are thevalues also used in Fig. 1(b). The corresponding periodic orbits ofperiod 1,2,3, . . . up to 6 are displayed in Figs. 3(a)–3(f), where theshaded portion in each diagram represents a part of the chaotic at-tractor in which they are embedded. In the next section we passover to the case of Hopf bifurcation originating from the complexconjugate eigenvalues of Eq. (6).

4. Hopf bifurcation and normal form

Consider a general nonlinear system written as:

x = Ax + F(x), x ∈ �n (12)

where F(x) = ©(‖x‖2) is a smooth function which can be ex-pressed as

F(x) = 1

2C1(x,x) + 1

6C2(x,x,x) + ©(‖x‖4) (13)

where C1 and C2 are bilinear and trilinear functions respectively. Ifthe matrix A has a pair of purely imaginary eigenvalues λ = ±iω0and (q, q) be the corresponding eigen vectors,

Aq = iω0q; Aq = −iω0q.

Furthermore let the p be the adjoint eigenvector satisfying,

AT p = −iω0p; AT p = iω0p,

with 〈p,q〉 = 1. The first Lyapunov coefficient of the system (if ori-gin is the point under consideration) can be written as [32];

l1(0) = 1

2ω0Re

[⟨p, C2(q,q, q)

⟩ − 2⟨p, C1

(q, A−1C1(q, q)

)⟩

+ ⟨p, C1

(q, (2iEω0 − A)−1C1(q,q)

)⟩].

The limit cycle is unstable that is there exists subcritical Hopf bi-furcation when l1(0) is greater than zero and when l1(0) < 0, weget supercritical Hopf bifurcation or stable limit cycle.

In the present situation the matrix A is given as⎛⎝ 1 + 2a∗

12 sin(θ∗) 4a∗

0a∗1 sin(θ∗) 2a∗

0a∗1

2 cos(θ∗)−2a∗

0a∗1 sin(θ∗) −γ0 − a∗

02 sin(θ∗) −a∗

02a∗

1 cos(θ∗)−4a∗

0(1 + cos(θ∗)) 4a∗1(1 + 2 cos(θ∗)) −2(2a∗

12 − a∗

02) sin(θ∗)

⎞⎠

for the fixed point (a∗0,a∗

1, θ∗) and γ = γ0. The term of the vectors

p and q are

p =(

1,0,1 + iω0 + 2a∗

12 sin(θ∗)

4a∗0(1 + cos(θ∗))

)T

and

q =(

1,iω0 − 1 − 2a∗

12 sin(θ∗)

4a∗0a∗

1 sin(θ∗),0

)T

.

The quantities C1(U,V), for any two vectors U = (u1, u2, u3) ∈ �3

and V = (v1, v2, v3) ∈ �3 is

C1(U,V) = (0,0,−8u1u2 + 12v1 v2)T

and

C2(U,V,W) = (0,0,0)T .

5332 A. Ray et al. / Physics Letters A 372 (2008) 5329–5335

(a) (b) (c)

(d) (e) (f)

Fig. 3. Unstable periodic orbits of period (a) one, (b) two, (c) three, (d) four, (e) five and (f) six for γ = 6.74 and δ = −6.015. The shaded region is the chaotic attractor.

Table 1Bifurcation continuation with respect to parameter γ where FLC means First Lyapunov Coefficient, BP means Bifurcation Point and HB means Hopf Bifurcation

a1 a2 φ γ Eigenvalues FLC BP

2.6406 1.6242 6.08 1.3215 −0.643,±22.7229i 4.0062 Subcritical HB2.5495 0.7071 4.7124 6.5 −11.0,±5.099i −0.088 Supercritical HB3.299 0.7467 4.2535 9.7587 −17.5175,±5.0818i −0.133 Supercritical HB14.90 1.738 3.3078 36.75 0,−35.75 ± 36.9977i – Limit point

After series of algebraic manipulations we get

l1(0) = 1

2ω0Re

[−2

1 + iω0 + 2a∗1

2 sin(θ∗)4a∗

0(1 + cos(θ∗))η

+ 1 + iω0 + 2a∗1

2 sin(θ∗)4a∗

0(1 + cos(θ∗))η′

]

where

η = 2(1 − iω0 + 2a∗1

2 sin(θ∗))a∗

0a∗1 sin(θ∗)

+ 12a13a23(1 + 2a∗1

2 sin(θ∗) + 5iω0)2

�21a∗

02a∗

12 sin2(θ∗)

,

η′ = 2(1 + iω0 + 2a∗1

2 sin(θ∗))a∗

0a∗1 sin(θ∗)

+ 12a′13a′

23(iω0 − 1 − 2a∗1

2 sin(θ∗))2

�22a∗

02a∗

12 sin2(θ∗)

,

�1 = det(A) and �2 = det(2iω0 E − A).

The roots of the characteristic equation (6) at γ = γ0 = 1.3215,δ = −6 are λ1,2 = ±iω0 = 22.7229i and λ3 = −0.643. At that pointRe( dλ

dγ |λ=iω0,γ =γ0 ) �= 0, which shows that the system (5) undergoesa Hopf bifurcation. The Hopf bifurcation of a system implies therearises a limit cycle. The stability of this limit cycle depends uponthe sign of first Lyapunov coefficient. The first Lyapunov coefficientat this parameter values is l1(0) = 4.0062 where a13 = 36.1555,

a23 = 22.2395, �1 = −332.0155, a′13 = 36.1555(1 − i), a′

23 =−22.2395 − 29.3895i, �2 = −995.99 − 70876.0i, η = 324.09 +55.908i and η′ = 4.0390 − 55.9078i. With increase of parameterγ we get another two Hopf points and a limit point. The differentsituations that arise and their corresponding Lyapunov coefficientsare shown in Table 1.

5. Conclusion

In our above analysis we have shown how a new type of dy-namical system is generated from the Galerkin type truncation ofthe DNLS equation describing the Alfven wave propagation in amagnetized plasma. This new set of equation has a rich chaoticstructure, showing both bifurcation (saddle and period-doubling)and Hopf bifurcation for selected values of the plasma parameters.As a consistency check we have also analyzed the same problemthrough the continuation program MATCONT [32,33]. The charac-teristics of limit cycle and general bifurcation may be exploredusing software package ‘MATCONT’. This package is a collectionof numerical algorithms implemented as a Matlab toolbox for thedetection, continuation and identification of limit cycles and var-ious types of bifurcation. This is actually done by studying thechange in the eigenvalue of the Jacobian matrix and also followingthe continuation algorithm. In this package we use prediction–correction continuation algorithm based on the pseudo inverse ofMoore–Penrose matrix for computing the curves of equilibria, limitpoints (LP), Hopf-bifurcations (H), along with fold bifurcation pointof limit cycles (LPC), etc.

A. Ray et al. / Physics Letters A 372 (2008) 5329–5335 5333

(a)

(b)

(c)

Fig. 4. Bifurcation continuation with respect to γ of (a) a1, (b) a2, (c) φ at δ = −6.0, where H = Hopf bifurcation, LP = Limit point.

5334 A. Ray et al. / Physics Letters A 372 (2008) 5329–5335

(a)

(b)

Fig. 5. (a) Family of limit cycle, starting from subcritical Hopf bifurcation point. (b) Variation of period with the change of bifurcation parameter γ , where LPC = Limit pointcycle.

In Fig. 4(a), we display one such output of MATCONT, show-ing the variation of the variable a1, with respect to γ , whichclearly exhibited the change from Hopf to limit point as γvaries. Figs. 4(b) and 4(c) show the bifurcation continuationof the variables a2 and φ respectively with respect to γ . Thelimit cycle starting from the Hopf bifurcation in the (φ,a1)

plane is shown in Fig. 5(a). Furthermore the variation of pe-riod can be obtained and shown in Fig. 5(b), with respect to γ .The chaotic scenario presented above is actually very muchuseful in the choice of plasma parameter values which willlead to a particular form of physical situation in actual experi-ments.

A. Ray et al. / Physics Letters A 372 (2008) 5329–5335 5335

Acknowledgements

The authors are very thankful to the anonymous referees fortheir significant comment and suggestions.

References

[1] F. Haas, L.G. Garcia, J. Goedert, G. Manfredi, Phys. Plasmas 10 (2003) 3858.[2] L.G. Garcia, F. Haas, L.P.L. de Oliveira, J. Goedert, Phys. Plasmas 12 (2005) 1230.[3] M. Marklund, Phys. Plasmas 12 (2005) 082110.[4] L. Stenflo, P.K. Shukla, M. Marklund, Europhys. Lett. 74 (2006) 844.[5] P.K. Shukla, L. Stenflo, New J. Phys. 8 (2006) 111.[6] P.K. Shukla, L. Stenflo, Phys. Lett. A 357 (2006) 229.[7] Y. Voitenko, M. Goossens, O. Sirenko, A.C.-L. Chian, Astron. Astrophys. 409

(2003) 331.[8] A.C.-L. Chian, F.A. Borotto, S.R. Lopes, J.R. Abalde, Planetary Space Sci. 48 (2000)

9.[9] A.C.-L. Chian, E.L. Rempel, F.A. Borotto, Nonlinear Process. Geophys. 9 (2002)

435.[10] R.Z. Sagdeev, A.A. Galeev, Nonlinear Plasma Theory, Benjamin, New York, 1969.[11] A. Hasegawa, C. Uberoi, The Alfven wave. DOE Review Series-Advances in Fu-

sion Science and Engineering, US Department of Energy, Washington, DC, 1982.[12] V.I. Petviashvili, O.A. Pokhotelov, Solitary Waves in Plasmas and in the Atmo-

sphere, Gordon and Breach, Philadelphia, 1992.[13] P.K. Shukla (Ed.), Phys. Scr. T (2004) 113.[14] V.N. Fedun, A.K. Yukhimuk, A.D. Voitsekhocskaya, J. Plasma Phys. 70 (2004)

699.

[15] Y.M. Voitenko, M. Goossens, Phys. Rev. Lett. 94 (2005) 135003.[16] P.K. Shukla, L. Stenflo, Phys. Plasmas 12 (2005) 084502.[17] G. Brodin, L. Stenflo, P.K. Shukla, physics/0605169, J. Plasma Phys., in press.[18] D. Ghosh, P. Saha, A. Roy Chowdhury, Commun. Nonlinear Sci. Numer. Simul. 12

(2007) 928.[19] A.P. Misra, K. Roy Chowdhury, A. Roy Chowdhury, Phys. Plasmas 14 (2007)

012110.[20] A.P. Misra, A. Roy Chowdhury, K. Roy Chowdhury, Phys. Lett. A 323 (2004) 110.[21] R.A. Miranda, E.L. Rempel, A.C.-L. Chian, F.A. Borotto, J. Atmos. Solar-Terrestrial

Phys. 67 (2005) 1852.[22] E.L. Rempel, A.C.-L. Chian, E.E.N. Macau, R.R. Rosa, Physica D 199 (2004) 407.[23] A.P. Misra, D. Ghosh, A. Roy Chowdhury, Phys. Lett. A 372 (2008) 1469.[24] A.P. Misra, A. Roy Chowdhury, Phys. Plasmas 13 (2006) 072305.[25] A.P. Misra, A. Roy Chowdhury, Eur. Phys. J. D 39 (2006) 49.[26] D. Ghosh, A. Roy Chowdhury, ASME, J. Compt. Nonlinear Dynam. 2 (2007) 267.[27] J.H. Curry, An Algorithm for Finding Closed Orbits, Lecture Notes in Mathemat-

ics, vol. 819, Springer, Berlin/Heidelberg, 1980.[28] A. Dhooge, W. Govaerts, Y.A. Kuznetsov, ACM Trans. Math. Software 29 (2003)

141.[29] A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Physica D 16 (1985) 285.[30] A.N. Sarkovskii, Ukr. Mat. Z. 16 (1) (1964) 61.[31] T.-Y. Li, J.A. Yorke, Am. Math. Mon. 82 (12) (1975) 985.[32] Yu.A. Kuznetsov, Elements of Applied Bifurcations Theory, Springer-Verlag,

1995.[33] W. Govaerts, Yu.A. Kuznetsov, MATCONT – Continuation Software in Matlab,

http://www.matcont.ugent.be.