# Singular Perturbation of Nonlinear Systems with Regular Singularity

Authors: Domingos H. U. Marchetti; William R. P. Conti
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

We extend Balser-Kostov method of studying summability properties of a singularly perturbed inhomogeneous linear system with regular singularity at origin to nonlinear systems of the formεzf′=Fε,z,f with F a Cν-valued function, holomorphic in a polydisc D-ρ×D-ρ×D-ρν. We show that its unique formal solution in power series of ε, whose coefficients are holomorphic functions of z, is 1-summable under a Siegel-type condition on the eigenvalues of Ff(0,0,0). The estimates employed resemble the ones used in KAM theorem. A simple lemma is applied to tame convolutions that appear in the power series expansion of nonlinear equations. Applications to spherical Bessel functions and probability theory are indicated. The proposed summability method has certain advantages as it may be applied as well to (singularly perturbed) nonlinear partial differential equations of evolution type.


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## Body

## 1. Introduction

We consider singularly perturbed nonlinear systems of the form (f′ means derivative of f with respect to z) (1)εzf′=Fε,z,f,withf=f1,…,fν and F=F1,…,Fν  ν-vector functions, Fi holomorphic in a polydisc, say D-ρ×D-ρ1×D-ρ1ν for some ρ1>ρ>0 (here, Dρ(z0)=z∈C:z-z0<ρ denotes an open disc of radius ρ>0, centered at z0, D-ρz0 denotes its closure and Dρ=Dρ0) such that the ν×ν matrix A0=Ff(0,0,0) is invertible, a condition that makes (1) possess a regular singularity at z=0.When (1) is linear, i.e., F=Af-b, where b=bε,z and A=Aε,z are, respectively, a ν-vector and a ν×ν matrix, whose entries are holomorphic in the polydisc DR×DR, R>0, such that A0,0-1 exists, Balser and Kostov [1] have established the following: (a) there exists a unique formal solution in the ring O(r)ε1 of formal power series (2)f^ε,z=∑i=0∞aizεiinε with coefficients aiz in the ring Or of holomorphic functions on Dr, continuous in its closure, satisfying (3)maxz≤raiz≤Cμii!,i=0,1,2,…for some positive constantsC, μ and 0<r<R; (b) provided the closed sector S-θ,γ;E of opening angle γ about the bisecting direction θ and radius E does not contain any ray on the direction of the eigenvalues λj’s of A0,0, (4)arg⁡λj-θ>γ2,j=1,…,n,f ^ ε , z is the 1-Gevrey asymptotic expansion as ε tends to 0 of a holomorphic function fε,z in Sθ,γ;E×Dr; (c) if γ>π is chosen so that (4) holds, then the formal series f^ε,z is, by an analogue of Borel-Ritt’s theorem for Gevrey asymptotic expansion (see, e.g., Section 3.2 of [2]), 1-summable in the direction θ and its sum equals fε,z.Summability of formal solutions to singularly perturbed linear systems with irregular singularity atz=0,(5)εz1+rf′=Af-b,with the Poincaré rankr≥1, has been investigated before in [3]. Contrarily to the previous case r=0, the unique formal power series solution f^ε,z is always 1-summable, irrespective of whether (4), the additional condition satisfied by the eingenvalues of A0,0, holds or not. The case of r=-1, on the other hand, has been studied in [4] for ν=1 and the summability of the formal series can be read from the properties of the initial data of (5). The case r=0 separates the two cases and we refer to [3] for an explanation on summability properties for each of distinct cases of simple examples in which A and b=bz are a scalar and a scalar function depending only on z. For recent investigations of the linear meromorphic system (5) with r≥1, Az=0≠0, and b≡0, about summable-resurgent of the Borel transform of its highest level’s reduced formal solutions and connection-to-Stokes formulas, see [5] and references therein.In the present article, all the three statements regarding the summability of the formal seriesf^ε,z, (a)–(c) above, will be extended for nonlinear differential equations (1).The nonlinear extension of Balser-Kostov’s statements has been requested by our investigation of renormalization group (RG) flow equation over probability measures inRN, starting from a measure restricted to the sphere SN-1 of radius N, as N goes to ∞. Let fε,z be the unique extension in S0,γ;E×Dr, with ε=2/N, of the meromorphic function (6)ϕε,z=i2zJN/2izNJN/2-1izNwhereJκx is the Bessel function of order κ. This function is the logarithmic derivative of the Fourier-Stieltjes transform σ^Nx of the uniform measure σN on the N-dimensional sphere of radius N. In [6], ϕε,z is shown to satisfy a continued fraction of Gauss, convergent in z≤1/4 uniformly in ε and this domain of analyticity is extended to the upper half-plane provided ε is real. In [7], fε,z given by (6) is the initial data whose summability properties are conjectured to be preserved under the RG dynamics. We refer to [6, 7] for the statistical mechanics context.Retrospectively, the development of summability methods in probability started with the classical Wiener’s Tauberian theory (see [8, 9] for a concise and, respectively, extensive overviews). The most common types of summation methods called matrix methods may be used as a criterion for asymptotic distribution functions (mod 1) of numerical sequences (see Section 7 in Chapter 1 of [10] for a review and [11] for necessary conditions on general dynamical systems, in connection with Gaussian processes). Summability methods are also used in the context of Lagrange interpolation of zeros of Jacobi polynomials and complete monotonicity of certain functions (see [12] and references therein). Related to the present work is the Borel summability of the 1/N expansion for the N-vector statistical model at high temperatures proved in [13] (see also [14] for similar results on related models). Our investigation considers a hierarchical version of this model at the critical temperature, from a dynamical point of view.TheN dependence in the argument of the r.h.s. of (6) is chosen in such way that ϕε,z attains, as ε goes to 0, a limit function (7)ϕ0,z=-11+1+4z(see Proposition 2.1 of [6]). ϕε,z satisfies an ordinary differential equation of Riccati type (8)εzϕ′+ϕ-2zϕ2+12=0which, despite being nonlinear, can be dealt with by Balser-Kostov’s method. Equation (8) is of the form (1) with ν=1 and (Statements (a)–(c) hold with 1/2 in (8) replaced by β(ε)/2 for any 1-summable βε=∑n≥0βnεn formal series in θ direction. In this case, the limit function (7) is replaced by ϕ0=-β0/1+1+4β0z) (9)Fε,z,f=-βε2-f+2zf2Balser-Kostov summability proof in [1] of the formal series solution f^ does not follow the usual route by which the (formal) Borel transform B^f^ of f^ is analytically continued along some sector of infinite radius (see, e.g., [2]). Their proof establishes instead Gevrey asymptotic expansion directly from (5), requiring for this an auxiliary lemma regarding an infinite system of linear equations of the same type whose coefficient matrix A=Aε,z is independent of z. Although (1) is nonlinear, the system of infinitely many equations obtained by taking derivatives of (1) with respect to ε is linear and Balser-Kostov’s method carries over to equation of the form (1). To prove these statements, suitable formulas and a simple but efficient way of estimating higher power of f are provided.The layout of this paper is as follows. In Section2 (Proposition 3), we prove existence of a unique solution of (1) in power series of z. In Section 3 (Proposition 6), we show that the formal power series in ε solution of (1) is Gevrey of order 1. In Section 4 (Proposition 7) Gevrey asymptotic is established. Our main result, the 1-summability of the formal solution of (1), is stated in Section 5 (Theorem 9) and proved using Propositions 3, 6, and 7 of the previous sections. The main ingredient (Lemma 4) is employed to tame arbitrarily large number of convolutions arisen in the expansion of F in powers of f. The advantage of the proposed summability method is that it can be applied to nonlinear partial differential equations of evolution type [7, 15].


## 2. Power Series inz

Under the hypothesis onF, the series(10)Fε,z,f=∑n,m=0:n+m≠0∞An,mεznf·mconverges (in norm) absolutely inD-ρ1×D-ρ1ν, uniformly in ε∈D-ρ, with the coefficients An,mε, regarded as a multilinear operator, (11)f·m∈Cν×⋯×Cν︸mcopies⟼An,mεf·m∈Cν(12)An,mεf·mi=∑i1,…,im=1νAn,mi,i1,…,imεfi1⋯fim,endowed with an operator norm induced by the Euclidean spaceCν:(13)An,mε=supv1,…,vm∈Cmνv1≠0,…,vm≠0An,mεv1⋯vmv1⋯vm,holomorphic inDρ as a function of ε.In (12) and from now on, f=f1,…,fν denotes a ν-vector with i-th component fi and Euclidean norm f2=f·f=∑i=1νf-ifi. The product f⋯f︸m-times is denoted by f·m to distinguish from the components fi’s of f.Since the left hand side of (1) vanishes for z=0, and A00ε≡0 is assumed in (10), a solution f=fε,z of (1) in power series reads (14)fε,z=∑k=1∞fkεzk.Remark 1.
Observe that0=Fε,0,f0=∑m≥1A0,m(ε)f0·m admits a trivial solution f0≡0 for f0=fε,0 which is unique by the implicit function theorem. For the F corresponding to example (9), A0,0ε=-βε/2 does not vanish identically but f and (9) may be replaced by f~=f+β/2 and (15)F~=-1+2βzf~+β22z+2zf~2,according to which A~0,0ε=0 and f~ε,0=0 hold. Equations (1), under the hypothesis of invertible Ff(0,0,0), can always be reduced to the same form with F satisfying (10). For this, by the implicit function theorem, Fε,0,f0=0 can be solved for f0(ε) and F~ε,z,f~=Fε,z,f0+f~ satisfies F~ε,0,0=A~0,0ε≡0.Substituting the power series (14) into (10) together with (1), we are led to a system of equations (16)εjI-A0,1εfj=gjε;f1,…,fj-1withgj=gj1,…,gjν given by (17)g1iε=A1,0iε,(18)gjiε;f1,…,fj-1=Aj,0iε+∑n≥0,m≥1:2≤n+m≤j∑i1,…,im=1νAn,mi,i1,…,imεfi1∗⋯∗fimj-nforj≥2; for any two sequences α=αkk≥1 and β=βkk≥1, their convolution product α∗β=(α∗β)kk≥1 is a sequence defined by α∗β1=0 and (19)α∗βk=∑l=1k-1αlβk-l,k≥2.The restrictionn+m≤j in (18) results from the fact that our sequence fi=fkik≥1 starts with k=1 and a convolution involving m sequences cannot have nonvanishing component j-n if j<n+m.Consequently, for anyk∈N arbitrary, (16) for 1≤j≤k forms aclosed system of ν·k equations, involving ν·k unknown functions which can be solved by iteration starting from(20)f1ε=εI-A0,1ε-1A1,0ε.If (16) for 1≤j≤k-1 and k≥2 have been solved, then(21)fkε=εkI-A0,1ε-1gkε;f1,…,fk-1.Regarding the inverse matrixεkI-A0,1(ε)-1, we have the following.Lemma 2 (see Lemma 1 of [1]).
Suppose (4) holds with θ=0 and λj, j=1,…,ν, eigenvalues of A0,1(0)=Ff0,0,0. One can always find E>0 such that, if kε≥c/c-1supε≤EA0,1ε for some c>1, the inverse matrix in (21), given by (22)εkI-A0,1ε-1=∑n=0∞1εkn+1A0,1εn,is bounded and satisfies εkI-A0,1(ε)-1≤c, uniformly in S0,γ;E. If kε<c/c-1supε≤EA0,1ε, let λjε, j=1,…,n, the eigenvalues of A0,1ε, be so that their distances from every ray η=reiτ intercepting S0,γ;E are bounded from below by a constant a>0: (23)a=inf⁡λjε-reiτ:0≤r<∞,τ≤γ,j=1,…,n,ε∈S0,γ;E.Then, (24)detεkI-A0,1ε=∏j=1νεk-λjε≥aν>0together with the formula A-1=AdjA/detA for inverse of a matrix A, where AdjA is the transposed of the cofactors matrix of A (see, e.g., [16]) and the boundedness in S0,γ;E of all cofactors of A0,1ε gives (25)εkI-A0,1ε-1≤c,uniformly in S0,γ;E for every k∈N.Proposition 3.
LetF be given by (10) with the eigenvalues of A0,10 obeying hypothesis (4). There exist γ, E, and σ such that (1) has a solution fε,z holomorphic in S0,γ;E×Dσ. The solution fε,z converges, as ε→0 in the sector S0,γ;E, to the unique solution f∗z of F0,z,f=0 in Dσ satisfying f(0)=0.Proof.Since (14) solves (1), its coefficients fkε satisfy the formal relations (16) whose solution depends on the existence of inverse matrix εkI-A0,1ε-1 for every k∈N and ε∈S0,γ,E. Assuming (4) holds for every eigenvalue of A0,1(0), let γ and E be such that (23), and consequently (25), holds. Hence, fkε given by (21) is bounded uniformly in S0,γ;E, uniquely defined for every k∈N and, in view of these, holomorphic in S0,γ;E.Letϕl and αn,m be the supremum in S0,γ;E of flε and An,mε, respectively: (26)ϕl=supε∈S0,γ;Eflε(27)αn,m=supε∈S0,γ;EAn,mε.By Cauchy formula (28)1n!m!F0,n,mε,0,0ff·m=12πi2∮∮Fε,ζ,ϕf/fζn+1ϕm+1dζdϕand there exists C<∞ (=supS-0,γ;E×D-ρ1×D-ρ1νFε,z,f, E≤ρ) such that(29)αn,m≤Cρ1nρ1m.Now, we prove that the majorant series ∑l=1∞ϕlσl converges and is bounded by ρ for some 0<σ<ρ<ρ1. For this, the following lemma will play an important role here and in the further sections. See Lemma 3.1 of [17] for similar result and Lemma 2.1 of Treves [18].L e m m a   4.Letλ≥0be given and letA=1+π2/3-1/2=0.1165536…. Consider the sequenceCll=0∞withC0=AorC0=0and(30)Cl=Al!λl2,∀l≥1.Then(31)∑l=0mClCm-l≤Cmholds for everym≥0.Proof.Since ml≥1, (32)1l+1m-l=mlm-land 0≤a-b2=2a2+b2-a+b2 holds for any real numbers a and b, we have(33)1Cm∑l=0mClCm-l≤A2+∑l=1m-1m2l2m-l2≤2A1+∑l=1m-11l2+1m-l2≤2A1+π23=1.It thus follows from (31) with C0=0 that (34)∑l1,…,lk≥1:l1+⋯+lk=mCl1⋯Clk≤Cmholds for any1≤k≤m and the sum is equal to 0 for k>m.Letc and A be as in Lemmas 2 and 4. Let C and ρ1 in (29) be such that (see Remark 5) (35)α0,m≤αAcρm,m≥1(36)αn,m≤αcCn1ρn+m,n≥1,m≥0for some α>0 and 0<ρ<ρ1. Suppose(37)ϕl≤αCl1κlholds for l≥1, with Cll≥1 the sequence in Lemma 4 with λ=0 and κ<ρ.Hence, by (20) together with (25), (27), and (36), we have (38)ϕ1≤cα1,0≤αA1κand, by (21) and (18) together with (25), (39)fkε≤cAk,0ε+c∑n≥0,m≥1:2≤n+m≤kAn,mεfε∗⋯∗fε︸mk-nwith fε denoting the sequence fjεj≥1. Taking the sup over ε∈S(0,γ;E) in both sides together with (27), (34), and (36), (40)ϕk≤cαk,0+c∑1≤m≤kα·,m∗ϕ∗⋯∗ϕ︸mk≤αCk1ρk+∑m≥1αρm1κk≤αCkκkρk+αρ-α1κk≤αCk1κkholds for k≥2, provided α<ρ/2 and (41)κ=ρ1-αρ-αWith α and κ satisfying these conditions, we conclude (42)ϕl=supε∈S0,γ;Eflε≤αAl21κl,∀l≥1and fl(ε)zll≥1 is a sequence of holomorphic functions, uniformly bounded in S(0,γ;E)×Dσ by ϕlσll≥1, whose sum f(ε,z)=∑l=1∞fl(ε)zl is bounded (in norm) by(43)∑l=1∞ϕlσl=αAκκ-σ=ρprovided σ<κ satisfies σ=κ(ρ-αA)/ρ=ρ-αA1-α/(ρ-α), by (41). Under this choice of σ, Fε,z,Dρν⊂Dσν uniformly in S(0,γ;E)×Dσ and the solution we have obtained by the formal expansions (16) and (18) acquires sense. The power series solution (14) of (1) thus converges to a unique analytic function fε,z in S0,γ;E×Dσ. The proof of uniqueness will be omitted.From the uniform convergence of (14) we conclude that, for any fixed z∈Dσ, the solution fε,z tends to (44)f0,z=limS0,γ;E∋ε→0∑l=0∞flεzl=∑l=0∞limS0,γ;E∋ε→0flεzl=f∗zwhere f∗(z) is the unique solution of equation (45)F0,z,f=0for f, by the analytic implicit function theorem (see, e.g., Section 2.3 of [19]). Note that the solution f∗z is regular at z=0 since, by (14), it must satisfy f(0)=0 and this concludes the proof of Proposition 3.Remark 5.
As observed at end of Section 1 of [1], the estimate σ for the (uniformly in ε∈S0,γ;E) radius of convergence of (14) can be much smaller than the radius of the largest disk in which detA0,1(0,z)≠0, the radius of convergence for the solution of the linear system F(0,z,0)+Ff(0,z,0)f=0 at ε=0. The Cauchy majorant method (see Section 3.3 of [19]) applied to (45) yields a majorant (46)ϕz=ρ121-1-z/σ11-z/ρ1of f∗z, holomorphic in a disc Dσ1, where σ1<ρ1 depends on A0,10-1·C, with C a constant defined in (29), and ρ1. Another (not sharp) method of this type, exploiting Lemma 4 to eliminate convolutions, yields a majorant holomorphic in a disc Dσ of radius possibly smaller than σ1. Despite this, since the latter method is undeniably practical and more suitable for extensions, we shall apply it in all further sections.


## 3. Formal Power Series inε

As in (10), the double series (47)Fε,z,f=∑n,mBn,mzεnf·mconverges (in norm) absolutely inD-ρ×D-ρν, uniformly in z∈D-ρ1, with the coefficients Bn,mε regarded as a multilinear operator fm∈Cmν↦Bn,mzfm∈Cν(48)Bn,mzf·mi=∑i1,…,im=1νBn,mi,i1,…,imzfi1⋯fim.By consistency,B00(0)=0 but B00z may not be identically zero. From here on, when no ambiguity arises, we drop the dot that precedes the power of f in f·m, introduced to distinguish from its components fi.Proposition 6.
Suppose the formal power series (2) satisfies (1), formally, with F=Fε,z,f obeying the hypotheses stated after (1). Then, the coefficients aizi≥0 of (2) are analytic functions of z in the open disc Dκ and there exist positive constants C and μ such that (49)aiz≤Ci!μiholds for all i≥0 and z∈D-σ, with σ<κ<ρ. In other words, the formal power series is of Gevrey order1; that is, f^ε,z∈Oσε1.Proof.
Substituting the power series (2) into (47), we are thus led to the following equations: for i=0, we have (50)0=∑m=0∞B0,mza0mz,which has already been solved for a0z. Recall that f∗z=a0x solves F0,z,f∗=0 and the coefficients a0,j of the power series a0z=∑j=1∞a0,jzj in z satisfy (51)a0,j≤αCj1κjas the estimates for ϕj in the previous section, uniformly in ε∈S0,γ;E, hold for ε=0. However, (52)B0,mz=∑n=0∞An,m0zntogether with (50) can be used, since the An,m=An,m0 do not depend on ε, to improve the disk Dσ of convergence of the series of a0z. See Remark 5.
Fori≥1, we have (53)zai-1′z=∑m=1∞mB0,mza0m-1zaiz+∑m=1∞B·,m∗az∗⋯∗az︸mi.Observe that the sum over m has no limit as the sequence a(z)=ak(z)k≥0 starts from k=0 and the convolution product, now defined by (54)α∗βk=∑l=0kαlβk-l,k≥0,for any two sequences α=αkk≥0 and β=βkk≥0, imposes no restriction on their number.
To isolateai, the largest index term in (53), we have to show that the matrix (recall B0,10=A0,10) (55)T0z=B0,1z+∑m=2∞mB0,mza0m-1=A010I+A010-1B0,1z-B0,10+∑m=2∞mB0,mza0m-1is invertible for every z∈Dκ for some κ≤ρ. For this, we take κ so small that (56)csupz∈Dκ0B0,1z-B0,10+∑m=2∞mB0,mza0m-1≤b<1and, consequently, T0(z)-1≤c/1-b holds uniformly in Dκ0.
It follows from (53) and (55) that (57)aiz=T0z-1zai-1′z-∑m=1∞B·,m∗az∗⋯∗az︸miand this relation determines uniquely aiz in terms of earlier coefficients. Note that aiz is holomorphic in Dκ and, by (51) and (43) (58)supz∈Dκ0a0z≤δA,for any δ>0, by letting κ small enough. Now, to obtain an estimate on the growth rate of |aiz|, let φi denote the i-th Nagumo norm (59)aii≔supz∈Dκ0dκziaiz,wheredκz=κ-z.of aiz and let βn,m the supremum in Dκ of Bn,mz. The properties we shall use on Nagumo’s norms is proved in [1] and references therein and are here summarized:
( 1 )   f + g k ≤ f k + g k;
( 2 )   f g k + l ≤ f k g l;
( 3 )   f ′ k + 1 ≤ e k + 1 f k;
( 4 )   f k ≤ κ f k - 1,
for any two functionsf and g holomorphic in Dκ and nonnegative integers k,l.
Let us assume that(60)φl≤δCl1νlholds for l=1,2,…,i-1 with Cl=Al!/l2, for some positive constants δ and ν to be determined. Similar to (29) and (36), (61)βn,m=B0≤C1ρ1nρm≤δ1-bcδCnρn+mholds for some C1<∞ and ρ1 large enough. Then, it follows by (57), (61), and (34) and the properties of Nagumo norms(62)φi≤c1-bz0ai-1′i+∑n=1i∑m=1∞βn,mκn∑i1,…,im≥0:i1+⋯+im=i-nφi1⋯φim≤c1-beκiφi-1+δ1νi∑n=1i∑m=1∞Ci-nκρnCnδρm≤2c1-beκν+δρ-δδCi1νi≤δCi1νi,where the last inequality holds provided δ<ρ/2 and (63)ν≤1-b2ceκ1-δρ-δand this completes the induction: (64)supz∈Dκ0dκzlalz≡all≤δAl!l21νl∀l≥1with δ and ν fixed so that (58) and (63) hold.
By definition (59) of Nagumo norm, (65)aiz≤1κ-σiaii≤Ci!μiholds for all i≥1 uniformly in D-σ0 for some σ<κ, with C=δA and μ-1=νκ-σ, which concludes the proof of Proposition 6.


## 4. Gevrey Asymptotic

In order to set up an equation involving derivatives off with respect to ε, we write(66)ϕiε,z=1i!∂if∂εiε,zϕi′ε,z=∂ϕi∂zε,zandϕε,z=ϕi(ε,z)i≥0 for the sequence of those functions defined on S0,γ;E×Dκ0; analogously to (10) and (47), we write (67)Fε,z,f=∑m=0∞Cmε,zfmFi,0,0ε,z,f=∑m=0∞Cmi,0ε,zfmwhereh[i] stands for the i-th derivative of h with respect to the first argument divided by i!. The i-th “total derivative” of F with respect to ε (F depends on ε explicitly and implicitly through f) can thus be written as (68)Giε,z,ϕ0,…,ϕi=1i!∂i∂εiFε,z,f=∑m=0∞Cm·,0ε,z∗ϕε,z∗⋯∗ϕε,z︸mi=Tε,zϕi+G~iε,z,ϕ0,…,ϕi-1where(69)Tε,z=∑n=0∞∑m=1∞mAn,mεznϕ0ε,zm-1is a linear operator (matrix) andG~i(ε,z,ϕ0,…,ϕi-1) depends only on derivatives of f with respect to ε of order lower than i.Differentiating (1) i times with respect to ε, dividing by i!, we have (70)εzϕi′-Tε,zϕi=Hiε,zfori≥1, where(71)Hiε,z=G~iε,z,ϕ0,…,ϕi-1-zϕi-1′may be considered as inhomogeneous holomorphic function ofε,z in S(0,γ;E)×Dσ(0), and for i=0 simply (1): (72)εzϕ0′=Fε,z,ϕ0.Proposition 7.
Letf(ε,z) be the unique holomorphic solution of (1) on S0,γ;E×Dσ0 with σ, γ, and E as in Proposition 3. There exist 0<σ1≤σ, 0<E1≤E, and positive constants C and μ such that (73)ϕiε,z≤Ci!μiholds for all i≥0 and every point ε,z in S0,γ;E1×D-σ10.Proof.
The casei=0 follows straightforwardly from Proposition 3. (70) which is a linear singular perturbation equation with regular singularity which can be dealt with in the following auxiliary result due to Balser-Kostov [1] (see Lemma 3 therein). For this, we drop temporarily all subindices i in (70).
LetT be expanded in power of z,(74)Tε,z-t0ε=∑n=1∞tnεzn=Sε,zand consider a sequence ψkε,zk≥0 satisfying the system(75)εzψ0′ε,z-t0εψ0ε,z=Hε,zεzψk′ε,z-t0εψkε,z=Sε,zψk-1ε,z,k=1,2,….
By (74) and linearity, the sum over all equations in (75) yields an equation of the form (70) satisfying by the sum ψε,z=∑k=0∞ψkε,z. We assume that Hε,z admits an expansion(76)Hε,z=∑n=0∞hnεznabsolutely convergent for z≤σ, uniformly in S0,γ;E. For H given by (68) and (71) this will actually be proven by induction when we resume the proof of Proposition 7. We write, in addition, fz≪Fz if fz=∑k=0∞ckzk is majorized by Fz=∑k=0∞Ckzk, i.e., if ck≤Ck holds for all k. If f is a ν-vector or a ν×ν matrix fz≪Fz means majorized relation for each component. For any hk=hk1,…,hkν∈Cν, let hk denote the vector hk1,…,hkν and analogous notation for the matrix tk=tki,j.
L e m m a   8 . There exists a unique sequence of functions ψ k ε , z k ≥ 0 , holomorphic in S 0 , γ ; E 1 × D - σ 0 , satisfying (75). Each ψ k ε , z has a zero of order k at z = 0 : ψ k 0 , k ε , 0 = 0 , and satisfies (77) ψ ε , z = ∑ k = 0 ∞ ψ k ε , z ≪ a I - a Γ z - 1 Ω z where (78) Ω z = ∑ n = 0 ∞ s u p ε ∈ S 0 , γ ; E h n ε z n (79) Γ z = ∑ n = 0 ∞ s u p ε ∈ S 0 , γ ; E t n ε z n holds for some a < ∞ and σ small enough. ψ ε , z is, in addition, the unique analytic solution in S 0 , γ ; E 1 × D σ 0 of (80) ε z ψ ′ ε , z - T ε , z ψ ε , z = H ε , z with ψ ε , 0 = 0 .
Proof. Let us assume that ψkε,z, as a function of z and for every ε, has a zero of order k at z=0 and is represented by (81)ψkε,z=∑n=k∞ϖn,kεzn.We will show that these assumptions lead to the actual solution of (75). Plugging (81) into (75) yields (82)εnI-t0εϖn,0ε=hnε,n≥0εnI-t0εϖn,kε=∑m=k-1n-1tn-mεϖm,k-1ε,for 1≤k≤n and n≥1. Observe that, by (69) and (74), together with the fact that ϕ0ε,0=∑j≥1aj0εj (recall a00=0), (83)εnI-t0ε=εnI-A0,1εI-εnI-A0,1ε-1∑m≥1mA0,mεϕ0ε,0m-1is invertible for every ε∈DE10 if we take E1≤E so small that (84)csupε∈D-E10∑m=2∞mA0,mεϕ0ε,0m-1≤d<1and εnI-t0(ε)-1≤c/1-d≡a<∞ holds uniformly in DE10.
From these relations, we have(85)ψ0ε,z=∑n=0∞εnI+t0ε-1hnεzn,ψkε,z=∑n=k∞εnI-t0ε-1∑m=k-1n-1tn-mεϖm,k-1εzn=∑m=k-1∞∑l=1∞εm+lI-t0ε-1tlεzlϖm,k-1εzm.Defining (86)Ψkz=∑n=k∞supε∈S0,γ;E1ϖn,kεzn,it follows, by (25), (78), and (79), that (87)Ψ0z≤aΩzΨkz≤aΓzΨk-1zfor k≥1. Since ψkε,z≪Ψkz for k≥1 and ψ0ε,z≪aΩz for k=0 hold for all ε,z∈S0,γ;E1×D-σ0, we conclude (77) provided the geometric series ∑k≥1akΓσk converges for some σ. By (74) and (69) (88)Γσ=∑n=1∞supε∈S0,γ;E1tn-1εσn<1aif σ is chosen small enough and, thence, ∑k=0∞ψk(ε,z)=ψε,z is a uniformly convergent series of analytic functions in S0,γ;E1×Dσ0 which solves (80). Since no other solution of (80), regular at z=0, exists, the proof of Lemma 8 is concluded.
We continue the proof of Proposition7. It remains to show that the series (76) is uniformly convergent in S0,γ;E1×Dσ0. This follows by induction. Clearly, h0ε,z is holomorphic in S0,γ;E1×Dσ0. Suppose that ϕjε,z is holomorphic in S0,γ;E1×Dσ0 for each 1≤j<i. Then, by (71), hiε,z is holomorphic in the same domain. By Lemma 8, ϕiε,z is holomorphic in S0,γ;E1×Dσ0 and, by (71), we conclude it also holds for hi+1ε,z, justifying its representation as a convergent series (76), uniformly in S0,γ;E1×Dσ0. By induction, ϕiε,z is holomorphic in S0,γ;E1×Dσ0 for each i≥1 and (89)ϕiε,z≪aI-aΓz-1Ωiz≪aI-aΓσ-1Ωiz,where Ωi depends on the ϕjε,z with j<i. For i=0, by (78), (90)fε,z=ϕ0ε,z≤aI-aΓσ-1Ω0z≤e0holds for all ε∈S0,γ;E1 and z∈Dσ0. For i≥1, we consider the modification of Nagumo norms: (91)fj=supz∈Dσ0dσ1zj∑n=0∞supε∈S0,γ;E1n!∂nf∂znε,0zn,with dσz=σ-z. Recalling (71), (76), and (78), it follows from (89) that(92)ϕii≤a1-aΓσ1Hii,where, by (68),(93)H11≤∑mCm1,01ϕ0m≤Cϕ0ρ-ϕ0and, together with the properties of Nagumo norms, for i≥2(94)Hii≤z0ϕi-1′i+∑m∑i0,…,im≥0:i0+⋯+im=1Cmi0,0i0ϕi1i1⋯ϕimim.From these, together with (92), a recursive relation of the same type studied in Section 3 may be derived for the ϕll (see (60)–(64)) and one may conclude that (the details for this estimate are left to the reader)(95)ϕll-1≤ΔAl!l2ωlholds for all l≥1 and some suitable constants Δ and ω. Picking σ1<σ together with the property of Nagumo norms yields (96)ϕiε,z≤σσ-σ1iϕii-1≤Ci!μifor all i≥1 uniformly in S0,γ;E1×D-σ10, with C=σΔA and μ=ω/σ-σ1. We choose C=max⁡e0,σ1ΔA in order to include the i=0 case. This concludes the proof of Proposition 7.


## 5. Summability of the Formal Series

Theorem 9.
Let (1) be considered with F given by (10) where the eigenvalues of A0,10 obey hypothesis (4) for ε,z in a domain S0,γ;E×Dσ0 with γ>π. Then, there exists a radius σ>0 such that for z∈D-σ0 the formal solution f^ε,z is1-summable in θ=0 direction.Proof.
By Taylor’s theorem(97)rIε,z=ε-Ifε,z-∑i=0I-1fizεi=IεI∫0εfIζ,zε-ζI-1dζ,where the integral is along a path from 0 to ε inside S0,γ;E. This, together with Proposition 7, implies (98)rIε,z≤CI!μIfor every I and ε,z∈S′×D-σ0, with S′ any proper subsector of S0,γ;E. In addition, Proposition 6 states that f^ε,z, a formal solution of (1), is an element of O(σ)ε1 and therefore is an element of Oσε1 for any σ1<σ. Take now σ1 and E1 sufficiently small. Hence, by definition (see Section 1.5 of [2]), f^ε,z is an asymptotic expansion of order 1, as ε→0 in the sector S0,γ;E1, of fε,z, which by Proposition 3 is an analytic solution of (1) in the domain S0,γ;E1×D-σ10. Then, as γ>π, by hypothesis, f(ε,z) is the only Gevrey order 1 asymptotic expandable function in S0,γ;E1 which has f^ε,z as its asymptotic expansion, and f^ε,z is 1-summable in θ=0 direction (see, e.g., Section 3.2 of [2]).


## 6. Summary of Results

Letεzf′=Fε,z,f be a singularly perturbed nonlinear system with F=F1,…,Fν holomorphic in a polydisc D-ρ×D-ρ1×D-ρ1ν for some ρ1>ρ>0 and the ν×ν matrix A0=Ff0,0,0 invertible. In Sections 3, 4, and 5, we have proven the following:(a)
There exists a unique formal solution in the ringO(r)ε1 of formal power series f^(ε,z)=∑i=0∞aizεi in ε with coefficients aiz in the ring Or of holomorphic functions on Dr, continuous in its closure, satisfying maxz≤raiz≤Cμii!,i=0,1,2,…, for some positive constants C, μ, and 0<r<R.(b)
Provided the closed sectorS-θ,γ;E of opening angle γ about the bisecting direction θ and radius E does not contain any ray on the direction of the eigenvalues λj’s of A0,0 (the Siegel-type condition (4)), f^ε,z is the 1-Gevrey asymptotic expansion as ε tends to 0 of a holomorphic function fε,z in Sθ,γ;E×Dr.(c)
Ifγ>π is chosen so that the eigenvalues λj’s of A0,0 are Siegel-type, then the formal series f^ε,z is 1-summable in the direction θ and its sum equals fε,z.To prove (a)–(c) we have extended Balser-Kostov summability method [1], establishing in Section 4 Gevrey asymptotic expansion directly from the equation. The ingredient for this is Lemma 8 (analogous to Lemma 3 of [1]), regarding an infinite system of linear equations of type (5) with r=0 and A=Aε,z independent of z. Although (1) is nonlinear, a system of infinitely many linear equations is obtained by taking as many derivatives of (1) with respect to ε as one wishes. Lemma 4 is the additional key ingredient that has provided to all the three statements suitable formulas and simple but efficient way of estimating higher power of f.The nonlinear extension of Balser-Kostov summability method has been requested by our investigation of renormalization group (RG) flow equation over probability measuresσN in RN as N goes to ∞. In terms of its Fourier transform σ^N(t,x)=exp-Ut,x the initial value problem of the RG flow equation reads(99)Ut+12ΔU-∇U2+3x·∇U-4U-12ΔU-∇U2x=0=0,t,x∈R+×RNwithσ^0Nx=ΓN/2JN/2-1βNxβNx/2-N/2-1=exp-U0,x, the Fourier transform of the uniform measure σ0N on the N-dimensional sphere of radius N. We observe that the Gaussian measure σ^N,Gauss(x)=exp-x2=exp-UGaussx is one stationary solution of the flow equation (99). The problem is to find β∈0,∞ so that the initial uniform measure σ0N is driven by RG flow to the Gaussian σN,Gauss stationary point, interpreted in probability theory as the Gaussian scaling limit of nonlinear process (99), starting from the uniform measure on the N-sphere. The paper [6] deals with the N=∞ case, for which the critical value is βc=4. The large N case is approached through resummation method [7] writing the Legendre transform of the RG flow and β in formal power series expansion in ε.We have used the radial coordinatez=-x2 to describe the flow: uNt,z=Ut,Nx/N. As explained in the introduction, the result of the present paper applies to the initial condition uN0,z. Let fε,z be the unique extension in S0,γ;E×Dr, with ε=2/N and β=4, of the meromorphic function (100)uN0,z′=i2zJN/22iβzNJN/2-12iβzN=ϕε,z.Then,ϕ satisfies the Riccati type differential equation (8), which is of the forma (1) with ν=1 and F given by (9), so the above summarized statements apply in this case. The usefulness of Balser-Kostov summability method combined with Lemma 4 is illustrated in [7, 15]. The method when applied to the RG flow equation (99) yields existence of formal series in ε together with Gevrey estimates to the coefficients, uniformly in t (global existence) at the critical value βcε.



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