Complex Envelope Variable Approximation in Nonlinear Dynamics
CComplex Envelope Variable Approximation in Nonlinear Dynamics.
Valeri V. Smirnov ∗ and Leonid I. Manevitch † Department of Polymers and Composite Materials,Federal Research Center for Chemical Physics, RAS (Dated: April 20, 2020)We present the Complex Envelope Variable Approximation (CEVA) as the very useful and com-pact method for the analysis of the essentially nonlinear dynamical systems. It allows us to studyboth the stationary and non-stationary dynamics even in the cases, when any small parameters areabsent in the initial problem. It is notable that the CEVA admits the analysis of the nonlinearnormal modes and their resonant interactions in the discrete systems without any restrictions onthe oscillation amplitudes. In this paper we formulate the CEVA’s formalism and demonstrate somenon-trivial examples of its application. The advantages of the method and possible problems arebriefly discussed.
I. INTRODUCTION
Modern level of scientific researches and technologiesmore frequently lead to the problems, the core of whichassociates with the nonlinear dynamical processes. Itconcerns with the macro- as well as micro- and nano-scale phenomena. If the first is the traditional scope ofthe nonlinear dynamics, the second is the new impetuousdeveloped direction of the nonlinear studies. Generallyspeaking the nonlinear dynamics takes part in so largenumber of the sciences and technologies that the factleads to formation of some inter-disciplinary ”nonlinearscience” . However, in spite of that the nonlinear dy-namics is required in everywhere, these problems remainvery complex and are far from the completion. Diversityand complexity of the nonlinear dynamics lead to the te-dious and to frequently ambiguous approximations. Moreone difficulty originates from that the numerical proce-dures do not allow to extrapolate the results obtained inone task into another one. This is one of the reasons whythe interest to the problems, which would seem multiplestudied, retain up to date.There are many analytical and numerical methods,which cover the analysis of the nonlinear problems ? –11 .The significant part of the most widely used approachesto the study of the nonlinear problems are the meth-ods, which based on the asymptotic expansion of thesolution into series of a small parameter. In particu-lar, the method of the multiscale expansion is based onthe separation of the time scales, the ratio of which isdetermined by a small parameter . An elementaryexample can be found in the beating phenomenon inthe system of the weakly coupled identical oscillators.The process of the energy transfer from one oscillatorto another one is determined by the frequency gap be-tween in- and out-of-phase modes. In such a formula-tion this problem nearly relates to the slowly varyingenvelope approximation, which has been considered byVan der Pol . If the oscillators have the nonlinearcharacteristics, not only the slow energy transfer, butthe energy capture on one of oscillators (energy local-ization) becomes possible . Here one should empha-size that the description of the energy transfer and lo- calization has been made in the terms of the complexrepresentation of the variables ? , that is near analogueto the second quantization formalism . Such an ap-proach in the combination with the multiscale expansionturns out to be very successful in the investigation of thewide class of the nonlinear problems: the coupled nonlin-ear oscillators , the forced nonlinear oscillators ,the energy transfer and localization in the 1D nonlinearlattices , the mode coupling and energy localizationin the carbon nanotubes , the synchronization of theself-excited oscillators and the classic analogue of thesuperradiant quantum transition , the nonlinear passivecontrol and energy sink , the problem of rotation sta-bility of coupled pendulums . In a number of cases thecomplex representation of the variables allows us to findthe stationary single-frequency solution (nonlinear nor-mal modes) for the essentially nonlinear systems with-out any assumptions about oscillation amplitudes andwithout using any small parameter . Thus, the largeadvantage of this approach is that we obtain the mainapproximation, which satisfies to the initial essentiallynonlinear problem. The non-stationary dynamics can bestudied in terms of the slowly changed envelopes. Oneshould note that no restriction on the varying amplitudeof the non-stationary oscillations arises, but the main re-quirement is a frequency closeness of the non-stationaryand stationary solutions. In particular, such conditionsallow us to study the nonlinear mode interactions in thediscrete extended systems, if the lengths of the latter arelarge enough . In such a case the slow time scale isnaturally appeared from the smallness of the inter-modefrequency gap.Generally speaking, the approach discussed below isa hybrid one. Actually, it is formally similar to theVan der Pol slow varying envelope approximation ,and, in some meaning, is close to the harmonic balancemethod . On the other hand the multiple scale expan-sion is the essential constituent of the consideration of thenon-stationary dynamics, but the post-factum revealingthe small parameter makes it close to semi-inverse meth-ods. Taking into account mentioned above we will re-fer to the discussed approach as the ”Complex EnvelopeVariable Approximation” (CEVA). The current work is a r X i v : . [ n li n . PS ] A p r aimed to describe the CEVA’s formalism in a general case(section II) and to demonstrate some examples, in whichthe CEVA’s advantages can be revealed (section III). Thesection IV contains the short discussion and conclusions. II. THE COMPLEX ENVELOPE VARIABLEAPPROXIMATION
Let us consider the nonlinear dynamical system theevolution of which is determined by the equation d udt + F ( u ) = 0 (1)where u is the function of time t and F is a nonlinearfunction of u . First of all we should note that we do notsingle out any parts, which contain a small parameter.As it was emphasized in Introduction we want to studythe slow processes which are not dictated by the externalfactors. In this way, it is convenient to introduce the newdynamical variables:Ψ = 1 √ (cid:18) √ ωu + i √ ω dudt (cid:19) , (2)where ω is the frequency, which should be determinedas the function of the oscillation amplitude. The inversetransformation is well known: u = 1 √ ω (Ψ + Ψ ∗ ) ; dudt = − i (cid:114) ω − Ψ ∗ ) (3)Using these variables one can rewrite equation (1) as fol-lows: i d Ψ dt − ω − Ψ ∗ ) − √ ω F (cid:18) √ ω (Ψ + Ψ ∗ ) (cid:19) = 0(4)In the number of the nonlinear systems we interest our-selves in the single-frequency stationary solution of equa-tion (1). In order to get it we suppose next form of thesolution: Ψ = ψe − iωt . (5)This representation corresponds to the first term of theFourier series and ψ is assumed as a constant. Expandingfunction F into the Taylor series and averaging equation(4) over the period 2 π/ω allows us to extract the secularterm. Finally we get the equation for function ψ in theform: ω ψ − √ ω ˜Φ ( ψ, ψ ∗ ; ω ) = 0 . (6)At first glance equation (6) is not more simple than theinitial one. However, for the number of the actual sys-tems function ˜Φ has the noteworthy structure:˜Φ ( ψ, ψ ∗ ; ω ) = ∞ (cid:88) k =0 c k (cid:32)(cid:114) ω (cid:33) k +1 | ψ | k ψ (7) FIG. 1. The amplitude-frequency dependence in accordanceto equation (12) (red solid curve) and exact value ω (blackdashed curve). As it is shown in Appendix, some ”good” nonlinearitiesadmit the representation of the infinite sum in equation(7) in the term of the special functions. Equation (6)should be considered as the amplitude-frequency rela-tion, taking into account the relationship between com-plex value ψ and the oscillation amplitude. The latterfollows from the first of equations (3) and expression (5).If A is the oscillation amplitude, the modulus of value ψ can be written as follows: | ψ | = (cid:114) ω A (8)In such a case, equation (6) should be written in the form: ω A − Φ( A ) = 0 . (9)This equation allows us to find oscillation frequency ω : ω = (cid:114) A Φ( A ) (10)This relation exhausts the stationary problem of the freeoscillations of the system with one degree of freedom.In order to illustrate this procedure efficiency, we findthe amplitude-frequency relation for the most knownnonlinear system - the pendulum. In such a case function F in equation (1) is sin u . It was shown early the re-spective ”secular” term in equation (6) is read as follows ω ψ − √ ω J (cid:32)(cid:114) ω | ψ | (cid:33) ψ | ψ | = 0 , (11)where J is the Bessel function of first order. It is easyto see that according to relation (8), the argument of theBessel function is the oscillation amplitude A . Therefore,expression (10) transforms into form ω = (cid:114) A J ( A ) . (12)Figure 1 shows the comparison of frequency (12) with theexact value ω = π/ K (sin ( A/ A = π . It is the expected result be-cause the motion with the amplitude A = π correspondsto the separatrix and evidently does not belong to thecategory of the single-frequency solutions (5).The stationary state with complex amplitude ψ andcorresponding frequency ω has the energy E = ω | ψ | − G ( ψ ; ω ) , (13)where function ˜Φ is coupled with G by the relation˜Φ = ∂G∂ψ ∗ (14)Let us consider energy (13) as the Hamilton functionof the system, which is parametrized by frequency ω : H = ω | ψ | − G ( ψ ; ω ) (15)The equation of motion can be obtained as follows: i dψdτ = − ∂H∂ψ ∗ . (16)The latter leads to the time-dependent version of equa-tion (6): i dψdτ + ω ψ − √ ω ˜Φ ( ψ ; ω ) = 0 (17)It is just the right time now to ask the question: what isa time scale of variable τ ? First of all, function ψ is theenvelope for single-frequency oscillations e − iωt , therefore,it should be slowly changing. In opposite case the aver-aging made above turns out to be invalid. So, we canonly consider the non-stationary motions with frequen-cies, those weakly distinct from the frequency of station-ary solutions. In such a sense, the development of non-stationary equations is similar to the slowly varying enve-lope approximation , and the slowness is determinedby the structure of the equations obtained. We shouldconsider a slow motion, but we are not bound by the val-ues of the amplitude variation. I.e., only the request isthat the specific time of the changing amplitude shouldbe essentially large than the oscillation period.Let us introduce the polar representation of function ψ . ψ = ae iδ (18)Variables a and δ form the canonical set for Hamil-ton function (15). Therefore, the equations of motion interms of polar variables should be represented in form: dadτ = − a ∂H∂δ ; dδdτ = 12 a ∂H∂a (19)We use the pendulum in order to make certain thatequations (19) actually describe the slow-time evolutionof the system under a small disturbance of the pendu-lum stationary oscillations. The Hamilton function ofthe pendulum can be written as follows: H = ω a − J (cid:32)(cid:114) ω a (cid:33) . (20) Let us ψ = ( a + α ) e iδ , where a is solution of equation(6) and α (cid:28) a is a disturbance. Taking into accountequations (19), we get dαdτ = 0 (21) dδdτ ≈ a (cid:32) ω a − √ ω J (cid:32)(cid:114) ω a (cid:33)(cid:33) +1 ωa J (cid:32)(cid:114) ω a (cid:33) α + J (cid:16)(cid:113) ω a (cid:17) √ aω / − J (cid:16)(cid:113) ω a (cid:17) a ω α The first of equations (21) shows that the amplitude ofthe disturbed motion is not change. The first term inthe right hand side of equations (21) is equal to zeroand varying of phase δ turn out to be proportional to∆ ω ∼ αdω/da . Due to that α = const , it means thatthe disturbed solution runs off with constant velocityfrom the initial solution: δ ∼ ∆ ω τ . One should em-phasize that the time scale of variable τ is controlled bythe smallness of the right hand side of equations (21).The latter can be determined by the smallness of eitherdisturbance’s amplitude ( α (cid:28) a ) or smoothness of theoscillation frequency ( dω/da (cid:28) ω/a ).As a conclusion of this section, let us check that equa-tions (19) correctly predict the frequency changing in thelimit of small amplitude a . Assuming a → ω → dδdτ ≈ α A . (22)The sign of the correction is positive because solution (5)contains e − iωt . Correction (22) accords with the expan-sion of exact pendulum frequency ω ≈ − A / III. EXAMPLES AND APPLICATIONSA. Forced damped oscillation of pendulum
The effect of the external forcing and dissipative pro-cesses often takes an important role in the dynamics ofthe nonlinear systems. Therefore, we start this sectionsconsidering the forced oscillations of the pendulum withthe viscous friction. Let us suppose that the pendulumundergoes the effect of external field F ( t ) = f cos ωt . Thestationary equation of motion can be written in terms ofcomplex variable ψ as follows: ω ψ − √ ω J (cid:32)(cid:114) ω | ψ | (cid:33) ψ | ψ | + i ν ψ = − f √ ω , (23)where ν is the coefficient of the viscous friction. Usingrelation (8), it is easy to find the amplitude-frequencyrelation for the non-dissipative system ( ν = 0): ω = 2 A ( J ( A ) − f ) . (24)However, function ψ becomes complex for non-zero fric-tion. Assuming ψ = x + iy , we should separate the realand imagine parts of equation (23). After some manipu-lations we can write the amplitude-frequency relation inthe form:2 ω (cid:0) x + y (cid:1) = A = f ν ω + ( ω − Ω ) (25) yx = tan δ = − νωω − Ω , where Ω = Ω( A ) = (cid:112) J ( A ) /A is the frequency of freeoscillations with the amplitude A . One should note thatthe first equation of (25) is the transcendent equationwith respect to amplitude A . It can be solved numeri-cally and the result is represented in figure 2. It is note-worthy that the amplitude-frequency relation (25) looksabsolutely similar to its linear analogue, with the dif-ference that the frequency of non-linear free oscillationswith amplitude A plays the role of own frequency of thelinear oscillator. The expression for phase shift δ also hasthe same form as for the linear system. B. Escape from a potential well
We illustrate the non-stationary nonlinear dynamicsconsidering the escape from potential well under effect ofsingle-frequency external field F = f cos ω t without fric-tion. The transition of the pendulum from oscillationsto the rotation gives the very clear example of such pro-cesses. Writing function ψ in polar form (18) one canascertain that the energy of forced oscillation is repre-sented as follows: H f = ω a − J (cid:32)(cid:114) ω a (cid:33) + af cos δ √ ω (26) FIG. 2. Amplitude-frequency relations (12, 24, 25) for thefree and forced oscillation of the pendulum without and withfriction (black, blue and red curves, respectively). Amplitudeof the external force f = 0 .
075 and friction ν = 0 . We would like to find what combinations of the force’sfrequency ω and amplitude f lead to the rotation of thependulum, if the initial conditions are zero. Such a prob-lem is non-trivial even in the case of the simple parabolicpotential . In order to determine the boundaries of os-cillations in the plane ”frequency - force amplitude” oneshould analyse the possible non-stationary trajectoriescorresponding to Hamilton form (26). Let us considerthe phase space in the terms { a, δ } assuming the fre-quency ω and force amplitude f as the parameters. Be-cause hamiltonian (26) does not contain the variables,which depends on the ”fast” time t , the stationary oscil-lations correspond to the stationary points on the phaseplane { δ, a } . There are three stationary states in the low-frequency region in Fig. 2 and only one stationary stateoccurs if the frequency is larger than some value ω ∗ . Thelatter can be found as the root of the equation: dωdA = f − AJ ( A ) √ A / (cid:112) J ( A ) − f = 0 (27)Figure 3 shows the phase portraits of the system (26)with the different values of frequency ω and the con-stant value of the force amplitude f . Figure 3(a) rep-resents the phase portrait with single stationary stateat the phase δ = 0. The thick blue curve, which passesthrough zero valued amplitude, separates the trajectoriesclosed around the stationary point from the transit-timeones. This trajectory is called the Limiting Phase Tra-jectory (LPT). In the problem under consideration theLPT describes the escape from the potential well, if themaximum of the LPT exceeds the limiting angle of theoscillations (i.e., π ). Such a case is observed in panel (b)of figure 3. The value of the threshold frequency can beevaluated from the condition H f ( a = 0 , δ = 0) = H f (cid:18) a = (cid:114) ω π, δ = 0 (cid:19) . (28)This condition corresponds to the high-frequency bound-ary of the escape from the well. One should notice that a bc dFIG. 3. The phase portraits of system (26) at force amplitude f = 0 . ω . Panels (a-d)correspond to ω = 0 . , . , . , .
75, respectively. The Limiting Phase Trajectories and separatrix are shown as thick blueand black dashed curves, respectively. Red points correspond to the stationary states. the phase portrait on panel (b) contains three stationarypoints, therefore, the respective frequency is smaller thenfrequency ω ∗ mentioned above. Decreasing frequency ω we can obtain the phase portrait depicted on panel (c) offigure 3. The specific feature of this phase portrait is thatthe LPT coincides with the separatrix crossed the unsta-ble stationary point at δ = π . No possibility to escapethe potential well at this frequency and at smaller ones(see fig. 3(d)) occurs because trajectory, which starts at a = 0 can not reach the limiting angle π . The frequencycorresponding to the phase portrait on panel (c) can beevaluated by solving of the equation H f ( a = 0 , δ = 0) = H f ( a = a u , δ = π ) , (29)where a u corresponds to the unstable stationary point.Thus solving equations (28) and (29), we can deter-mine the domain of the force’s frequency and amplitudes,where the escape from the potential well is possible. Itis noteworthy that, in order to do it, we do not needin solving the non-stationary equations (19), but we canfind the domain’s boundaries analysing the variation ofthe phase portrait at various values of f and ω . Nevertheless, if we want to estimate the time of theescape from the well, we need in the integration of thenon-stationary equation along the LPT. Actually, the es-cape time can be estimated as follows: T = (cid:90) T dt = (cid:90) √ ω π dada/dt (30)From the first of equations (19) we get dadt = f sin δ √ ω (31)Taking into account that the LPT passes through zeroamplitude, the expression for cos δ can be found fromenergy (26):cos δ = √ ωaf (cid:32) − ω a − J (cid:32) √ a √ ω (cid:33)(cid:33) (32)Finally, combining equations (30 – 32), we can write theperiod of the escape from the well as follows: T = (cid:90) π √ ωAdA (cid:114)(cid:16) − (cid:0) ω A (cid:1) − J ( A ) + f A + 1 (cid:17) (cid:16)(cid:0) ω A (cid:1) + J ( A ) + f A − (cid:17) (33) FIG. 4. The time of the escape from potential well on the ω − f plane. Contours correspond to the constant periods ofthe escape from the well, which are signed in the plot legendin right. The last expression can be estimated numerically at fixedvalues ω and f . Figure 4 shows the contours of the con-stant escape time, which have been calculated accord-ingly to expression (33). The uncoloured region corre-sponds to ( ω − f ) domain, where the transition to rota-tion of pendulum is unreachable. C. Instability of the rotation of coupled pendula
In this section we would like to perform the stabilityanalysis for the in-phase rotation of two coupled pendula.More detail description should be looked over .The energy of two coupled pendula is determined asfollows: H = (cid:88) j =1 , (cid:32) (cid:18) dq j dt (cid:19) + σ (1 − cos q j ) + β − cos ( q j − q − j )) (cid:33) . (34)(In order to allow the mutual rotation of the pendula, weassume 2 π − periodical interpendulum potential, which issimilar to the interaction of the coaxial arranged dipoles.)The equations of motion have the form d q j dt − β sin ( q − j − q j ) + σ sin q j = 0; j = 1 , . (35)Let us introduce the in-phase mode θ = ( q + q ) / θ = ( q − q ) /
2. The respec-tive equations of motion d θ dt + σ cos θ sin θ = 0 , (36) d θ dt + β sin 2 θ + σ cos θ sin θ = 0 . admit the exact solutions ( θ = θ ( t ) , θ = 0) and( θ = 0 , θ = θ ( t )).If energy E of in-phase motion exceeds the value 2 σ ,the pendula undergo the synchronous rotation with pe-riod T = 2 √ K (cid:0) σE (cid:1) √ E , (37)where K is the complete elliptic integral of the first kind.However, the numerical simulations show that the in-phase rotation turn out to be unstable at some valuesof coupling parameter β . The difference of the rotation velocities of the pendula exhibits some periodic pertur-bations, an example of which is shown in figure 5(a).The panel (b) of figure 5 shows the range of couplingparameter β with unstable rotations of the pendula independence on the rotation energy. At the first glancethe existence of instability in the in-phase rotation con-tradicts to the limiting cases of the extremely large andextremely small coupling. Actually, if the coupling is neg-ligible, the pendula are independent and no instability inthe rotation occurs. From the other end, when couplingconstant β → ∞ , two pendula can be considered as a sin-gle pendulum with the doubled mass and no instabilityoccurs again.Let us assume that the energy of the in-phase rotationis large enough ( E (cid:29) σ ). In such a case we can representthe obvious solution of the first of equation (36) as follows θ = ω r t + λ sin ωt, (38)where ω r = 2 π/T is the rotation frequency and λ = σ/ω r (cid:28) d θ dt + (2 β + σ cos ω r t ) sin θ = 0 . (39)(Considering θ as a small perturbation we assume thatcos θ ≈ a bFIG. 5. (a) The data of numerical simulations of the in-phase rotation of coupled pendula. The behaviour of the difference ofthe pendulums velocities in the unstable range of the in-phase rotation. (b) The range of the coupling constant β ( E ), wherethe unstable in-phase rotation occurs, is coloured gray. Blue and red points were obtained in the direct numerical simulations.The black lines show the threshold values, corresponding to relations (42, 43). Equation (39) is the well-known equation of the para-metrically excited pendulum. It is common knowledgethat the first parametric resonance occurs at the fre-quency, which is one-half of the own one. Thus we shouldestimate the perturbation with frequency Ω = ω r /
2. Representing out-of-phase perturbation in the complexform accordingly expression (7), extracting the ”carrier”exponential e − i Ω t and discriminating the secular term,we can write the stationary equations for the modulusand the phase of the perturbation: σa J (cid:16)(cid:113) a (cid:17) sin 2 δ = 0 (40) Ω2 a − √ βJ (cid:16)(cid:113) a (cid:17) − σ √ (cid:104) J (cid:16)(cid:113) a (cid:17) − √ a J (cid:16)(cid:113) a (cid:17)(cid:105) cos 2 δ = 0 , where J n is the Bessel function of order n . These equa-tions describe the stationary out-of-phase oscillationscoupled with the in-phase rotation of the pendula. Theamplitude and phase of such oscillations should be deter-mined numerically. However, we can write the energy ofsuch oscillations as follows H = Ω2 a − β (cid:32) − J (cid:32)(cid:114) a (cid:33)(cid:33) − σJ (cid:32)(cid:114) a (cid:33) cos δ. (41)Taking into account this expression, one should analysethe non-stationary trajectories on the phase plane at thedifferent values of coupling parameter β .Figure 6 shows the phase portraits of the system withrotation energy E = 5 . β . Fig.6(a) represents the phase plane of the system when thecoupling parameter β is smaller than the bottom thresh-old of the instability. One can see that there is onlystationary solution a = 0 for any values of phase δ . Anytrajectories, which are close to the stationary state cannot rise and the rotation is stable.However, if coupling constant β exceeds some thresh-old, new stationary point with a (cid:54) = 0 and phase δ = 0appears. Simultaneously the trajectory, which separatesthe sets of the closed and transit-time trajectories, forms.The fact is important that this trajectory pass throughzero value of the amplitude a , therefore any perturba-tions, which start from zero amplitude evolve along this a bcFIG. 6. (Color online) The phase plane of system (41) atthree values of coupling parameter β (a) β = 0 .
7, (b) β = 0 . β = 1 . E = 5 . σ = 1. trajectory. This trajectory is the aforementioned LPT.Fig. 6(b) shows the phase portrait after formation ofthe stationary solution and the LPT. One should noticethat from the viewpoint of the rotational instability, itis not the fact of the existence of the stationary pointthat matters, but the appearance of the Limiting PhaseTrajectory.In order to determine the threshold of the instability,one should note that the latter arises via the formation ofthe stationary solution at the point ( a = 0 , δ = 0). Thisneeds in the condition ∂ H /∂a = 0 at this point. Solv-ing this equation with respect to the coupling parameter,we get the bottom threshold value as follows β l = 14 (cid:0) − σ (cid:1) . (42)While the coupling parameter grows the amplitude ofthe stationary solution as well as the LPT increase too.However, when the LPT’s low edges reach the phase value δ = ± π/
2, the next bifurcation happens. Namely, thesymmetrical pair of the unstable stationary solutions ap-pear in the points ( a = 0 , δ = ± π/ δ = ± π/
2, that leads to β t = 14 (cid:0) + σ (cid:1) . (43)The thresholds (42, 43) are shown in figure 5(b) by theblack lines. One can notice the excellent fit these valueswith the data of the numerical simulations.Three problems discussed above exhaust the list of theexamples, by which we would like to illustrate the appli-cations of the CEVA, but many another applications ofthe method discussed above can be found in . IV. DISCUSSION
In spite of the list of the possible applications of theCEVA can be continued we would like to discuss themethod’s peculiarities. First of all, one should notice thatthe CEVA is close to such widely used methods as theVan der Pol one, the multiscale expansion, and the har-monic balance approximation. Like two first methods theCEVA uses the envelope functions for the description ofthe stationary as well as non-stationary dynamics. Nev-ertheless, due to application of the complex variables thefinal results are more clear and understandable. More-over, in the number of cases we can get the frequencyspectrum (or the frequency-amplitude relation) withouttedious calculations for large amplitude oscillations (see,for example, Appendixes), and we can do it even in thepresence of the external forces and the friction. In fact,no assumption about the smallness of amplitude or thepresence of any small parameter is needed. The naturalrestriction for this procedure arises from the requirementof the sufficient smoothness of the the frequency as the function of the oscillation amplitude. In other words, thesingle-frequency approximation has to be sufficient andthe influence of higher harmonics should be neglegible.Really, the accounting the third harmonic in the frame-work of the CEVA is possible , but above we restrictedby the single-harmonic description. The natural sequenceof the single-frequency description is that the CEVA isthe insufficient in the neighbourhood of separtrix solu-tions. The disadvantage of the CEVA is that the usingthe stationary solution in the form (5) and averaging theequations (4) lead to that only symmetric part of the po-tential makes the contribution into oscillation frequency.Solving this problem is the subject of further researches.While the stationary equations are clear enough, thedevelopment of the non-stationary equations in theframework of the CEVA need in additional accuracy. Thereason is that the ”slowness” of the non-stationary equa-tions depends on a smallness of the right hand side ofequations (16,19). At the same time, the amplitude of thedisturbed solution can be essentially different than theamplitude of the stationary oscillation, but the frequencyshould be approximately the same. By what manner wecan satisfy so opposite requirements? As an example weshould point the resonant interaction of the nonlinearnormal modes. Actually, the frequencies of modes withclose wave number are close (see Appendix A) if theiramplitudes are equal. However, amplitude of their sumin dependence of the phase difference can be almost re-double or turns out to be small. Due to smallness ofthe frequency difference the transition from one state toanother one describes by the slow non-stationary dynam-ical equations. (This process is an analogue of the wellknown beating in the system of two weakly coupled iden-tical oscillators.) Thus, we need in the careful controlof the time scale of the non-stationary equations. An-other problem with the non-stationary equations, whichare deduced accordingly to relations (16,19), associateswith the complexity of them. They admit the analyti-cal solution only for the exceptional cases even for thesimplest power potentials. However, some simplificationoriginates from the existence of the additional integral ofmotion, which is analogue of the quantum-mechanical oc-cupation number. For example, in the case of N-particlelattice this integral is X = (cid:80) | ψ j | . In the number ofcases the presence of this integral allows us to reduce thedimensionality of the phase space and to do the analysisby the phase plane method.One should emphasize that the using of Hamilton func-tion (15) in the space of the envelopes turns out to be verysuccessful, because it allows us to make some principalfindings without solving the non-stationary equations. Itwas demonstrated in the analysis of the in-phase rota-tion instability of the coupled pendula. The analogousresults were obtained in by essentially large efforts. Inthis work we do not discuss the analysis of the inter-action of the nonlinear normal modes, because of thisproblem has been described in the number of papers forthe discrete dynamical systems of various origin (cou-pled oscillators, coupled self-excited oscillators, nonlinearchains, carbon nanotubes). Only one remark should bemade about transition to the infinite degrees-of-freedomsystems. It was demonstrated that the continualizationof the non-stationary equations for the discrete systemsleads to the specific nonlinear Schr¨odinger equation. Thelatter turns out to be the complex analogue of the well-known sine-Gordon equation in the case of the Frenkel-Kontorova or the Sine-lattice models , or describesthe specific breather solution for the circumferential flex- ure oscillations of the carbon nanotubes . Appendix A: The dispersion relation for 1DSine-lattice
In the Appendix we clarify the derivation of the sta-tionary equation in terms of complex variables. We willuse the chain of the interacting particles with 2 π -periodicpotential (so called sine-lattice ? ). The Hamiltonfunction of the chain can be written as follows: H = N (cid:88) j =1 (cid:32) (cid:18) dϕ j dt (cid:19) + β (1 − cos ( ϕ j +1 − ϕ j )) + σ (1 − cos ϕ j ) (cid:33) , (A1)where ϕ j is the displacement j − th particle from groundstate, N is the number of the particles, β and σ are theconstants.The respective equations of motion are read as d ϕ j dt − β (sin ( ϕ j +1 − ϕ j ) − sin ( ϕ j − ϕ j − )) (A2)+ σ sin ϕ j = 0 . Let us expand the trigonometric functions into Taylorseries and replace the variables ϕ j accordingly to expres-sions (3): i d Ψ j dt − ω (cid:0) Ψ j − Ψ ∗ j (cid:1) + 1 √ ω ∞ (cid:88) k =0 ( − k (2 k + 1)! (cid:18) √ ω (cid:19) k +1 × (A3) (cid:16) β (cid:16) (Ψ j +1 − Ψ j + cc ) k +1 − (Ψ j − Ψ j − + cc ) k +1 (cid:17) − σ (cid:0) Ψ j + Ψ ∗ j (cid:1) k +1 (cid:17) = 0In order to find the stationary single-frequency solutionone should represent functions Ψ j ( t ) = ψ j e − iωt withmodulus ψ j which does not depend on the time. It is easy to show that substituting this expression into equa-tion (A3) and averaging it over period 2 π/ω leads to theequation ω ψ j + β √ ω (cid:32) J (cid:32)(cid:114) ω | ψ j +1 − ψ j | (cid:33) ψ j +1 − ψ j | ψ j +1 − ψ j | − (A4) J (cid:32)(cid:114) ω | ψ j − ψ j − | (cid:33) ψ j − ψ j − | ψ j − ψ j − | (cid:33) − σ √ ω J (cid:32)(cid:114) ω | ψ j | (cid:33) ψ j | ψ j | = 0 , where J is the Bessel function of the first order. As-suming the periodic boundary conditions and taking intoaccount relation (8) one can see that the plane wave ψ j = χe iκj with κ = 2 πk/N , k = 0 , , . . . , N/ χ = (cid:112) ω/ A is the exact solution of equation (A4), if frequency ω is determined by expression ω = 2 A (cid:16) σJ ( A ) + 2 βJ (cid:16) A sin κ (cid:17) sin κ (cid:17) (A5)0 Appendix B: Estimation of the frequencies of simplenon-linear oscillator
In order to demonstrate the efficiency of the CEVAin the estimation of the amplitude-frequency relation forthe stationary oscillations, we made the calculation forthe simple one degree-of-freedom system: d xdt + x + x + x = 0 (B1)Following the procedure of section 2, it is easy to showthat the frequency of the stationary oscillations is deter-mined as follows: ω = (cid:114) A + 58 A (B2)These results we compare with the data, which have beenobtained in works and by different methods. Thefinal expression for the frequency in work was obtainedby some intuitive considerations: ω = (cid:112) . A + 0 . A (B3)In spite of that this expression is extremely similar toequation (B2), we suppose that the choice of the numeri- cal coefficients is rather intuitive . The respective valuesof the oscillation frequency are shown in Table 1 as ω .The method of high-order harmonic balance was usedin work . After some tedious calculations the valuesof the oscillation frequency for different amplitudes havebeen obtained. They are represented in Table 1 as ω .The main problem of the last two work is that evena little changing the initial equation (B1) (for example,a varying of the numerical coefficients) can cause ardu-ously predictable effect on the final result and, in order toestimate it, we should perform the full calculation moretimes. Table 1 and figure 7 represent the comparativeanalysis of the data, which were calculated by expres-sions (B2) and reprinted from works and . The rel-ative errors have been calculated with respect to exactvalues ( ω e ) (see Table 1). δ j = ω j − ω e ω e , ( j = 1 , ,
3) (B4)
ACKNOWLEDGMENTS
Authors are grateful to Russia Science Foundation(grant 16-13-10302) for the financial supporting of thiswork. ∗ [email protected]; permanent address: 4 KosyginStreet, Moscow, Russia † [email protected] P. L. Christiansen, M. P. Sorensen, and A. C. Scott,eds.,
Nonlinear Science at the Dawn of the 21st Century (Springer-Verlag Berlin Heidelberg New York, Heidelberg,2000) p. 458. A. Scott,
Nonlinear Science: Emergence and Dynamics ofCoherent Structures , 2nd ed. (Oxford University Press, Ox-ford, 2003) p. 480. R. Z. Sagdeev, D. A. Usikov, and G. M. Zaslavsky,
Nonlin-ear Physics: From the Pendulum to Turbulence and Chaos (Harwood Acad. Publ., New York, 1988) p. 315. N. Krylov and N. Bogoluibov,
Introduction to Non-LinearMechanics (Princeton Univ. press, Princeton, NJ USA,1943) p. 106. A. H. Nayfeh and D. K. Mook,
Nonlinear oscillations (John Willey & Songs, New York, 1979). J. Sanders, F. Verhulst, and J. Murdock,
Averag-ing Methods in Nonlinear Dynamical Systems , 2nd ed.,Applied Mathematical Sciences, Vol. 59 (Springer Sci-ence+Business Media, LLC, New York, 2007) p. 431. R. E. Mickens,
Truly nonlinear oscillators: An Introduc-tion to Harmonic Balance, Parameter Expansion, Itera-tion, and Averaging Methods (World Scientific PublishingCo. Pte. Ltd, Singapore, 2010). E. Esmailzadeh, D. Younesian, and H. Askari,
AnalyticalMethods in Nonlinear Oscillations. Approaches and Ap-
TABLE I. The frequencies of the stationary oscillations forsystem (B1) at different amplitude A . ω e corresponds to theexact value, ω is determined by expressions (B2), and ω ,and ω reprinted from works and , respectively. Relativeerrors δ j are calculated accordingly to expression (B4). A ω e ω ω ω δ δ δ plications , Solid Mechanics and Its Applications, Vol. 252(Springer Nature B.V., Dordrecht, The Netherlands, 2019)p. 286. L. Cveticanin,
Strong Nonlinear Oscillators. Analytical So-lutions , 2nd ed., Mathematical Engineering (Springer In-ternational Publishing AG, Cham, Switzerland, 2018) p.317. J.-H. He, International Journal of Modern Physics B ,1141 (2006). J. Leon and M. Manna, Journal of Physics A: Mathemat- FIG. 7. The relative errors for the oscillation frequencies.Black, blue and red lines show the results of current work,and the data from work and , respectively.ical and General , 2845 (1999). J. Kevorkian and J. Cole, “The method of multiple scalesfor ordinary differential equations,” in
Multiple Scale andSingular Perturbation Methods. , Applied MathematicalSciences, Vol. 114 (Springer, New York, New York, 1996)pp. 267–409. B. Van der Pol, The London, Edinburgh and Dublin Phil.Mag. and J. of Sci. , 978 (1927). B. Van der Pol and J. Van der Mark, Nature , 363(1927). L. Manevitch, Arch. Appl. Mech. , 3011 (2007). L. I. Manevitch, Nonlinear Dynamics , 95 (2001). P. A. M. Dirac,
The Principles of Quantum Mechanics , 4thed., International Series of Monographs on Physics, Vol. 27(Oxford University Press, Oxford, 1978). H. Haken,
Quantum field theory of solids (North-HollandPub. Co., Amsterdam, New York, 1976). L. I. Manevitch and V. V. Smirnov, “Resonant energy ex-change in nonlinear oscillatory chains and limiting phasetrajectories: from small to large system.” in
AdvancedNonlinear Strategies for Vibration Mitigation and SystemIdentification , CISM Courses and Lectures, Vol. 518, editedby A. F. Vakakis (Springer, New York, 2010) pp. 207–258. L. I. Manevitch and A. I. Musienko, Nonlinear Dynamics , 633 (2009). L. I. Manevitch, V. V. Smirnov, and F. Romeo, Cyber-netics and Physics , 130 (2016). A. Kovaleva and L. I. Manevitch, Phys Rev E , 024901(2013). L. I. Manevitch and V. V. Smirnov, Phys. Rev.
E 82 ,036602 (2010). V. V. Smirnov and L. I. Manevitch, Phys. Rev. E ,022212 (2017). V. Smirnov, L. Manevitch, M. Strozzi, and F. Pellicano,Physica D: Nonlinear Phenomena , 113 (2016). V. V. Smirnov and L. I. Manevitch, Nonlinear Dynamics , 205 (2018). M. Kovaleva, L. Manevich, and V. Pilipchuk, J. Exp.Theor. Phys. , 369 (2013). L. Manevitch and M. Kovaleva, Dokl. Phys. , 428 (2013). D. Bitar, A. Ture Savadkoohi, C.-H. Lamarque, E. Gour-don, and M. Collet, Nonlinear Dyn , 1433 (2020). L. I. Manevitch and A. F. Vakakis, SIAM J. Appl. Math. , 1742 (2014). V. V. Smirnov, “Revolution of pendula: Rotational dy-namics of the coupled pendula,” in
Problems of NonlinearMechanics and Physics of Materials , Advanced StructuredMaterials, Vol. 94 (Springer International Publishing AG,Berlin, 2019) pp. 141–156. L. I. Manevitch, V. V. Smirnov, and F. Romeo, Cyber-netics and Physics , 91 (2016). O. V. Gendelman and Karmi, Nonlinear Dyn , 2775(2019). L. I. Manevitch, A. Kovaleva, V. Smirnov, andY. Starosvetsky,
Nonstationary Resonant Dynamics of Os-cillatory Chains and Nanostructures (Springer Nature,2018). L. A. Smirnov, A. K. Kryukov, G. V. Osipov, andJ. Kurths, Regul. Chaotic Dyn. , 849 (2016). S. Takeno and S. Homma, J Phys Soc, Jpn , 65 (1986). J.-H. He, Results in Physics , 102546 (2019). M. Chowdhury, M. A. Hosen, K. Ahmad, M. Ali, andA. Ismail, Results in Physics7