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Differential Equations, Vol. 40, No. 6, 2004, pp. 780–788. Translated from Differentsial’nye Uravneniya, Vol. 40, No. 6, 2004, pp. 731–739. c 2004 by Kalyakin. Original Russian Text Copyright


ORDINARY DIFFERENTIAL EQUATIONS

Asymptotic Solution of the Threshold Phenomenon Problem for the Principal Resonance Equations L. A. Kalyakin Institute of Mathematics and Computer Center, Russian Academy of Sciences, Ufa, Russia Received September 8, 2003

INTRODUCTION Autoresonance is one of the most interesting phenomena in nonlinear oscillations. The energy of forced oscillations of a nonlinear system substantially increases in the course of time, although the driving force remains small. This phenomenon, observed in various physical systems [1–8], is used, for example, for accelerating relativistic particles [9–12]. Numerical experiments based on various mathematical models have shown that, for autoresonance to appear, it is necessary that the pump amplitude exceed some threshold value [13, 14]. In the present paper, we obtain a result of this sort analytically by an asymptotic analysis of the equations


   2   −y 1 d x 2 + θ− x +y λ = (λ = const). (1) dθ y x 0 These so-called principal resonance equations [15, 16] are the main object of study in autoresonance theory [13, 14, 16]. They contain one dimensionless parameter λ. We pose the problem on the relationship between the integration constants in the asymptotics as θ → +∞ and θ → −∞. This problem is considered in the asymptotic approximation in two versions, as λ → 0 and as λ → +∞. The following assertion is a corollary to our results. Theorem 1. (1) There exists a λ0 > 0 such that for all λ < λ0 , the solution tending to zero as t → −∞ remains bounded as θ → +∞; (2) there exists a λa > 0 such that for all λ > λa , the solution tending to zero as θ → −∞ infinitely grows as θ → +∞; (3) these properties are structurally stable under perturbations of the asymptotics at −∞. The statement that is most important for the autoresonance problem is contained in assertion (2) of the theorem: the condition λ > λa corresponds to pump amplitudes exceeding the threshold, and the corresponding solution with increasing amplitude describes the initial stage of the system’s entry into autoresonance [16, 17]. For λ < 0, there are no solutions increasing as θ → +∞ [18]. Therefore, only positive values λ > 0 are of interest, and it is this case that is to be discussed below.1 1. THE STRUCTURE OF THE ASYMPTOTICS AT INFINITY AND STATEMENT OF THE PROBLEM To state the problem precisely, we need results on the asymptotics obtained in [18]; it is convenient to represent them in the form of rigorous assertions. The main objects in these results are asymptotic series in inverse powers of θ, for example,

  ∞ sin S cos S θ −n xn (S; λ, c1 , c2 ) , (1.1) + c2 + x (θ; λ, c1 , c2 ) ∼ = c1 cos S − sin S n=1 1

Note that the equations are invariant under the transformation (x, y, θ, λ) ⇒ −(x, y, θ, λ); thus if the coefficient λ changes its sign, then the asymptotics at the two infinities are exchanged. c 2004 MAIK “Nauka/Interperiodica” 0012-2661/04/4006-0780 

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with coefficients depending on the “fast” variable S = θ 2 /2−(c21 + c22 ) λθ. Here and in the following, we write x(θ; λ) = (x, y)T as a column vector.   − Theorem 1.1. Let λ > 0. System (1) has a two-parameter family of solutions x− θ; λ, c− 1 , c2 − with arbitrary constants c− 1 and c2 . These solutions have an asymptotic expansion in the form of the series (1.1) as θ → −∞. This family exhausts all solutions of Eq. (1). Recall that the coefficients xn are found from the recursion system of equations obtained by the substitution of the series into (1). The constants c1 , c2 ∈ R remain arbitrary parameters of the solution. The asymptotic structure of the series is provided by the boundedness of the coefficients xn (S; λ, c1 , c2 ), which prove to be periodic vector functions of S. For c1 = c2 = 0, the entire construction substantially degenerates, and the asymptotic series become purely power series with constant coefficients:
  ∞ −1 0 ∼ + x0 (θ; λ) = −θ θ −n xn,0 (λ). (1.2) 1 n=2 − − Corollary 1.1. The parameter values c− 1 = c2 = 0 correspond to the unique solution x0 (θ; λ) with purely power-law decreasing asymptotics in the form of the series (1.2) as θ → −∞.

Similar asymptotics can be constructed at +∞.   + Theorem 1.2. Let λ > 0. System (1) has a two-parameter family of solutions x+ θ; λ, c+ 1 , c2 + with arbitrary constants c+ 1 and c2 . These solutions admit an asymptotic expansion in the form of the series (1.1) as θ → +∞. This family exhausts all bounded solutions of Eq. (1). + + Corollary 1.2. The values c+ 1 = c2 = 0 correspond to the unique solution x0 (θ; λ) with the purely power-law decreasing asymptotics in the form of the series (1.2) as θ → +∞.

Unlike the first case, not all solutions are bounded in the asymptotics as θ → +∞. There is an ample family of growing solutions, whose asymptotics can be conveniently represented in polar coordinates after the change of variables x = r cos ψ and y = r sin ψ. Theorem 1.3. Let λ > 0. System (1) has a two-parameter family of growing solutions r(θ; λ, c, s), ψ(θ; λ, c, s) with arbitrary constants c and s. These solutions can be expanded in the asymptotic series
1/2 ∞  θ + θ −1/4 θ −n/8 rn (S; λ, c), r(θ; λ, c, s) = λ n=1 (1.3) ∞  π θ −n/8 ψn (S; λ, c), θ → +∞. ψ(θ; λ, c, s) = + 2 n=1 The coefficients are 2π-periodic functions of the phase variable 4  √ 4 sn (λ, c)θ n/4 + s0,1 (λ, c) ln θ + s. S = θ 5/4 λ1/4 2 + 5 n=1

(1.4)

√ In particular, r1 = c sin S and ψ1 = −c 2 cos S.

For c = 0, the dependence on the second parameter s disappears, and the asymptotic solution (1.3) can be represented by a series in powers of θ −1/2 with constant coefficients. This asymptotics corresponds to a unique solution xa (θ; λ), which will be written out with subscript a :
1/2  ∞ ∞ θ π  −n/2 + θ −n/2 n (λ), ψa (θ; λ) = + θ ψn (λ), θ → +∞. (1.5) ra (θ; λ) = λ 2 n=1 n=2 DIFFERENTIAL EQUATIONS

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One can construct another purely power series in powers of θ −1/2 by taking the different value ψ(θ) ∼ = −π/2 of the phase in the leading term of the asymptotics. Such a construction corresponds to a one-parameter family of solutions exponentially close as θ → +∞ [18]. In the present paper, we do not use such solutions. Remark 1.1. All solutions and their asymptotic expansions are infinitely differentiable with respect to the variables θ and λ as well as with respect to the integration constants c1 and c2 . This important property follows from the corresponding uniform estimates in the contraction mapping method, which was used in the proof of the asymptotics in [19]. An analog of the scattering problem for system (1) is as follows:  + find formulas  relating the − − − − integration constants for various λ in the form c+ = f , c ; λ , c = f , c ; λ . So far, one c c 1 2 1 1 2 2 1 2 has not managed to solve this problem (of finding the functions f1 and f2 ) in the exact statement for the nonlinear equations (1).2 However, the following problem is of interest from the viewpoint of the autoresonance problem. A Simplified Statement of the Scattering Problem (the Threshold Phenomenon Problem) For a solution that tends to zero as θ → −∞, we should find the dependence of the form of the asymptotics as θ → +∞ on the parameter λ; namely, for what λ is the solution bounded, and for what λ is it growing? In the present paper, we answer this question in the two critical cases: for small and for large values of the parameter λ. 2. ASYMPTOTIC SOLUTION OF THE SCATTERING PROBLEM FOR A WEAK NONLINEARITY 2.1. For each c0 > 0, there exists a λ0 > 0 such that for all λ ∈ [0, λ0 ) and     −Theorem − − −  c1  , c− θ; λ, c− determined by the constants c− ≤ c 0 , the solution x 2 1 , c2 1 and c2 in the asymptotics as θ → −∞ remains bounded as θ → +∞. Proof. The proof is based on the boundedness of an arbitrary solution of the limit linear problem (for λ = 0). In this case, the solution admits the integral representation     

θ θ 2 2 2 2   cos (θ /2)  − η η sin (θ /2)  − − c1 + c2 + sin dη  + cos dη  , (2.1) = x− θ; 0, c− 1 , c2 2 2 2 2 cos (θ /2) − sin (θ /2) −∞

−∞

and the relationship between the constants can be written out in closed form:   − − π/2, c+ π/2. c+ 1 = c1 + 2 = c2 + In the nonlinear problem (with λ > 0), the main difficulty is the absence of closed-form expressions for the exact solution or at least for first integrals. Moreover, even for the asymptotic expansions as λ → 0, we have no formula that can be used on the entire real line θ ∈ R. This well-known disadvantage of asymptotic approximations prevents one from using methods like the averaging method or the multiscale method to write out the asymptotic relations between the constants as λ → 0. Our solution of the scattering problem is essentially an existence theorem, in which the smallness of the parameter λ is needed only for the corresponding mapping to be a contraction.   −  − − − − − , c , c θ, λ; c , y θ, λ; c determined by the asymptotics Consider the family of solutions x 1 2 1 2   +  + + + + + as θ → −∞. Of solutions x θ, λ; c1 , c2 , y θ, λ; c1 , c2 with bounded asymptotics at the other infinity, we should find the solution that is a continuation of the solution given on the left. For this, it is sufficient that the matching conditions         + − + − = x− θ0 ; λ, c− y + θ0 ; λ, c+ = y − θ0 ; λ, c− x+ θ0 ; λ, c+ 1 , c2 1 , c2 , 1 , c2 1 , c2 2

The problem on the relationship between the constants involves severe difficulties even for integrable Painlev´e equations [20]. DIFFERENTIAL EQUATIONS

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be satisfied at some time θ0 . These two relations should be considered as algebraic equations for + the constants c+ 1 and c2 . Since there exist solutions unbounded to the right and bounded to the left, such equations are not necessarily solvable for λ > 0. However, it was shown above that in the limit case λ = 0, these equations are solvable.  + +It  follows from explicit formulas like (2.1) for the + + + ∂ (x , y )/∂ c1 , c2 does not vanish for λ = 0. Since the functions solution x that   the Jacobian  + + + + , c and y θ, λ; c are continuous in all variables, it follows from the implicit x+ θ, λ; c+ 1 2 1 , c2 + function theorem that the equations for c+ 1 and c2 are solvable in some neighborhood of each − λ ∈ [0, λ0 ). The neighborhood depends on the chosen compact set of the parameters c− 1 and c2 . The proof of Theorem 2.1 is complete. 3. ASYMPTOTIC SOLUTION OF THE SIMPLIFIED SCATTERING PROBLEM FOR A STRONG NONLINEARITY The case of large values of the parameter λ proves to be much more complicated than the preceding one. There arise difficulties in the solution of the scattering problem even if we wish to prove an existence theorem. In particular, these difficulties are due to the fact that one cannot find an appropriate limit problem as λ → +∞. This becomes most obvious after the scale transformation x(θ; λ) = λ−1/3 X(τ ; ε), θ = λ1/3 τ , λ−2/3 = ε, which results in equations with a small parameter ε either multiplying the derivative,


   2  −Y d X 1 2 ε + τ − X +Y = , (3.1) dτ Y X 0 or occurring in the coefficient τ = εt if the equations are written out in the fast time t = τ /ε. Attempts to consider the limit equations for ε = 0 result in either algebraic equations or an autonomous system with bounded solutions, which have little to do with our problem. Needless to say, the asymptotics of solutions of such equations as ε → 0 can be constructed with the use of various versions of the nonlinear WKB method. The leading terms of this asymptotics are sometimes referred to as “adiabatic approximations.” However, none of the known approximations to the solution is uniform on the entire real line θ ∈ R, and hence they cannot be used for deriving the desired relationship (e.g., see [15, 17, 21–23]). We restrict our considerations to the simplified scattering problem and prove the following assertion for it. Theorem 3.1. There exists a λa > 0 such that for all λ > λa , the solution of Eqs. (1) tending to zero as θ → −∞ admits the asymptotics    −   x0 (θ; λ) = θ/λ + O θ −3/8 , θ → +∞, (3.2) growing to the right. The proof is based on the representation of the equations in the form (3.1) with a small parameter ε. When studying the simplified scattering problem, we use two distinct solutions of Eq. (3.1). One of them, X0 (τ ; ε) = λ1/3 x− 0 (θ; λ), is determined by a purely power-law decreasing asymptotics (1.2) at −∞. The other, Xa (τ ; ε) = λ1/3 xa (θ; λ), is determined by a purely power-law growing asymptotics (1.5) at +∞. For each of these solutions, we construct a uniform asymptotics on the corresponding half-line as ε → 0. Both asymptotics are represented by the same power series. The main goal of the forthcoming considerations is to show that these two solutions are asymptotically close3 uniformly on the entire real line θ ∈ R. The most difficult part of the proof is the justification of the asymptotics. Lemma 3.1. Both solutions X0 (τ ; ε) and Xa (τ ; ε) admit an asymptotic expansion in the form of the same power series X0 (τ ; ε) ∼ = Xa (τ ; ε) ∼ =

∞ 

εn Xn (τ ),

ε → 0.

n=0 3

It is known that the coincidence of asymptotics does not guarantee the coincidence of solutions. DIFFERENTIAL EQUATIONS

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(3.3)

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For each m ≥ 0, this asymptotics is uniform with respect to τ on the half-line τ ∈ (−∞, ε−m ] for X0 (τ ; ε) and on the half-line τ ∈ [−ε−m , +∞) for Xa (τ ; ε). T

Proof. The components of the vectors Xn = (Xn , Yn ) are found from the algebraic equations obtained by matching the coefficients of like powers εn after the substitution of the series (3.3) into Eq. (3.1). In the leading term, the first component is zero, X0 ≡ 0, and for the second component Y0 (τ ), one takes the unique smooth root of the cubic equation4 Y 3 − τ Y = 1 defined on the entire real line. The subsequent coefficients Xn , Yn (τ ), n ≥ 1, are found from linear equations by recursion formulas. From these formulas, one can readily derive the asymptotics of the coefficients at infinity. Although one can write out the entire series expansions, we restrict our exposition to the structure of the leading terms:  −τ −1 + O (τ −4 )   as τ → −∞ X0 (τ ) ≡ 0, Y0 (τ ) = √ τ + (1/2)τ −1 + O τ −5/2 as τ → +∞,  (3.4) −1−2n O (τ )  as τ → −∞  Xn , Yn (τ ) = O τ (1−n)/2 as τ → +∞. The above-described construction of a formal solution in the form of the series (3.3) seems to be obvious. However, it should be complemented by a justification; i.e., it remains to prove the existence of exact solutions with this asymptotics as ε → 0. The corresponding proofs are not trivial and will be carried out on each of the two half-lines separately. Since the two cases are practically the same, we restrict our considerations to the left half-line τ ∈ (−∞, ε−m ]. By U(τ ; ε) = (U, V )T we denote the segment of length N + 1 of the series (3.3), which, after the substitution into Eq. (3.1), produces a discrepancy that is O (εN +1 ) uniformly on the entire real line τ ∈ R. The exact solution is sought in the form of the sum X0 = U + εN u. For the remainder vector u(τ ; ε) = (u, v)T , we obtain equations in which all nonlinear terms contain small factors:


 −v  2N 2 −v N 2 −V N . (3.5) − ε |u| − ε · 2(u, U) εu + M (τ ; ε)u = εF(τ ; ε) − ε |u| u u U The known discrepancy vector F(τ ; ε) has a rapidly decreasing asymptotics at each of the points at infinity:  −2−2N O (τ ) as τ → −∞  (1−N )/2  F(τ ; ε) = O τ as τ → +∞ uniformly with respect to ε on each compact set. The matrix of the linear part can be expressed via known functions:   2U V −τ + U 2 + 3V 2 . M (τ ; ε) = τ − 3U 2 − V 2 −2U V The equations are supplemented by the zero conditions at −∞ : u(τ ; ε) → 0

as

τ → −∞.

(3.6)

In the problem (3.5), (3.6) for the remainder, one can prove the existence theorem in the class of functions that rapidly tend to zero at −∞ and are bounded on the half-line τ ∈ (−∞, ε−m ] uniformly with respect to ε for all sufficiently small ε ∈ (0, ε0 ). The proof is based on the approximate inversion of the linear part of the operator by the WKB method [24, 25]. Here the eigenvalues ±iµ(τ ; ε) of the matrix M (τ ; ε) play an important role. Since the asymptotics of the entries of the matrix as ε → 0 and τ → ±∞ are known, we know also the asymptotics of the eigenvalues. In particular, in the leading term, we have µ2 = 1/Y0 (τ ) + O (ε), ε → 0, for all τ ∈ R. Since Y0 (τ ) > 0 for all τ , it follows from the asymptotics (3.4) that the eigenvalues are pure imaginary for all sufficiently small ε ∈ (0, ε0 ) and for all τ ∈ R. 4

The additional pair of roots, which exist for τ ≥ (3/2)2/3 , is not used in the construction. DIFFERENTIAL EQUATIONS

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For the linear homogeneous part of Eq. (3.5), we construct a pair of WKB approximations   U± (τ ; ε) = R± (τ ; ε) exp ±iε−1 ϕ(τ ; ε) . The singular part of the solution with respect to ε is represented by the rapidly oscillating exponential with phase function τ ϕ(τ ; ε) = µ(ζ; ε)dζ. 0

The amplitude vectors R± (τ ; ε), smooth with respect to ε, are found from the condition that the discrepancies be small [24]. For our purpose, it suffices to require the second-order smallness ε

  dU± + M (τ ; ε)U± = ε2 F± (τ ; ε) exp ±iε−1 ϕ(τ ; ε) dτ

with discrepancy F± (τ ; ε) = O (1), ε → 0, for all τ ∈ R. Omitting well-known details of the construction of the amplitudes [24], we only give their asymptotics at infinity:  O (1)   as τ → −∞ R± (τ ; ε) = O τ 3/8 as τ → +∞. These WKB approximations are used to invert the leading linear part in Eq. (3.5) [25]. This inversion is performed with the use of a substitution similar to the variation of constants method: T u = C+ U+ + C− U− . For the vector C(τ ; ε) = (C+ , C− ) of new unknown functions, the equations preserve their structure, but the matrix of the linear part occurs with a small parameter: dC/dτ = H0 (τ ; ε) + εH1 (τ ; ε)C + εN −1 H2 (C, τ ; ε) + ε2N −1 H3 (C, τ ; ε).

(3.7)

Here H0 is obtained from the discrepancy F in the original equations (3.5). This known vector function is rapidly decreasing at infinity uniformly with respect to ε :  −1−2N O (τ ) as τ → −∞  (3−N )/2  H0 (τ ; ε) = O τ as τ → +∞. The matrix of the linear part has a tempered decay at infinity: H1 = O (τ −2 ), τ → ±∞. The nonlinear terms include a quadratic form H2 and a cubic form H3 in C+ , C− whose coefficients are majorized by linear functions in coarse estimates as τ → ±∞. The differences in solving the problems for the remainder on different half-lines begin at the following stage, where the differential equations are replaced by integral equations. For solutions that tend to zero as τ → −∞, we obtain integral equations in the form τ C(τ ; ε) =

  H0 (ζ; ε) + εH1 (ζ; ε)C + εN −1 H2 (C, ζ; ε) + ε2N −1 H3 (C, ζ; ε) dζ.

(3.8)

−∞

With regard to the properties of Hk listed above, one can readily prove the solvability of Eq. (3.8) in the class of decreasing functions C(τ ; ε) = O (τ −2N ), τ → −∞. For the formal proof, one can use the Banach space of continuous vector functions with the weighted norm C = sup−∞ 0. By virtue of the periodic dependence on s, the set of corresponding trajectories DIFFERENTIAL EQUATIONS

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is a tube in the extended phase space x, y, t. The trajectory of the solution Xa (τ ; ε) with constant c = 0 lies inside the tube. Using the expansion (3.10), we can estimate the difference of solutions from below with the use of the triangle inequality, and the estimate in the asymptotic approximation has the form   |X(τ ; ε, c, s) − Xa (τ ; ε)| ≥ cτ −3/8 + O τ −1/2 , τ → +∞. Since the asymptotics are uniform with respect to ε and s, it follows that, for all sufficiently large τ ≥ τ0 , the estimate can be reduced to the form |X(τ ; ε, c, s) − Xa (τ ; ε)| ≥ τ −3/8 c/2. In particular, if τ = ε−1 , then we obtain |X(τ ; ε, c, s) − Xa (τ ; ε)| ≥ ε3/8 c/2,

τ = ε−1 ,

for all ε ∈ (0, ε0 ). This inequality, together with the estimate (3.9), implies that the trajectory of the solution X0 (τ ; ε) decreasing to the left enters the tube at time τ = ε−1 for all sufficiently small ε ∈ (0, ε0 ). By virtue of the uniqueness of the solution of the Cauchy problem, this trajectory remains in the tube for all τ ∈ R. Therefore, the solution in question grows at infinity:   √ τ → +∞. |X0 (τ ; ε)| = τ + O τ −3/8 , This implies the desired assertion (3.2). Although the estimate for the remainder in the variable θ is not uniform with respect to λ, this plays no role for the growth of the solution as θ → +∞ for −3/2 each given λ > ε0 . The proof of the theorem is complete. Remark 3.1. The amplitude constant can be chosen to depend on the small parameter c = εN for all N ≥ 0 so as to ensure that the tube is arbitrarily narrow of the order O (εN ). For the asymptotic expansion, this provides a justification everywhere on the real line τ ∈ R. Therefore, the difference of the solutions has the super-power order X0 (τ ; ε)−Xa (τ ; ε) = O (εN ), ε → 0, for all N uniformly on the entire real line τ ∈ R. Remark 3.2. Considerations given in the end of the proof show that the pencil of trajectories that can be determined  −an  asymptotics of the form (1.1) as τ → −∞ with sufficiently small  −  by    amplitude constants c1 , c2  ≤ c0 (ε) (which may be different for different ε) enters the tube going to infinity. This implies the stability (under perturbations of the boundary condition at −∞) of the solution with asymptotics growing at +∞. CONCLUSION Theorems 2.1 and 3.1, together with Remark 3.2, imply the assertion of Theorem 1. This answers Friedland’s question on the existence of the threshold phenomenon in autoresonance. Results obtained by asymptotic methods do not seem to be exhaustive even for the simplified scattering problem. In particular, they do not give an algorithm for the computation of the critical parameter value λcr that separates the solution x− 0 (θ; λ) in the case of bounded asymptotics (for λ < λcr ) from the case of a growing asymptotics (for λ > λcr ). This illustrates the fact that the approach based on asymptotic expansions has restricted possibilities. The critical parameter value can be approximately found from numerical experiments, which suggest that 0.1685 < λcr < 0.1687 (e.g., see [14]). By combining numerical and asymptotic results, one can readily prove the uniqueness of λcr . The problem on the relationship between parameters for the system of principal resonance equations (1) remains open; it has not been solved even in the asymptotic approximation with respect to the parameter λ. Obviously, from the WKB approximations, one can extract the relationship between the oscillation amplitudes at different infinities. However, it is impossible to relate the phase shifts even in the case of a weak nonlinearity (in the asymptotics as λ → 0). DIFFERENTIAL EQUATIONS

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All results remain valid for the more general equations considered in [18]. The proofs are essentially the same as above, since they are based on asymptotics of the same form (1.1), (1.3). ACKNOWLEDGMENTS The work was financially supported by the Russian Foundation for Basic Research (project no. 03-01-00716, the Program “Scientific Schools” (project no. 1446.2003.1), and INTAS (project no. 03-51-4286). REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

Andronov, A.A. and Gorelik, G.A., Dokl. Akad. Nauk SSSR, 1945, vol. 49, no. 9, pp. 664–666. Golovanivsky, K.S., Phys. Scr., 1980, vol. 22, p. 126. Golovanevskii, K.S., Fizika Plazmy, 1985, vol. 11, no. 3, pp. 295–299. Golovanivsky, K.S., IEEE Trans. Plasma Sci., 1983, PS–11, p. 28. Meerson, B. and Friedland, L., Physical Review A, 1990, vol. 41, pp. 5233–5236. Fajans, J., Gilson, E., and Friedland, L., Phys. Plasmas, 1999, vol. 6, pp. 4497–4503. Fajans, J., and Friedland, L., Am. J. Phys., 2001, vol. 69, no. 10, pp. 1096–1102. Friedland, L., Astophys. J., 2001, vol. 547, part 2, pp. L75–L79. Veksler, V.I., Dokl. Akad. Nauk SSSR, 1944, vol. 43, no. 8, pp. 346–348. Veksler, V.I., Dokl. Akad. Nauk SSSR, 1944, vol. 44, no. 9, pp. 393–396. Livingston, M.S., High-Energy Particle Accelerators, New York, 1954. Kolomenskii, A.A. and Lebedev, A.N., Teoriya tsiklicheskikh uskoritelei (Theory of Cyclic Accelerators), Moscow, 1962. Friedland, L., Physics of Plasmas, 2000, vol. 7, no. 5, pp. 1712–1718. Friedland, L., Physical Review E , 2000, vol. 61, no. 4, pp. 3732–3735. Haberman, R. and Ho, E.K., J. Appl. Mech., 1990, vol. 62, pp. 941–946. Kalyakin, L.A., Dokl. RAN , 2001, vol. 378, no. 5, pp. 594–597. Kalyakin, L.A., Russ. J. Math. Phys., 2002, vol. 9, no. 1, pp. 84–95. Kalyakin, L.A., Dokl. RAN , 2003, vol. 388, no. 3, pp. 305–308. Kalyakin, L.A., Proceedings of the Steklov Institute of Mathematics, 2003, supp. 1, pp. S108–S125. Its, A.R. and Novoksheniv, V.Yu., Lecture Notes in Mathematics, vol. 1191, Berlin, 1980. Neishtadt, A.I., Prikl. Mat. Mekh., 1975, vol. 39, no. 4, pp. 621–632. Neishtadt, A.I., Differents. Uravn., 1987, vol. 23, no. 12, pp. 2060–2067. Glebov, S.G. and Kiselev, O.M., Russ. J. Math. Phys., 2002, vol. 9, no. 1, pp. 60–83. Fedoryuk, M.V., Asimptoticheskie metody dlya lineinykh obyknovennykh differentsial’nykh uravnenii (Asymptotic Methods for Linear Ordinary Differential Equations), Moscow, 1983. Fedoryuk, M.V., Zh. Vychislit. Mat. Mat. Fiz., 1986, vol. 26, no. 2, pp. 198–210.

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