A remark on renormalization group theoretical perturbation in a class of ordinary differential equations
aa r X i v : . [ m a t h - ph ] S e p A REMARK ON RENORMALIZATION GROUP THEORETICAL PERTURBATIONIN A CLASS OF ORDINARY DIFFERENTIAL EQUATIONS
ATSUO KUNIBA
Abstract
We revisit the renormalization group (RG) theoretical perturbation theory on oscillator type secondorder ordinary differential equations. For a class of potentials, we show a simple functional relationamong secular coefficients of the harmonics in the naive perturbation series. It leads to an inversionformula between bare and renormalized amplitudes and an elementary proof of the absence of secularterms in all order of the RG series. The result covers non-autonomous as well as autonomous casesand refines earlier studies including the classic examples as Van der Pol, Mathieu, Duffing and Rayleighequations. 1.
Introduction
In the second order ordinary differential equations such asVan der Pol : d ydt + y + ε ( y − dydt = 0 , (1)Mathieu : d ydt + y + ε ( g + 2 cos t ) y = 0 , (2)Duffing : d ydt + y + ε ( dydt + gy ) = 0 , (3)Rayleigh : d ydt + y + ε dydt (cid:16) (cid:0) dydt (cid:1) − (cid:17) = 0 , (4)naive perturbation around ε = 0 leads to a series of the form y = X k ≥ X n ∈ Z (polynomial in t ) ε k e n i t . (5)The polynomiality of the coefficients is called secular and invalidates the effective description beyond thetime scale typically like O ( ε − ). The renormalization group (RG) theoretical approach is a successfulexample of singular perturbation (cf. [1, 5, 7]) which circumvents the difficulty and offers an effectiveresummation of the divergent series. The basic strategy is to absorb the secular t -dependence of (5) intorenormalized amplitudes and describe the slow dynamics of the latter by the so called RG equation. Themethod has rich background and perspectives. See for example [2, 8, 10, 11] and the references therein.This short note is an elementary and modest addition to the well-developed machinery. We focus onthe equations of the form d ydt + y = εV, V = arbitrary polynomial in ε, e ± i t , y and dydt , (6)which are bit specific but well cover (1)–(4). The linear and dydt -free case like Mathieu equation (2) mayalso be viewed as a stationary Schr¨odinger equation in one dimensional periodic potentials expressiblewith finitely many Fourier components.Let P n ∈ Z P n ( ε, t, A, B )e n i t be the naive perturbation series (5) that reduces to A e i t + B e − i t at ε = 0.Here P n ( ε, t, A, B ) is a power series in ε which we call secular coefficient. Especially P ± ( ε, t, A, B ) areimportant in that the relevant e ± i t are the resonant harmonics. Our main finding is the functional relation(Corollary 5) P n ( ε, t, A, B ) = P n ( ε, t − s, P ( ε, s, A, B ) , P − ( ε, s, A, B )) ( ∀ n ∈ Z ) , (7) and the resulting refinements in the conventional RG approach. For example, absence of secular termsin all order of the RG series follows immediately, and the manifest bijection (30) between the bare andrenormalized amplitudes makes it unnecessary to introduce the so called renormalization constants andto resort to the implicit function theorem.Our proof of (7) is elementary and elucidates why it works naturally for V of the form (6). It wouldbe interesting to explore a geometric and/or holographic interpretation of it (cf. [3, 9]), and to seek asimilar structure in a wider class of differential equations.In Section 2 a precise definition of P n ( ε, t, A, B ) is given. In Section 3 the main result (7) is proved. InSection 4 renormalized amplitude is introduced and the RG equation is derived in one line calculation.In Section 5 the classical examples (1)–(4) are treated along the scheme of the paper. Similar analysesare available in many literatures and we have no intention to claim the originality, not to mention thebasic idea and flow of the RG analysis. The last subsection 5.5 includes an exercise on a nonlinear andnon-autonomous example. 2. Naive perturbation
We study the second order ordinary differential equation for y = y ( t ) of the form d ydt + y = εV (cid:16) ε, e i t , e − i t , y, dydt (cid:17) . (8)Here ε is a parameter with respect to which the perturbation series is to be constructed. The function V = V (cid:16) ε, e i t , e − i t , y, dydt (cid:17) , which we call the potential , is a polynomial in the indicated five variables.Namely, we assume that V has the form V = X k ∈ Z , l,m,n ∈ Z ≥ C klmn ε n e k i t y l (cid:16) dydt (cid:17) m , (9)where C klmn is the coefficient independent of ε, t and is nonzero only for finitely many quartets ( k, l, m, n ).We work in the generic complex domain, so C klmn = C ∗− klmn need not be imposed. Thus for example, V = e t y + (4 + ε e − t ) y (cid:0) dydt ) + 7e − t is covered but V = ty is not. We would like to remove thetrivial linear case V = C y + C dydt + P k D k e k i t . So the existence of nonzero C klmn with l + m ≥ | k | ≥ max(2 − l − m,
0) is assumed.In naive perturbation we will be concerned with the solutions of the form y ( ε, t ) = X n ∈ Z X k ∈ Z ≥ ε k f n,k ( t )e n i t , f n,k ( t ) = polynomial in t. (10)It is a formal power series in ε and is also a formal Laurent series in e i t . As we will see, our constructionalways leads to f n,k ( t ) = 0 for sufficiently large | n | for each fixed k . Therefore any product, say y (cid:0) dydt (cid:1) ,makes sense as a formal Laurent series in e i t .Consider the formal power series expansion y ( ε, t ) = y ( t ) + εy ( t ) + ε y ( t ) + · · · , (11)which corresponds to setting y k ( t ) = P n ∈ Z f n,k ( t )e n i t in (10). Substituting (11) into (8), we get anequation for each power of ε : d y dt + y = 0 , (12) d y dt + y = V (cid:16) , e i t , e − i t , y , dy dt (cid:17) , (13) · · · · · · (14) d y k dt + y k = h V (cid:16) ε, e i t , e − i t , k − X j =0 ε j y j , k − X j =0 ε j dy j dt (cid:17)i ε k − ( k ≥ . (15) General proof of this basic fact seems lacking in the literature. The form of V in (9) is a sufficient but not the necessary condition for the approach in this paper to work. See theremark after the proof of Lemma 3. G THEORETICAL PERTURBATION 3
Here and in what follows, (cid:2) · · · (cid:3) x denotes the coefficient of x . The general solution to (12) is y ( t ) = A e i t + B e − i t , (16)where A and B are two independent complex parameters. Starting from it, one can successively determine y k ( t ) with k = 1 , , . . . , uniquely up to the freedom of adding α k e i t + β k e − i t for arbitrary α k , β k for each k . Let us choose these α k , β k so that the constant term of f ± ,k ( t ) becomes zero for all k ≥
1. Namely,we demand f ± ,k ( t = 0) = 0 ( ∀ k ≥ . (17)This completely fixes α k , β k , again successively, for k = 1 , , , . . . . From the construction, it is easy tosee the property f n,k ( t ) = 0 ( | n | ≫ , k fixed) mentioned after (10).We define Y ( ε, t, A, B ) to be the resulting formal solution y ( ε, t ) (11). It can also depend on the otherparameters in the potential V like g in (2) and (3). This last class of parameters inherent in V will besuppressed in the notation below. By using the Y ( ε, t, A, B ), define the quantity P n ( ε, t, A, B ) to be thecoefficients occurring in the expansion into harmonics: Y ( ε, t, A, B ) = X n ∈ Z P n ( ε, t, A, B )e n i t . (18)In terms of (10), this means setting P n ( ε, t, A, B ) = P k ≥ ε k f n,k ( t ). We call A, B the bare amplitudesand P n ( ε, t, A, B ) the secular coefficient of the harmonics e n i t . The special case P ± ( ε, t, A, B ), which iscalled the resonant secular coefficient (cf. [10]), will play a key role in what follows. By the definitionthey satisfy P ± ( ε, , A, B ) = (cid:18) AB (cid:19) , (19) P n (0 , t, A, B ) = Aδ n, + Bδ n, − . (20)In contrast to (19), P n ( ε, , A, B ) with n = ± n -sum in (18) can be some subset of Z depending on thepotential V . For instance in the list (1) – (4), the sums are actually P n ∈ Z +1 except (2). In anotherexample V = y e t , the sum reduces to P n ∈ Z ≥ +1 . Example 1.
Consider Van der Pol equation (1), which corresponds to taking the potential as V =(1 − y ) dydt in (8). Then the above definition leads to P n ( ε, t, A, B ) = 0 for n even as mentioned. For odd n we have the following table, where the k ( ≥
0) th row and the n ( ≥
1) th column from the NW cornershows the polynomial f k, n − ( t ) in (10) corresponding to Y ( ε, t, A, B ). We have set C = AB to save thespace. e it e it e it e it y A y At (1 − C ) i A y At ( −
2i + 8i C − C +2 t − Ct + 6 C t ) − i A ( − − i C − t + 12 Ct ) − A y − At (96 C − C + 111 C +24i t − Ct + 444i C t − C t − t + 104 Ct − C t + 120 C t A (4i − C + 29i C − Ct + 104 C t +72i t − Ct + 120i C t ) A ( − − C − t + 60 Ct ) − A ... ... ... ... P ( ε, t, A, B ) P ( ε, t, A, B ) P ( ε, t, A, B ) P ( ε, t, A, B ) For instance one has P ( ε, t, A, B ) = − A ε
192 + 5 A ε − − C − t + 60 Ct ) + O ( ε ) . (21)In this example, P n with negative n is obtained by P − n ( ǫ, t, A, B ) = P n ( ǫ, t, B, A ) | i →− i . Such a relationis not valid in general for the potentials that are not e i t ↔ e − i t symmetric like V = y e t + 2 dydt e − i t . There is no loss of generality compared with setting f ± ,k ( t = t ) = 0 with another parameter t . This degree offreedom is essentially incorporated into the forthcoming (26). The present convention was employed implicitly in [8]. ATSUO KUNIBA
As this example demonstrates, the secular coefficients are formal power series in ε whose leading orderbehaves as P n ( ε, t, A, B ) ∼ O ( ε d n ) , d n → ∞ as | n | → ∞ . (22)3. Properties of secular coefficients
Lemma 2.
Let s be an arbitrary prameter. The formal series y = y ( ε, t ) of the form (10) that satisfiesthe differential equation (8) and the conditions (i) y (0 , t ) = A e i t + B e − i t , (ii) (cid:2) y ( ε, t ) (cid:3) e ± i t | t = s = P ± ( ε, s, A, B ) (23) is unique and given by y ( ε, t ) = Y ( ε, t, A, B ) in (18). (The LHS of (ii) means the value of the resonantsecular coefficients evaluated at t = s .)Proof. From the construction in the previous section, we know that the solution exists uniquely bychoosing the parameters α k , β k appropriately so as to fit (ii) in each order of ε . It is obvious that asolution y ( ε, t ) = Y ( ε, t, A, B ) fulfills the both (i) and (ii). (cid:3) The next lemma is the point where the specific form (9) of the potential V is used. Lemma 3.
For arbitrary parameters s, C and D , the formal series X n ∈ Z P n ( ε, t − s, C, D )e n i t (24) is also a solution to the differential equation (8).Proof. The only nontrivial claim is that shifting t to t − s in P n without changing e n i t keeps it to be asolution. To show this, regard (18) as a formal Laurent series in e i t . Then the original equation (8) isequivalent to the family of equations for the coefficient of each harmonics e m i t : ∂ P m ( ε, t, A, B ) ∂t + 2 im ∂P m ( ε, t, A, B ) ∂t + (1 − m ) P m ( ε, t, A, B )= " εV (cid:16) ε, e i t , e − i t , X n ∈ Z P n ( ε, t, A, B )e n i t , X n ∈ Z (cid:0) ∂P n ( ε, t, A, B ) ∂t + in (cid:1) e n i t (cid:17) e m i t ( ∀ m ∈ Z ) . (25)In general, the RHS contains infinite sums like P n + n + n = m +5 P n P n ∂P n ∂t when V = e − t y dydt forexample. However, thanks to (22), they are actually convergent and make sense as formal power seriesin ε . The point here is that (25) is totally an autonomous equation, i.e., all the t -dependence is via P n and its derivatives. Therefore the shifted equation t → t − s is equally valid. (cid:3) The crux of the above argument is that (25) is well-defined and autonomous. It clarifies why thenon-autonomous part of V has to be Laurent polynomials as in (9). Non e Z i t type dependence like V = (2 t e i t + 3 t ) y dydt spoils the autonomous nature of the RHS of (25), hence invalidates the proof andthe statement. We leave the consideration on a wider class of potentials like V = e y + y − i t as a futureproblem.The main result of this paper is the following. Theorem 4.
For any s, t, A and B , the following identity between the formal series is valid: X n ∈ Z P n ( ε, t, A, B )e n i t = X n ∈ Z P n ( ε, t − s, P ( ε, s, A, B ) , P − ( ε, s, A, B ))e n i t . (26) Proof.
From Lemma 2, it suffices to verify that the RHS of (26) is a solution to the equation (8) andsatisfies the conditions (i) and (ii) in (23). The fact that it is a solution is assured by Lemma 3. To check(23) is straightforward by using (20). (cid:3)
Theorem 4 tells that the RHS of (26) is independent of s . In the RG context, it implies the independenceof the choice of the initial time by a suitable renormalization of the amplitudes. The novelty of (26) isthat the required normalization is exactly achieved by P ± itself. Corollary 5.
The secular coefficients satisfy the functional relation: P n ( ε, t, A, B ) = P n ( ε, t − s, P ( ε, s, A, B ) , P − ( ε, s, A, B )) ( ∀ n ∈ Z ) . (27) G THEORETICAL PERTURBATION 5
Further specializing (27) | n = ± to t = 0 and applying (19) (with the reset s → − t ), we obtain an“inversion” formula: P ± ( ε, t, P ( ε, − t, A, B ) , P − ( ε, − t, A, B )) = (cid:18) AB (cid:19) . (28)Although our derivation of Theorem 4 and Corollary 5 has been quite elementary, their consequencesare rather nontrivial. For instance, the first nontrivial assertion of (27) about (21) is P ( ε, t, A, B ) ≡ P (cid:0) ε, t − s, A + 12 εAs (1 − C ) , B + 12 εBs (1 − C ) (cid:1) mod O ( ε ) . (29)4. Renormalized amplitude and RG equation
We introduce A r ( t ) and B r ( t ) by either one of the following two sets of relations: (cid:18) A r ( t ) B r ( t ) (cid:19) = P ± ( ε, t, A, B ) ←→ (cid:18) AB (cid:19) = P ± ( ε, − t, A r ( t ) , B r ( t )) . (30)Their equivalence is assured by the inversion relation (28). Now Corollary 5 is stated as the identity: P n ( ε, t, A, B ) = P n ( ε, t − s, A r ( s ) , B r ( s )) . (31)In particular the case s = t reads P n ( ε, t, A, B ) = P n ( ε, , A r ( t ) , B r ( t )) . (32)This relation proves that the secular t -dependence in the LHS can be eliminated totally by switchingfrom the bare amplitude A, B to the new ones A r ( t ) , B r ( t ). In this sense the variables A r ( t ) , B r ( t ) arecalled the renormalized amplitudes [2, 10, 8]. They allow us to rewrite the naive perturbation series (18)as Y ( ε, t, A, B ) = X n ∈ Z P n ( ε, , A r ( t ) , B r ( t ))e n i t = A r ( t )e i t + B r ( t )e − i t + X n ∈ Z \{± } P n ( ε, , A r ( t ) , B r ( t ))e n i t . (33)By the construction, the RG series (33) is free of secular terms to all order of ε .The remaining task is to describe the dynamics or “modulation” of the renormalized amplitudes A r ( t ) , B r ( t ) entering (33). The left relation in (30) is certainly an answer, but there is no point in sub-stituting it into (33) since it just brings us back to the original expansion (18) which is full of secularterms. So one should devise an alternative maneuver which suppresses the secular (non-autonomous) t -dependence totally and t always “stays within the house A r ( t ) , B r ( t )”. Now with the exact renormal-ization (32) at hand, this can be done in a single line: ddt (cid:18) A r ( t ) B r ( t ) (cid:19) (30) = ∂P ± ∂t ( ε, t, A, B ) (31) = ∂P ± ∂t ( ε, t − s, A r ( s ) , B r ( s )) s → t = ∂P ± ∂t ( ε, , A r ( t ) , B r ( t )) . (34)In the last step we have changed s to t . This is allowed by the s -independence due to ∂∂s (cid:0) ∂P ± ∂t ( ε, t − s, A r ( s ) , B r ( s ) (cid:1) = ∂∂t (cid:0) ∂P ± ∂s ( ε, t − s, A r ( s ) , B r ( s ) (cid:1) (31) = 0. From this maneuver it is clear that the t -derivativein the last expression of (34) does not touch A r ( t ) , B r ( t ). The differential equation (34) is called the RGor amplitude equation. One sees that the dynamics of the renormalized amplitude is governed by theresonant secular coefficients P ± to all order of ε .Let us isolate the top term of the power series P ± ( ε, t, A, B ) and name the other part as Q ± ( ε, t, A, B ): P ± ( ε, t, A, B ) = (cid:18) AB (cid:19) + εQ ± ( ε, t, A, B ) . (35)Then Q ± ( ε, t, A, B ) = P k ≥ ε k − f ± ,k ( t ) is still a power series in ε such that Q ± ( ε, , A, B ) = 0 (36)because of (19). Now the RG equation (34) is simplified slightly as ddt (cid:18) A r ( t ) B r ( t ) (cid:19) = ε ∂Q ± ∂t ( ε, , A r ( t ) , B r ( t )) , (37) At the time of writing this, the number of COVID-19 infected in the world is 32356828.
ATSUO KUNIBA where, as in (34), the t -derivative in the RHS does not concern A r ( t ) , B r ( t ). This representation indicatesthat the RG dynamics is indeed “slow” in the sense that the RHS is at last of order O ( ε ).In the earlier works [10, 8], the right relation in (30) was conventionally formulated as A = A r ( t ) Z a ( ε, t, A r ( t ) , B r ( t )) , B = B r ( t ) Z b ( ε, t, A r ( t ) , B r ( t )) (38)by further introducing the so called the renormalization constants Z a , Z b . Moreover, reversing theserelations had to be attributed to the implicit function theorem. One of the main achievements in thispaper is the manifest bijection (30) between the bare and the renormalized amplitudes that untanglesthese issues and to have elucidated its elegant origin in the functional relation (27). The renormalizationconstants mentioned in the above, although they can now be dispensed with, acquire a “closed formula”as Z a ( ε, t, A r ( t ) , B r ( t )) = A r ( t ) − P ( ε, − t, A r ( t ) , B r ( t )) = 1 + εA r ( t ) − Q ( ε, − t, A r ( t ) , B r ( t )) , (39) Z b ( ε, t, A r ( t ) , B r ( t )) = B r ( t ) − P − ( ε, − t, A r ( t ) , B r ( t )) = 1 + εB r ( t ) − Q − ( ε, − t, A r ( t ) , B r ( t )) . (40)5. Examples
Van der Pol equation.
We consider (1), which was also treated in Example 1. Introduce thevariables R = R ( t ) and θ = θ ( t ) connected to the renormalized amplitudes A r ( t ) = R ( t )e i θ ( t ) , B r ( t ) = R ( t )e − i θ ( t ) . (41)Set τ = t + θ ( t ). Then the renormalized expansion (33) reads y = 2 R cos τ − εR τ − ε R (cid:0) τ + R (3 cos 3 τ + 5 cos 5 τ ) (cid:1) − ε R (cid:0)
36 sin 3 τ − R (27 sin 3 τ + 5 sin 5 τ ) + R (261 sin 3 τ −
15 sin 5 τ −
28 sin 7 τ ) (cid:1) + O ( ε ) . (42)The RG equation (37) is given by d log Rdt = ε (1 − R )2 − ε R (32 − R + 37 R )128 + ε R ( − R − R + 4589 R )36864 − ε R (2950992 − R + 28047688 R − R − R + 4493323 R )21233664 + O ( ε ) , (43) dθdt = ε ( − R − R )16 + ε ( − − R + 1020 R − R + 497 R )3072+ ε ( − − R + 181872 R − R + 121432 R + 417540 R − R )1769472 + O ( ε ) . (44)By postulating d log Rdt = 0, one finds the values on the limit cycle:2 R c = 2 + ε − ε − ε O ( ε ) , (45) (cid:0) dθdt (cid:1) c = − ε
16 + 17 ε ε O ( ε ) . (46)The approximate leading value 2 R c = 2 is well-known from the energy balance argument that the totalwork by the friction term during a period should be zero, i.e., by requiring R π ( y − (cid:0) dydt (cid:1) dt = 0 for y = 2 R c cos t . This change of variables is optional. The original A r ( t ) and B r ( t ) equally suit the numerical work. The same featureapplies to the other equations in this section. G THEORETICAL PERTURBATION 7
Mathieu equation.
We consider (2) with g dependent on ε as g = g + g ε + g ε + · · · . Then Q ( ε, t, A, B ) defined in (35) is given by Q ( ε, t, A, B ) = i Ag t − εt (cid:0) i(8 A + 12 B + 3 Ag − Ag ) + 3 Ag t (cid:1) + ε t (cid:0) i(88 Ag + 72 Bg + 9 Ag − Ag g + 72 Ag ) + 3 Ag (8 + 3 g − g ) t − Ag t (cid:1) + O ( ε ) . (47)The other one is obtained by Q − ( ε, t, A, B ) = Q ( ε, t, B, A ) | i →− i . This example is exceptional among(1)–(4) in that it is the only equation which is linear and moreover non-autonomous. Reflecting theformer feature, the RG equation also becomes linear. In fact, differentiation of (37) can be combined andsplit into the two identical equations d A r ( t ) dt = − ω A r ( t ) , d B r ( t ) dt = − ω B r ( t ) , (48)where the constant ω is given by ω = ε g − ε g (8 + 3 g − g ) − ε
576 (80 − g − g + 192 g + 216 g g − g − g g ) − ε − g + 3920 g + 189 g − g g − g g + 1296 g g + 1152 g + 1296 g g − g g − g g ) + O ( ε ) . (49)The stable region ω > ω < A r ( t ) , B r ( t ) are separated by the curve ω = 0. Solving it order by order in ε with respect to g , g , g , . . . yields two branches. Let us presentthem for the combination a := 1 + εg which is the usual coupling constant in the conventional setting ofMathieu equation (2) as d ydt + ( a + 2 ε cos t ) y = 0. Then the branches are a = a ± , where a − = 1 − ε ε
216 + · · · , a + = 1 + 5 ε − ε
216 + · · · . (50)These curves in the ( a, ε ) plane specify the boundaries of the unstable region a − < a < a + and thestable region in the vicinity of the so called second resonance point ( a, ε ) = (1 , n th resonance is( a, ε ) = ( n ,
0) [1, sec.11.4].) The result a ± (50) agrees with the zeros of the determinants ∆ ± ( ε, a ) = 0of the semi-infinite Jacobi matrices around the resonance [6, sec.7-1], where∆ − ( ε, a ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a − εε a − εε a − εε a − εε . . . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , ∆ + ( ε, a ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a εε a − εε a − εε a − εε . . . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . It will be an interesting exercise to see how the RG series fits the exact solution [4].5.3.
Duffing equation.
Consider (3) with g = 1 which can be attained by y → y/ √ g . We haveincluded a dydt term since otherwise the equation is integrable by an elliptic function. Introduce thevariables R = R ( t ) and θ = θ ( t ) connected to the renormalized amplitudes by (41) and set τ = t + θ ( t ).Then the renormalized expansion (33) reads y = 2 R cos τ + εR τ + ε R (cid:0) τ + R (cos 5 τ −
21 cos 3 τ ) (cid:1) + ε R (cid:0) −
36 cos 3 τ − R (567 sin 3 τ −
19 sin 5 τ ) + 3 R (417 cos 3 τ −
43 cos 5 τ + cos 7 τ ) (cid:1) + O ( ε ) . (51) Symmetric, tri-diagonal matrices with positive off-diagonal elements. We imagine ε is positive to reply on this nomen-clature. ∆ − ( ε, a ) and ∆ + ( ε, a ) correspond to S e ( x ) and C e ( x ) on [6, p176], respectively. ATSUO KUNIBA
The RG equation (37) is given by d log Rdt = − ε ε R − ε R
64 + 5931 ε R
512 + ε R (16092 − R )4096 + O ( ε ) , (52) dθdt = 3 εR − ε (2 + 15 R )16 − ε R (8 − R )128 + ε ( − R − R )1024 − ε R (8 + 21305 R − R )2048 + O ( ε ) . (53)5.4. Rayleigh equation.
We consider (4). Introduce the variables R = R ( t ) and θ = θ ( t ) connected tothe renormalized amplitudes by (41) and set τ = t + θ ( t ). Then the renormalized expansion (33) reads y = 2 R cos τ + εR
12 sin 3 τ + ε R (cid:0) − τ + R (9 cos 3 τ − cos 5 τ ) (cid:1) + ε R (cid:0) −
36 sin 3 τ − R (63 sin 3 τ + 17 sin 5 τ ) + R (111 sin 3 τ + 51 sin 5 τ − τ ) (cid:1) + O ( ε ) . (54)The RG equation (37) is given by d log Rdt = ε (1 − R )2 + ε R (22 − R )128 − ε R (2268 − R − R + 1603 R )36864 + O ( ε ) , (55) dθdt = ε ( R − ε ( −
24 + 156 R − R + 65 R )3072+ ε ( − − R + 305208 R − R − R + 84627 R )1769472 + O ( ε ) . (56)5.5. A non-linear and non-autonomous example.
Finally we consider a nonlinear and non-autonomousexample: d ydt + y = 2 ε dydt y cos t. (57)Introduce the variables R = R ( t ) and θ = θ ( t ) connected to the renormalized amplitudes by (41). Thenthe renormalized expansion (33) reads y = 2 R cos( θ + t ) + 14 εR sin(2 θ + 3 t ) − ε R (cid:0) θ cos(2 θ + 3 t ) + cos(3 θ + 5 t ) (cid:1) + ε R (cid:0) −
24 sin(2 θ + 3 t ) −
33 sin(4 θ + 7 t ) + 14 sin θ cos(3 θ + 5 t ) + 72 sin 2 θ cos(2 θ + 3 t )+ 288 cos 2 θ sin(2 θ + 3 t ) −
146 cos θ sin(3 θ + 5 t ) (cid:1) + O ( ε ) . (58)The RG equation (37) is given by d log Rdt = 12 εR cos θ − ε R sin 2 θ + 516 ε R cos θ − ε R (cid:18) θ + 132 sin 4 θ (cid:19) + O (cid:0) ε (cid:1) , (59) dθdt = 12 εR sin θ − ε R (cid:18)
14 cos 2 θ + 38 (cid:19) + 18 ε R sin θ + 1128 ε R (9 cos 2 θ − θ −
3) + O (cid:0) ε (cid:1) . (60)Unlike Van de Pol, Duffing and Rayleigh equations, one has the essential mixture of R and θ in the RHSof the RG equation reflecting the non-autonomous and nonlinear nature of (57). As exemplified in thefollowing plot, there seems only one peak in the envelop R ( t ) in a certain parameter range. G THEORETICAL PERTURBATION 9
20 40 60 80 100 - - Figure 1.
Plot of y ( t ) vs t by the direct numerical solution of (57) (blue) and the RGexpansion (red) started from the same initial condition R (0) = 0 . , θ (0) = − . ε = 0 .
25. We have kept only ε , ε terms in (58) and the leading ε term in (59), (60).Taking ε term in (59) and (60) into account already makes it too difficult to observethe discrepancy. Acknowledgements.
The author thanks Yoshitsugu Oono for a communication around 2007. Thiswork is supported by Grants-in-Aid for Scientific Research No. 16H03922, 18H01141 and 18K03452 fromJSPS.
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