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See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/231154201 Renormalization Group for nonlinear oscillators in the absence of linear restoring force Article in EPL (Europhysics Letters) · October 2010 DOI: 10.1209/0295-5075/91/60004 CITATIONS READS 12 197 2 authors: Amartya Sarkar Jayanta Bhattacharjee Fritz Haber Institute of the Max Planck Society Harish-Chandra Research Institute 12 PUBLICATIONS 104 CITATIONS 303 PUBLICATIONS 3,481 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: Interfacial Instability View project Non-linear Dynamics View project All content following this page was uploaded by Amartya Sarkar on 11 August 2016. The user has requested enhancement of the downloaded file. SEE PROFILE Home Search Collections Journals About Contact us My IOPscience Renormalization Group for nonlinear oscillators in the absence of linear restoring force This content has been downloaded from IOPscience. Please scroll down to see the full text. 2010 EPL 91 60004 (http://iopscience.iop.org/0295-5075/91/6/60004) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 141.14.64.64 This content was downloaded on 11/08/2016 at 08:16 Please note that terms and conditions apply. September 2010 EPL, 91 (2010) 60004 doi: 10.1209/0295-5075/91/60004 www.epljournal.org Renormalization Group for nonlinear oscillators in the absence of linear restoring force A. Sarkar(a) and J. K. Bhattacharjee Department of Theoretical Sciences, S. N. Bose National Centre for Basic Sciences - Salt lake, Kolkata 700098, India received 7 July 2010; accepted in ﬁnal form 8 September 2010 published online 14 October 2010 PACS PACS PACS 05.10.Cc – Renormalization group methods 47.20.Ky – Nonlinearity, bifurcation, and symmetry breaking 02.30.Mv – Approximations and expansions Abstract – Perturbative Renormalization Group (RG) has been very useful in probing periodic orbits in two-dimensional dynamical systems (Sarkar A., Bhattacharjee J. K., Chakraborty S. and Banerjee D., arXiv:1005.2858v1 (2010)). The method relies on ﬁnding a linear center, around which perturbation analysis is done. However it is not obvious as to how systems devoid of any linear terms may be approached using this method. We propose here how RG can be done even in the absence of linear terms. We successfully apply the method to extract correct results for a variant of the second-order Riccati equation. In this variant the periodic orbit disappears as a parameter is varied. Our RG captures this disappearance correctly. We have also applied the technique successfully on the force-free Van der Pol-Duﬃng oscillator. c EPLA, 2010 Copyright Renormalization Group in the study of nonlinear ordinary diﬀerential equations was introduced by Chen et al. [1]. It has been a very eﬀective tool, since it works by analyzing a direct perturbation expansion. Being perturbative in nature, it works around a linear dynamical system which has a center [2] —a family of initial condition-dependent periodic orbits. We ask the following question: What if the dynamical system does not have a linear part? As an example, for the usual nonlinear oscillator, having the equation of motion ẍ + ω 2 x + λx3 = 0, the part ẍ + ω 2 x = 0 serves as the linear system with a center, about which the perturbation is done. The oscillator ẍ + λx3 = 0, on the other hand, has no linear part about which perturbation can be carried out. We ﬁrst establish how the RG can be used in this case, where apparently no perturbative technique is feasible. Having done that, we consider a generalization of the second-order Riccati equation [3,4]. We write this damped oscillator as ẍ + βλxẋ + λ2 x3 = 0, where λ is a coupling constant and β a dimensionless number. This diﬀerential equation, with slight variations has occurred widely in literature under several names and in several studies, like the generalized Emden equation [5–8], Lienard-type oscillator [9] and has been the subject of extensive study by Leach et al. [10], Mahomed and Leach [11], Bouquet (a) E-mail: et al. [12], Lemmer et al. [13], etc. The second-order Riccati equation has β = 3. This oscillator has no linear part and consequently not amenable to perturbation theory. For β = 3 the equation is exactly solvable and does not show oscillatory behavior [14]. For β 1, however, it is known to have periodic solutions [14]. We use our modiﬁed RG to predict that the oscillatory behavior seen for β 1 disappears through a vanishing of the frequency of oscillation (time period diverges). This happens at a critical value βc , for which a second-order calculation yields the value βc = 75/11. We carry out a numerical integration which yields βc = 2.82, in very good agreement with our predictions. We further show that for β close to βc , the time period diverges as (β − βc )−1/2 . The numerical results bear out our analytical results. We further illustrate the eﬀectiveness of our modiﬁed RG methodology applying it on a variant of the so-called Van der Pol-Duﬃng oscillator (ẍ + ẋ(x2 − 1) + λx3 = 0), which again is devoid of any linear terms. We successfully apply the modiﬁed RG method to extract the correct dependence of the time period on the parameter λ. We begin by demonstrating how the perturbative RG works for the nonlinear oscillator, ẍ + ω 2 x + λx3 = 0. (1) Perturbation theory begins with the expansion x = x0 + λx1 + λ2 x2 + · · · . amarta345@bose.res.in 60004-p1 (2) A. Sarkar and J. K. Bhattacharjee Inserting the above series in eq. (1) and equating the which integrate to A = A0 and Θ = 3λA2 τ /8ω + Θ0 . coeﬃcient of λn to zero for all n, we have Inserting in eq. (12), we have O(λ0 ) : ẍ0 + ω 2 x0 = 0, (3) 3λA30 τ + Θ0 x(t) = A0 cos ωt + 8ω O(λ1 ) : ẍ1 + ω 2 x1 = −x30 , (4) 2 3A − 0 λ(t − τ ) sin(ωt + Θ) (5) O(λ2 ) : ẍ2 + ω 2 x2 = −3x20 x1 . 8ω 3 λA 0 If we take the initial condition as x(t0 ) = A0 and ẋ(t0 ) = 0, + {cos 3(ωt + Θ) − cos(ωt + Θ)}. (15) 32ω 2 we have x0 = A0 cos(ωt + Θ0 ) and solving for x1 from ẍ1 + ω 2 x1 = −A30 cos3 (ωt + Θ0 ) A3 = − 0 (3 cos(ωt + Θ0 ) + cos 3(ωt + Θ0 )) , (6) 4 The ﬁnal removal of the τ -dependence is done through the choice τ = t and this gives λA3 x(t) = A0 cos(Ωt + Θ0 ) + cos 3(Ωt + Θ0 ) 2 32Ω (16) − cos(Ωt + Θ0 ) + O(λ2 ), we get as solutions of the system of eqs. (3) and (4) up to O(λ) 3 A0 where x = A0 cos(ωt + Θ0 ) + λ {cos 3(ωt + Θ0 ) 3λA2 32ω 2 Ω = ω + + O(λ2 ). (17) 8ω 3A20 Θ0 − cos(ωt + Θ0 )} − t+ sin(ωt + Θ0 ) , (7) This is the standard result of amplitude-dependent oscilla8ω ω tor frequency [15] for this oscillator, which as shown above where Θ0 = −ωt0 . The perturbative series solution is easily obtained by perturbative RG calculations. not important from here diverges when t t0 . In order to handle this divergence, The next issue that we need to address is what happens we introduce an arbitrary time τ and split the interval when ω = 0 and there is no obvious way doing perturbation t + Θ0 /ω as t − τ + τ + Θ0 /ω. We now exploit the fact theory. A direct integration allows us to obtain the that for a given trajectory, the present solution is inde- oscillation frequency [15] in this case which turns out to pendent of where one puts the initial condition. Any be point on the trajectory could have served as the initial √ Γ(3/4) 1/4 1/4 condition and accordingly, we take a new initial condition (λE) 1.18 (λE) , (18) Ω0 = (2 π) Γ(1/4) at t = τ , writing A0 (t0 ) = A(τ )Z1 (τ, t0 ), (8) Θ0 (t0 ) = Θ0 + Z2 (τ, t0 ), (9) where E is total energy of the oscillating particle. Our aim is to modify the perturbative RG used in eq. (1) to handle this situation. To this end, we write eq. (1) (with ω = 0) as (19) ẍ + αλx2 x + λ x3 − αλx2 x = 0, where the renormalization factors Z1 and Z2 are designed to remove the divergence coming from the “past”. This is done perturbatively by expanding the renormalization where x2 is the average of x(t)2 over a cycle. In the factors as above equation α is a constant number which we need to (10) ﬁnd out. If we ignore the term in parenthesis, we have an Z1 = 1 + λα1 + λ2 α2 + · · · , equivalent linear oscillator with frequency (11) Z2 = λβ1 + λ2 β2 + · · · . The choice of α1 = 0 and β1 = (3A2 τ /8ω + Θ0 /ω) removes the divergent contribution of the past and we are left with Ω2 = αλx2 . (20) If A is the amplitude of motion, i.e. ẋ = 0 at x = A, then E = 14 λA4 . Clearly, our task is to ﬁnd “α” and we proceed to do it by treating λ(x3 − αλx2 x) as the perturbation. The introduction of the term λαx2 x has the advantage of making the parameter “α” dimensionless and further makes the perturbation term λ(x3 − αλx2 x) “small”, Imposing the condition that x(t) has to be independent of in a sense i.e. a certain part of x3 is subtracted. This τ leads to the ﬂow equations replacement of x3 by x2 x is generally called equivalent linearization [15,16]. In most cases it is used as a basis dA = 0, (13) for self-consistent treatment, however, here we use it for a dτ RG-based perturbation theory. Now, rewriting eq. (19) as dΘ 3λA2 = , (14) ẍ + Ω2 x + λ x3 − αλx2 x = 0, (21) dτ 8ω 3A3 λ(t − τ ) sin(ωt + Θ) x(t) = A(τ ) cos(ωt + Θ) − 8ω λA3 + {cos 3(ωt + Θ) − cos(ωt + Θ)} . (12) 32ω 2 60004-p2 Renormalization Group for nonlinear oscillators in the absence of linear restoring force we expand exactly as in eq. (2) and obtain at diﬀerent We will now show that carried out to the next order orders of λ this method actually gives us an improvement. In the process, the method of carrying out higher-order calcuẍ0 + Ω2 x0 = 0, (22) lation becomes apparent. At O(λ2 ), we write eq. (24) as 2 (23) ẍ1 + Ω2 x1 = − −x30 − αx20 x0 , A3 A 2 (1 + cos 2Φ) x = − 3 (cos 3Φ − cos Φ) ẍ + Ω 2 2 32Ω2 ẍ2 + Ω2 x2 = − 3x20 x1 − αx1 x20 − 2αx0 x1 x0 . (24) A3 With the initial condition x = A0 and ẋ = 0 at t = t0 , we −α (cos 3Φ − cos Φ) 32Ω2 have A4 (25) x0 = A0 cos(Ωt + θ0 ). −2αA cos Φ (cos 3Φ − cos Φ) 32Ω2 At O(λ), we get αA5 3A5 αA5 3 = − cos Φ + + cos Φ 3A0 A3 64Ω2 64Ω2 32Ω2 cos(Ωt + θ0 ) + 0 cos 3(Ωt + θ0 ) ẍ1 + Ω2 x1 = − 4 4 + higher harmonics αA30 cos(Ωt + θ0 ) , − (26) 2 3(1 − α)A5 = cos Φ + higher harmonics, (33) 64Ω2 leading to where Φ = Ωt + θ. It should be clear from the derivation of A3 3 α the amplitude equation in eq. (13) that only the resonating x1 = − 0 − ) t sin(Ωt + θ0 2Ω 4 2 terms matter and not the higher harmonics. Also noting 2 2 A30 + [cos(Ωt + θ0 ) − cos 3(Ωt + θ0 )] . (27) that Ω = αλA /2 to the lowest order, we can write the 32Ω2 solution to eq. (33), right up to the second order as In exact analogy with eq. (1), we can now write x(t), up 3 1−α A30 3 α − − cos(Ωt + θ) − λ x(t) = A 0 to O(λ) 2Ω 4 2 32 α 3 ) + · · · (34) × cos(Ωt + θ A 0 {cos(Ωt + θ0 ) x(t) = A0 cos(Ωt + θ0 ) + λ 2 32Ω which leads to A3 3 α θ0 − − cos 3(Ωt + θ0 )} − t+ 3 1 3 1 2Ω 4 2 ω + α2 = 1+ . (35) 4 32 2 24 (28) × sin(Ωt + θ0 ) + O(λ2 ). The actual frequency to this order comes from The renormalization constants are introduced in an identical fashion as before and the requirement that x(t) has to be independent of τ yields dA = 0, dτ 3 α dθ = − . dτ 4 2 (29) (30) The frequency Ω as deﬁned has to remain unaﬀected. And this happens if eq. (30) has a ﬁxed point which is achieved for α = 3/2. Thus the solution x(t) is A cos Ωt + O(λ) and hence x2 = A2 /2 = (E/λ)1/2 . The frequency of the nonlinear oscillator at this order is found from Ω2 = (3/2)λ(E/λ)1/2 and is given by 1/2 3 (λE)1/4 . Ω= 2 Ω2 = αλx2 = αλx20 + 2λx0 x1 A2 + 2αλ2 x0 x1 = αλ 2 A2 αλA2 = αλ − λA2 using eqs. (25) and (32) 2 32Ω2 A2 λA2 − using eq. (20) αλ 2 16 αλA2 1 1− 2 12 1 1 1/2 3 = (λE) 1+ 1− 2 24 12 3 1 +··· , (36) (λE)1/2 1− 2 24 leading to (31) Ω= This is 3% above the exact result shown in eq. (18). As in eq. (16), the ﬁnal form of x1 is A3 x1 = [cos 3(Ωt + θ0 ) − cos(Ωt + θ0 )]. 32Ω2 (32) 3 1 (λE)1/4 1 − +··· , 2 48 (37) within 1% of the actual result. This shows how our perturbation theory can be systematically improved and how eﬀective it is in giving reasonable answers. 60004-p3 A. Sarkar and J. K. Bhattacharjee We now turn to the generalized second-order Riccati equaAs before we deﬁne the renormalization constants Z1 tion. This equation has been widely studied by both and Z2 as in eqs. (8) and (9). And a procedure identical mathematicians and physicists for more than a century. to that which yielded eqs. (13) and (14), now leads to This equation arises in a variety of mathematical probdA lems [8,17]. And various mathematicians have studied this = 0, (50) dτ equation and its variants for speciﬁc choices of parame ters [17–20]. Time and again it has been shown to arise 3 α β 2 A2 dθ in a number of physical problems —like modeling fusion =λ − − . (51) dτ 4 2 12 2Ω of pellets [21], one-dimensional analogue of Yang-Mill’s 2 boson gauge theory [3], etc. We write down the equation as The frequency remains at Ω, provided, α = 32 1 − β9 . (38) By deﬁnition α > 0 and hence the restriction β < 3 for a ẍ + βλxẋ + λ2 x3 = 0 periodic state to exist. This is the lowest non-trivial order. and carry out the perturbation calculation as we have We need to take into consideration the next order as we detailed before. The expansion of x is did for the value of α when β was equal to zero. For this we turn to eq. (44) and using eqs. (46), (47) 2 3 4 x = x0 + λx1 + λ x2 + λ x3 + λ x4 + · · · . (39) and (49) obtain As done for the earlier example, we introduce a dimenβ(1 − β 2 )A4 βA4 2 sionless parameter α and rewrite eq. (38) as α − 3 + 2β sin 2Φ + sin 3Φ x3 = 36Ω3 32Ω3 3 2 2 2 2 ẍ + βλxẋ + λ αx x + λ x − αx x = 0, or 13β 2 βA4 1 3 2 2 2 + − sin 4Φ. (52) ẍ + Ω x = −βλxẋ − λ x − αx x , (40) Ω3 144 × 15 80 where Ω2 = λ2 αx2 . We ﬁnd at diﬀerent orders 2 ẍ0 + Ω x0 = 0, ẍ2 + Ω x2 = − (44) (45) In the second term of the RHS of the above equation, we insert the lowest-order value of λA2 /Ω2 , namely 2/α. With this substitution and noting that, to the lowest order α is (9 − β 2 )/6, we ﬁnally arrive at (41) ẍ1 + Ω2 x1 = −βx0 ẋ0 , 2 Turning to eq. (45) and ﬁnding the coeﬃcient of cos(Ωt + θ), it is immediately possible to write the RG phase ﬂow for eq. (38) correct to O(λ2 ) as 3 α β 2 A2 dθ =λ − − dτ 4 2 12 2Ω λ2 A4 α 37β 2 (23β 2 + 9)(9 − β 2 ) + − 3+ . (53) 2Ω3 64 9 96 × 18 x30 − αx20 x0 (42) − βx1 ẋ0 − βx0 ẋ1 , (43) ẍ3 + Ω2 x3 = − 3x20 x1 − αx1 x20 − 2αx0 x1 x0 −β(x2 ẋ0 + ẋ2 x0 + ẋ1 x1 ), ẍ4 + Ω2 x4 = β(x0 ẋ3 + x1 ẋ2 + x2 ẋ1 + x0 ẋ3 ) + − 3x0 x21 − 3x20 x2 + αx2 x20 +2αx1 x1 x0 + αx0 x0 x2 . We can now write down the solutions at the ﬁrst two orders as x0 = A0 cos(Ωt + θ0 ), x1 = −β (46) A20 [sin 2(Ωt + θ0 ) − 2 sin(Ωt + θ0 )]. (47) 6Ω The divergence comes at the next order 3 α β2 ẍ2 + Ω2 x2 = A30 − + + cos(Ωt + θ0 ) 4 2 12 + non-resonating terms. (48) dθ λA2 9 − β 2 − 6α 23β 2 + 9 3 37β 2 = − − + . (54) dτ 2Ω 12 144 32 288 The critical value βc of β up to which a positive α can be found to give dθ/dτ = 0 is βc = 75/11 2.61. We thus conclude from a calculation good to the ﬁfth order in amplitude, that a periodic state will be found in eq. (38) for β < βc (= 2.61) and that the time period will start to diverge as βc is approached. From the ﬁxed point of eq. (54) we ﬁnd α = (225 − 33β 2 )/144 and hence a time period of To order O(λ2 ), we have λβA20 [sin 2(Ωt + θ0 ) 6Ω λ2 A30 3 α β2 − 2 sin(Ωt + θ0 )] + − + + 2Ω 4 2 12 θ0 × t+ (49) sin(Ωt + θ0 ). Ω 2π 2π = ∝ T= 1/2 Ω [αλx2 ] x = A0 cos(Ωt + θ0 ) − 75 −β 11 −1/2 , (55) which diverges in a characteristic fashion, (βc − β)−1/2 , as β approaches βc . We have carried out the numerical simulation of eq. (38) for diﬀerent values of β. As β is increased, the time period 60004-p4 Renormalization Group for nonlinear oscillators in the absence of linear restoring force 0.2 Now we consider the so-called Van der Pol-Duﬃng oscillator without the forcing term, given by the equation β=0.65 β=1.00 β=1.30 β=1.95 β=2.60 0.15 ẍ + ẋ(x2 − 1) + λx3 = 0, Y 0.1 0.05 0 -0.05 -0.1 -1.5 -1 -0.5 0 0.5 1 1.5 X (56) where and λ are both small. This is often referred to as the Van der Pol-Duﬃng oscillator [22], in our case though there is no forcing term. This equation is yet another example of an oscillator without any linear term. As outlined above, to carry out perturbative RG we have to rewrite eq. (56) as ẍ + ẋ(x2 − 1) + αλx2 x + λ x3 − αx2 x = 0, or ẍ + Ω2 x = −ẋ(x2 − 1) − λ x3 − αx2 x , (57) where Ω is given by Fig. 1: Solutions in phase space for diﬀerent values of β. Ω2 = λαx2 . (58) We expand x as in eq. (39) and ﬁnd at diﬀerent orders increases as well and, as can be concluded from ﬁg. 1, it becomes extremely large as β approaches the critical value, 2.61. Figure 1 shows the phase space evolution of the oscillator for ﬁve diﬀerent values of β. We notice that the smaller the value of β, the larger is the amplitude of the periodic orbit. For ẋ < 0, i.e. the lower part of the orbit has a convexity, the tangent to which approaches asymptotically towards the x-axis as the value of β approaches the critical value βc = 2.61. As β approaches βc the velocity becomes slower and slower while crossing the y-axis (x = 0), and this ultimately leads to the time period divergence, through which the periodic solution vanishes. Leach et al. [10] have considered the Riccati equation in the form ẍ + xẋ + αx3 = 0 and have carried out extensive studies on its solution for diﬀerent values of α. They have used a self-similar transformation to obtain a diﬀerent form of the equation of motion in the new phase space (ζ = xt; η = ẋt2 ), where the potential (V (ζ) = αζ 4 /4 − ζ 3 /3 + ζ 2 ) is much more transparent to analysis. Further, for a value of α = 1/8, V (ζ) is seen to have a stationary point of inﬂection which corresponds to the critical value where periodic solution vanishes. Recently, Chandrasekhar et al. [8] have found the general solutions to the second-order Riccati equation, ẍ + αxẋ + βx3 = 0, for arbitrary values of α and β. Choosing suitable canonical transformations for each of the three cases, α2 < 8β, α2 = 8β, and α2 > 8β, they write down their respective general solutions. From their results as well, it is apparent that the transition from periodic to non-periodic solutions occurs at α2 = 8β. Noting that α as deﬁned by Leach et al. is related to β deﬁned here as β = 1/α, we see that the critical value according to both them √ and Chandrasekhar et al. is βc = 8 = 2.83. Numerical simulations put the critical value of β at 2.82. Keeping in mind that our approach is perturbative, the value of βc = 2.61 obtained through perturbative RG up to O(λ2 ) is in fairly good agreement with previous literature and numerics. O(0 λ0 ) : ẍ0 + Ω2 x0 = 0, O(1 λ0 ) : ẍ1 + Ω2 x1 = −ẋ0 (x20 − 1), 0 1 O( λ ) : 2 ẍ2 + Ω x2 = − (59) x30 − αx20 x0 (60) . (61) Now we can write down the solutions to the above set of equations and up to ﬁrst order in perturbation parameters we have 3 A0 (sin 3Ωt − 3 sin Ωt) x(t) = A0 cos Ωt − 32Ω 3 A0 A0 A20 + −1 +λ (cos 3Ωt − 3 cos Ωt) 2 4 32Ω2 A3 3 α − − 0 t sin Ωt . (62) 2Ω 4 2 Proceeding from now on as earlier, we arrive at the RG ﬂow equations, right up to ﬁrst order in the perturbation parameters and λ, given by A0 A20 dA = − −1 , (63) dτ 2 4 dθ A30 3 α =λ − . (64) dτ 2Ω 4 2 The frequency remains at Ω, provided the RHS of eq. (64) vanishes and we have, α = 3/2. From the amplitude ﬂow, eq. (64), it can be concluded that the oscillator exhibits limit cycle oscillations with √radius 2. So from eq. (58), we have Ω = λα(A2 /2) = 3λ. Numerical simulations of the oscillator conﬁrm the above results as is quite evident from ﬁg. 2. The ﬁgure shows time periods for diﬀerent values of λ. And it is clear that the simulated data compares extremely well with the 1st-order perturbative RG result. To conclude, we again emphasize that although doing perturbative RG in systems with linear terms is obvious, its not so in cases without any linear terms. We 60004-p5 A. Sarkar and J. K. Bhattacharjee 60 Simulated Data Theoritical Data 55 T (in arbitrary units) 50 45 40 35 30 25 20 15 10 0 0.01 0.02 0.03 0.04 0.05 λ 0.06 0.07 0.08 0.09 0.1 Fig. 2: Dependence of time period on λ. Comparison between RG result and simulations. have in this paper borrowed ideas from equivalent linearization [14–16,23] and combined with RG approach to successfully reproduce results in few such cases. We have shown that the disappearance of the periodic solution to the generalized second-order Riccati equation is satisfactorily explained with the results obtained from the RG analysis. We also capture the correct behavior of the limit cycle appearing in a force-free Van der Pol-Duﬃng oscillator. Here it can be said that this technique of doing perturbative RG in the absence of linear terms may ﬁnd applications in a wide range of problems where the absence of linear terms makes them inaccessible to perturbative techniques. REFERENCES [1] Chen L. Y., Goldenfeld N. and Oono Y., Phys. Rev Lett., 73 (1994) 1311; Chen L.Y., Goldenfeld N. and Oono Y., Phys. Rev. E, 54 (1996) 376. [2] Sarkar A., Bhattacharjee J. K., Chakraborty S. and Banerjee D., arXiv:1005.2858v1 (2010). [3] Chisholm J. S. R. and Common A. K., J. Phys. A: Math. Gen., 20 (1987) 5459. [4] Carinena J. F. and Ranada M. F., J. Math. Phys., 46 (2005) 062703. [5] Leach P. G. L., J. Math. Phys., 26 (1985) 2510. [6] Chandrasekar V. K., Senthilvelan M., Anjan Kundu and Lakshmanan M., J. Phys. A: Math. Gen., 39 (2006) 9743. [7] Chandrasekar V. K., Pandey S. N., Senthilvelan M. and Lakshmanan M., Chaos, Solitons Fractals, 26 (2005) 1399. [8] Chandrasekhar V. K., Senthilvelan M. and Lakshmanan M., J. Phys. A: Math. Theor., 40 (2007) 4717. [9] Chandrasekhar V. K., Senthilvelan M. and Lakshmanan M., Phys. Rev. E, 72 (2005) 066203. [10] Leach P. G. L., Feix M. R. and Bouquet S. E., J. Math. Phys., 29 (1988) 2563. [11] Mahomed F. M. and Leach P. G. L., Quaest. Math., 8 (1985) 241. [12] Bouquet S. E., Feix M. R. and Leach P. G. L., J. Math. Phys., 32 (1991) 1480. [13] Lemmer R. L. and Leach P. G. L., J. Phys. A: Math. Gen., 26 (1993) 5017. [14] Jordan D. W. and Smith P. A., Nonlinear Ordinary Diﬀerential Equations: An Introduction to Dynamical Systems (Oxford University Press, New York) 1999. [15] Bhattacharjee J. K., Malik A. K. and Chakraborty S., Indian J. Phys., 81 (2007) 1115. [16] Banerjee K., Bhattacharjee J. K. and Mani H. S., Phys. Rev. A, 30 (1984) 1118. [17] Ince E. L., Ordinary Diﬀerential Equations (Dover, New York) 1956. [18] Davis H. T., Introduction to Nonlinear Diﬀerential and Integral Equations (Dover, New York) 1962. [19] Painleve P., Acta Math., 25 (1902) 1. [20] Golubev V. V., Lectures on Analytical Theory of Diﬀerential Equations (Gostekhizdat, Moscow) 1950. [21] Erwin V. J., Ames W. F. and Adams E., Wave Phenomena: Modern Theory and Applications, edited by Rogers C. and Moodie J. B. (North-Holland, Amsterdam) 1984. [22] Chandrasekar V. K., Senthilvelan M. and Lakshmanan M., J. Phys. A: Math. Gen., 37 (2004) 4527. [23] Strogatz Steven H., Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering (Westview Press, Reading, Mass.) 1994. 60004-p6 View publication stats