new development of asymptotic analysis and dynamical

New Development of Asymptotic Analysis andDynamical Systems

(漸近解析と力学系の新展開)

Date: 2009, June 15(Mon) − June 19(Fri.)Place: RIMS Kyoto University, Room Number 115

Program

June 15 (Monday)13:30 − 14:30 OUCHI, Sunao (大内　忠) (Sophia University)

Multisummability of solutions of formal power seriesof differential equations I

14:45 − 15:45 YAMAZAWA, Hiroshi (山澤浩司) (Caritas University)On construction of true solution with logarithm termand multisummability for some linear partial differential equations

16:00 − 17:00 MIYAKE, Masatake (三宅正武) (Nagoya University)Newton polygon and Gevrey hierarchy in index formulasof a singular system of ODOps on formal Gevrey spaces

June 16 (Tuesday)9:45 − 10:45 YOSHINO, Masafumi (吉野正史) (Hiroshima University)

Resummation and analytic continuation of an asymptoticsolution of a small denominator problem

11:00 − 12:00 OUCHI, Sunao (大内忠) (Sophia University)Multisummability of solutions of formal power seriesof differential equations II

13:30 − 14:30 CHIBA, Hayato (千葉逸人) (Kyoto University)Fast-slow systems with codimension 2 fold points

14:45 − 15:45 MATSUOKA, Chihiro (松岡千博) (Ehime University)Borel-Laplace transform of a difference equation associatedwith Henon maps (joint work with HIRAIDE, Koichi)

16:00 − 17:00 ITO, Hidekazu (伊藤秀一) (Kanazawa University)Generalized action-angle coordinates near singularities ofsuperintegrable systems

18:00 − banquet (The place is to be announced.)

June 17 (Wednesday)10:00 − 11:00 COSTIN, Ovidiu (Ohio State University)

Behavior of lacunary series at the natural boundary(joint work with M. Huang)

11:15 − 12:15 COSTIN, Rodica D. (Ohio State University)Differential systems with Fuchsian linear part:correction and linearization, normal forms and matrix valuedorthogonal polynomials

free time in the afternoon1

June 18 (Thursday)9:45 − 10:45 COSTIN, Ovidiu (Ohio State University)

The one-dimensional Schrodinger equation for potential wellsand barriers. Borel summation and Gamow vectors.(joint work with M. Huang)

11:00 − 12:00 SHUDO, Akira (首藤啓) (Tokyo Metropolitan University)Role of natural boundaries of KAM curves in dynamicaltunneling problems

13:30 − 14:30 KAMIMOTO, Shingo (神本晋吾) (University of Tokyo)On a Schrodinger operator with a merging pair of a simple poleand a simple turning point, I− WKB theoretic transformation to the canonical form

14:45 − 15:45 KOIKE, Tatsuya (小池達也) (Kobe University)On a Schrodinger operator with a merging pair of a simple poleand a simple turning point, II− Computation of Voros coefficients and its consequence

16:00 − 17:00 UMETA, Youko (梅田陽子) (Hokkaido University)Multiple-scale analysis for the first Painleve hierarchy(joint work with Takashi Aoki and Naofumi Honda)

June 19 (Friday)9:30 − 10:30 TAKAHASHI, Kinya (高橋公也) (Kyushu Institute Technology )

Semiclassical analysis of multi-dimensional barrier tunneling10:40 − 11:40 SASAKI, Shinji (佐々木真二) (RIMS Kyoto University)

On the classification of Stokes graphs for second orderFuchsian-type equations

11:15 − 12:15 TAKEI, Yoshitsugu (竹井義次) (RIMS Kyoto University)Summability of formal solutions of PDEs and the geometryof Stokes curves

free discussions in the afternoon

The up-to-date program and the abstracts are also available at

http://www.math.sci.hiroshima-u.ac.jp/~yoshino/rims0906.html

Supported by RIMS Kyoto University and partially supported by Yoshitsugu Takei(RIMS , Grant-in-Aid for Scientific Research, No. 21340029) and Masafumi Yoshino(Hiroshima Univ. Grant-in-Aid for Scientific Research, No. 20540172).

On construction of true solution with logarithmterm and multisummability for some linear

partial differential equations

Hiroshi YAMAZAWADepartment of Language and Culture, Caritas Junior College

In this tale, let us study a construction of a true solution for some linear partialdifferential equations by using Theory of Multisummability. For example let usconsider the following linear partial differential equation:

(t, x) ∈ C × C,

(t∂

∂t− ρ(x)

)u(t, x) =

2∑

j=0

ajt(t∂

∂t

)j( ∂∂x

)2−ju(t, x) + f(t, x)

(0.1)

where

· aj ∈ C for j = 0, 1, 2

· a function ρ(x) is holomorphic in a neighborhood of x = 0 with Reρ(0) > 0,ρ(0) �∈ N = {0, 1, . . .}

· a function f(t, x) is holomorphic in a neighborhood of (t, x) = (0, 0)with f(0, x) ≡ 0.

For this equation, there exists the following formal solution:

U(ϕ)(t, x) =∞∑

i=1

ui(x)ti + ϕ(x)tρ(x) + tρ(x)

∑

k≤2ii≥1

ϕi,k(x)ti(log t)k(0.2)

where {ui(x)}i≥1, ϕ(x) and {ϕi,k(x)}k≤2i,i≥1 are holomorphic functions in a commonneighborhood of x = 0. Moreover ϕ(x) is any holomorphic function and a functionϕi,k(x) depends on ϕ(x). For this formal solution U(ϕ), we have that

∞∑

i=1

ui(x)

Γ(i)ti + ϕ(x)tρ(x) + tρ(x)

∑

k≤2ii≥1

ϕi,k(x)

Γ(i)ti(log t)k(0.3)

is converges for a sufficiently small t.

1

In the case a0 �= 0 or a1 �= 0, we can show that we have a true solution with theasymptotic expansion U(ϕ) by [O1] (Ouchi ’95).

At next we consider the case a0 = a1 = 0 and a2 �= 0. If we take ϕ(x) ≡ 0, thenwe have ϕi,k(x) ≡ 0 for all (i, k), that is, the formal solution U(0) becomes a formalpower series solution.

For this case, we have a true solution by using Theory of Multisummability by[O2] (Ouchi ’02). Many mathematicians study Theory of Summability; W. Balser,M. Miyake, Y. Sibuya and M. Hibino and so on.

In this talk in the case ϕ(x) �≡ 0 we study a construction of a true solution withan asymptotic expansion U(ϕ) by using Theory of Multisummability. In fact, theformal function

ϕ(x)tρ(x) + tρ(x)∑

k≤2ii≥1

ϕi,k(x)ti(log t)k(0.4)

satisfies the equation

(t∂

∂t− ρ(x)

)u(t, x) =

2∑

j=0

ajt(t∂

∂t

)j( ∂∂x

)2−ju(t, x).(0.5)

Then as generally we will unfold Theory of Summability for the following equation:

(t∂

∂t− ρ(x)

)φ(t, x) =

∑

j+|α|≤m

aj,α(t, x)(t∂

∂t

)j( ∂∂x

)αφ(t, x)(0.6)

where (t, x) ∈ C×Cd and aj,α(t, x) is holomorphic in (t, x) = (0, 0) with aj,α(0, x) ≡0.

Suppose that it states in detail at the time in a lecture.

References

[1] Ouchi, S., Genuine solutions and formal solutions with Gevrey type estimatesof nonlinear partial differential equations, J. Math. Sci. Univ Tokyo 1 (1995),375–417.

[2] , Multisummability of Formal Solutions of Some Linear Partial Differ-ential Equations, J. Diff. Equ. 185 (2002), 513–549.

2

Newton polygon and Gevrey hierarchy in index formulas of

a singular system of ODOps on formal Gevrey spaces

Masatake MIYAKE (Nagoya University)

1. IntroductionWe study a system of ordinary differential operators L ≡ L(z,D), which is singular

at z = 0, of the following form,

(1) L(z,D) = zp+1D − A(z), A(z) = (aij(z)) ∈ MN(R{z}),

where p ∈ N := {0, 1, 2, 3, · · · }, z ∈ C, D = d/dz and MN(R{z}) denotes the set of N×Nmatrices with entries of meromorphic functions at z = 0 which we denote by R{z}.

The purpose of this study is to define the Newton polygon N(L) of the system L(z,D),and to prove an index formula on formal Gevrey space which is explicitly calculated fromthe Newton polygon N(L). This gives an extension of the theory for a single operator byJ.-P. Ramis [Ram] to a system of operators.

List of Notations :

1. C[z], C{z}, C[[z]] : polynomials, convergent series, formal poer series of z over C,respectively.

2. R[z] := C[z]/C[z] = {f(z)/g(z) ; f(z), g(z) ∈ C[z]}, R{z} := C{z}/C{z}, R[[z]] :=C[[z]/C[[z]] : : rational functions, meromorphic functions, formal meromorphic func-tions which may have pole singularities at z = 0, respectively.

3. MN(R), GLN(R) : N × N -matrices, invertible matrices of entries in a ring R,respectively.

2. Review of Newton polygon and index formula by J.-P. RamisWe give a short review of Newton polygon for a single operator and an index formula

on formal Gevrey space proved by J.-P. Ramis [Ram].Let a single operator P ≡ P (z,D) be given by

(2) P = a0(z)(zD)m +m∑

j=1

aj(z)(zD)m−j, aj(z) ∈ C{z}, a0(z) ≡ 0.

We denote by O(aj) = rj ∈ N ∪ {+∞} the order of zeros at z = 0, and consider thefollowing correspondence between an operator and a figure,

aj(z)(zD)m−j ←→ Q(m − j, rj) := {(x, y) ∈ R2 ; x ≤ m − j, y ≥ rj}.

Then the Newton polygon N(P ) is defined by

(3) N(P ) := Convex − hull

{m∪

j=0

Q(m − j, rj)

}.

1

The Newton polygon N(P ) is characterized by its vertexes which we denote by{(m − ji, rji

)}ki=0 (k ≥ 0) with

(4) 0 = j0 < j1 < · · · < jk ≤ m, rj0 > rj1 > · · · > rjk.

Let the slopes of k + 2 sides of N(P ) denoted by {ρi}ki=−1 be given by

(5) ρi =ri − ri+1

ji+1 − ji

, ρ−1 := ∞ > ρ0 > ρ1 > · · · > ρk−1 > 0 =: ρk.

We note that the operator P (z,D) is regular singular at z = 0 if and only if k = 0.J.-P. Ramis showed that these verteces and slopes characterize the operator P (z,D)

on formal Gevrey space Gs (1 ≤ s ≤ ∞), where Gs ⊂ C[[z]] is defined by

(6) u(z) =∑∞

n=0unzn ∈ Gs def.⇐⇒

∑∞

n=0un

ζn

(n!)s−1∈ C{ζ} (1 ≤ s < ∞),

and G∞ := C[[z]].Then he proved that the mapping P (z,D) : Gs → Gs defines a Fredholm operator for

every s ∈ [1,∞], and its index χ(P ;Gs) := dimC ker(P ;Gs)−dimC coker(P;Gs) is obtainedas follows: The index χ(P ;Gs) is a right continuous step function on s ∈ [1, +∞] with kdiscontinuous points {si}k−1

i=0 given by

(7) si = 1 + 1/ρi, s−1 := 1 < s0 < s1 < · · · < sk−1 < +∞ =: sk,

and the index is given by

(8) χ(P ;Gs) = −rji, si−1 ≤ s < si, (0 ≤ i ≤ k).

When i = k the formula holds for sk−1 ≤ s ≤ sk = +∞.

Remark 1. The index formula was proved by Malgrange [Mal] in cases s = 1 and ∞,and by Komatsu [Kom] for a normalized system of higher order operators in case s = 1.For a general system of operators, the formula was proved in a joint paper [M-Y] with M.Yoshino in case s ∈ R for operators with polynomial coefficients, where the determinanttheory of matrices of differential operators on formal Gevrey space was employed. Wenote that the restriction of polynomial coefficients is removed for 1 ≤ s ≤ ∞ by compactperturbation argument.

3. Statement of ResultsWe define the order of a meromorphic function as follows. For a non zero formal

meromorphic function a(z) ∈ R[[z]],

(9) O(a) := r ∈ Z ⇐⇒ a(z) = zrb(z), b(z) ∈ C[[z]], b(0) = 0.

and if a(z) = 0 we define O(a) := +∞. The order O(A) of a matrix function A(z) isdefined similarly.

Def.3.1 (Volevic’s weight) For a matrix function A(z) = (aij(z)) ∈ MN(R[[z]]), letrij = O(aij) ∈ Z ∪ {+∞}. Then Volevic’s weight V (A) ∈ Q ∪ {+∞} is defined by

(10) V (A) := min1≤n≤N

min1≤i1<···<in≤N

minσ∈Sn

1

n

∑n

k=1rik,iσ(k)

.

2

Def.3.2 (Full rank system of irregular singular type) A system L = zp+1D−A(z)(A(z) ∈ MN(R[[z]]) is called a full rank system of irregular type if

(11) V (A) < p, O(det A(z)) = NV (A).

The definition implies for the characteristic polynomial det(λI−A(z)) = λN−∑N

j=1 aj(z)λN−j

thatO(aj) ≥ jV (A), O(aN) = NV (A),

but the converse does not hold.

Def.3.3 (Regular singular type) A system L is called of regular singular type ifV (A) ≥ p.

Theorem A (Reduction to a canonical form) A singular system L with A(z) ∈MN(R{z}) is reduced into a canonical form by a formal meromorphic transformation asfollows: ∃ P (z) ∈ GLN(R[[z]]), ∃ N = N1 + · · · + Nk + Nk+1 (Nj ≥ 0, k ≥ 0) such that

(12) P−1(z)L(z,D)P (z) = zp+1D − Triang(A1(z), · · · , Ak(z), Ak+1(z))

where Triang(· · · ) denotes a block wise diagonal matrix with j-th diagonal block Aj(z),where

Lj(z,D) := zp+1D − Aj(z), Aj(z) ∈ MNj(R[[z]]),

is full rank system of irregular singular type for 1 ≤ j ≤ k and Lk+1(z,D) is of regularsingular type with a property

(13) V (A1) < V (A2) < · · · < V (Ak) < p ≤ V (Ak+1).

Def.3.4 (Newton polygon) For a singular system L with A(z) ∈ MN(R{z}), theNewton polygon N(L) is defined by

(14) N(L) := N(det(L1(z, ζ)) + · · · + N(det Lk(z, ζ)) + N(det Lk+1(z, ζ)).

We note that the Newton polygon N(p) for a polynomial p(z, ζ) =∑m

j=0 aj(z)(zζ)j isdefined similarly with an operator p(z,D).

Theorem B (Index formula) A singular system L with A(z) ∈ MN(C{z}) defines aFredholm operator on Gs (1 ≤ s ≤ ∞), and the index χ(L;Gs) is obtained similarly witha single operator by using the Newton polygon N(L).

Remark 2. In a recent paper with K. Ichinobe [M-I], we defined and characterized theirregularity of solutions of L(z,D)u(z) = 0 as a maximal rate of exponential growth whenz → 0, which we denoted by ρ(L). Now we know that this problem is equivalent tocharacterize the steepest slope of the Newton polygon N(L). The arguments developedin the paper play important and powerful roles in this study.

References

[Kom] Komatsu, H., J. Fac. Sci. Univ. Tokyo Sect. IA 18 (1971), 379-398. [Mal] Mal-grange, B., Enseignement Math. 20 (1970), 146-176. [M-I] Miyake, M. and Ichinobe,K., Funkcial. Ekvac., 52 (2009), 53 - 82 . [M-Y] Miyake, M. and Yoshino, M., Funkcial.Ekvac., 38 (1995), 329-342. [Ram] Ramis, J.-P., Mom. Amer. Math. Soc., 48 (1984).

3

Fast-slow systems with codimension 2 fold pointsDepartment of Applied Mathematics and Physics

Kyoto University, Kyoto, 606-8501, Japan

Hayato CHIBA *1

Abstract. The existence of stable periodic orbits and chaotic invariant sets of singularlyperturbed problems of fast-slow type with codimension two fold points is proved by meansof the boundary layer technique and the blow-up method. In particular, the blow-up methodis effectively used for analyzing the flow near the fold points in order to show that an orbitnear the fold points is extended along a solution of the first Painleve equation in the blow-upspace.

1 Introduction

Singularly perturbed ordinary differential equations of the form

x1 = f1(x1, · · · , xn, y1, · · · , ym, ε),...

xn = fn(x1, · · · , xn, y1, · · · , ym, ε),y1 = εg1(x1, · · · , xn, y1, · · · , ym, ε),...

ym = εgm(x1, · · · , xn, y1, · · · , ym, ε),

(1.1)

are called a fast-slow system, where the dot ( ˙ ) denotes the derivative with respect to time t,

and where ε > 0 is a small parameter. The unperturbed system of this system is given by

x1 = f1(x1, · · · , xn, y1, · · · , ym, 0),...

xn = fn(x1, · · · , xn, y1, · · · , ym, 0),y1 = 0,...

ym = 0.

(1.2)

The set of fixed points of the unperturbed system is called a critical manifold, which is defined

by

M = {(x1, · · · , xn, y1, · · · , ym) ∈ Rn+m | fi(x1, · · · , xn, y1, · · · , ym, 0) = 0, i = 1, · · · , n}. (1.3)

Typically M is an m-dimensional manifold. It is known that the shape and stability of the

critical manifold determine global behavior of the flow of the fast-slow system.

*1 E mail address : [email protected]

1

2 Main results

In this talk, we investigate the 3-dimensional fast-slow system of the form

x = f1(x, y, z, ε),y = f2(x, y, z, ε),z = εg(x, y, z, ε),

(2.1)

with a small parameter ε > 0. Note that the critical manifold

M = {(x, y, z) ∈ R3 | f1(x, y, z, 0) = f2(x, y, z, 0) = 0} (2.2)

gives curves on R3 in general. Under some assumptions for the shape and stability of the

critical manifold (see Fig.1), we will prove that there exists a stable periodic orbit if ε > 0

is sufficiently small. Further, we will prove that if ε increases, a succession of the period-

doubling bifurcations occurs and it induces a chaotic invariant set.

L+

S+

+

L--

-a

S+r

Sa

-Sr

Fig. 1 Critical manifold M = S +a ∪ S +r ∪ S −a ∪ S −r and the flow of the unperturbed

system of Eq.(2.1).

To show the existence of a stable periodic orbit or a chaotic invariant set, we have to

calculate a succession of the transition maps (Poincare maps) along the flow of Eq.(2.1) as

Fig.2. It is shown that if ε > 0 is sufficiently small, the global Poincare map is contractive and

thus a stable periodic orbit exists (Fig.3(a)). On the other hand, if ε > 0 increases, the global

Poincare map proves to have Smale horseshoe and thus the existence of a chaotic invariant

set is proved (Fig.3(b)).

When calculating the transition maps near the fold points L± of the critical manifold, the

blow-up method is effectively used in order to “de-singularize” the fold points. It is shown

that in the blow-up space, a solution orbit of the system (2.1) is extended along a solution of

the first Painleve equation.

2

+

_

_

_

_

SR a

Sa

+

+

_L

Lout

out

+in

+II

II

in

Fig. 2 Poincare sections and a schematic view of the images of the rectangle R under

a succession of the transition maps.

(a) (b)

R

Fig. 3 Positional relationship between the rectangle R and its image under the Poincare map.

References

[1] H. Chiba, Fast-slow systems with codimension 2 fold points, (submitted;http://yang.amp.i.kyoto-u.ac.jp/˜chiba/fast slow1.pdf)

3

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Summability of formal solutions of PDEs

and the geometry of Stokes curves

Yoshitsugu TAKEIResearch Institute for Mathematical Sciences

Kyoto UniversityKyoto, 606-8502 Japan

Let us consider the following initial value problem for a partial differential equa-tion in two complex variables (t, z) ∈ C:

(1)

(∂

∂t− a(z)

∂2

∂z2

)u(t, z) = 0,

u(0, z) = ϕ(z).

As one can readily observe, the initial value problem (1) has a solution of the form

(2) u(t, z) =

∞∑

j=0

(a(z)

d2

dz2

)j

ϕ(z)tj

j!,

which is in general divergent for an analytic initial value ϕ(z). More generally, sucha formal (divergent) solution u(t, z) also exists for the following equation as well:

(3)

(∂

∂t− P (z,

∂

∂z)

)u(t, z) = 0,

u(0, z) = ϕ(z).

Since the pioneering work of Lutz-Miyake-Schafke for the heat equation ([LMS]),many works have been done on the summability of the formal solution u(t, z) forthese initial value problems. See, e.g., [B], [M], [CT], [BL], . . . . In these works thecase of partial differential equations with constant coefficients are mainly discussed.

In the case of equations with variable coefficients, on the other hand, very fewthings are known even for such a simple equation as (1). In this talk, through theinvestigation of several typical concrete equations, we would like to show that thegeometry of Stokes curves is related to the summability of u(t, z) in the variable

1

coefficients case. To be more specific, we apply the Laplace transformation Lt→τ

for the variable t to Equation (1) and employ the scaling τ = η2 to obtain a one-dimensional Schrodinger equation

(4)

(∂2

∂z2− η2

1

a(z)

)ψ(z, η) = 0 with ψ = Lu.

Then we see that the WKB solutions and the Stokes geometry of (4) play an im-portant role in the study of the resummation of the formal solution u(t, z) of theoriginal partial differential equation (1). The details will be discussed in the talk.

References

[B] W. Balser: Multisummability of formal power series solutions of partialdifferential equations with constant coefficients, J. Differential Equations,201(2004), 63-74.

[BL] W. Balser and M. Loday-Richaud: Summability of solutions of the heat equa-tion with inhomogeneous thermal conductivity in two variables, preprint,2009.

[CT] O. Costin and S. Tanveer: Nonlinear evolution PDEs in R+× Cd: existenceand uniqueness of solutions, asymptotic and Borel summability properties,Ann. Inst. H. Poincare Anal. Non Lineaire, 24(2007), 795-823.

[KT] T. Kawai and Y. Takei: Algebraic Analysis of Singular Perturbation Theory,AMS, 2005.

[Ko1] T. Koike: On a regular singular point in the exact WKB analysis, Toward

the Exact WKB Analysis of Differential Equations, Linear or Non-Linear,Kyoto Univ. Press, 2000, pp. 39-53.

[Ko2] : On the exact WKB analysis of second order linear ordinary dif-ferential equations with simple poles, Publ. Res. Inst. Math. Sci., 36(2000),297-319.

[LMS] D.A. Lutz, M. Miyake and R. Schafke: On the Borel summability of divergentsolutions of the heat equation, Nagoya Math. J., 154(1999), 1-29.

[M] M. Miyake: Borel summability of divergent solutions of the Cauchy problemto non-Kowalevskian equations, Partial Differential Equations and Their Ap-

plications, World Sci. Publ., River Edge, NJ, 1999, pp. 225-239.

2

Resummation and analytic continuation of anasymptotic solution of a small denominator problem

Masafumi YOSHINO,Graduate School of Science, Hiroshima University

1. Linearization problem. Let y = (y1, . . . , yn) ∈ Cn, n ≥ 2 be the variable in C

n. Weconsider a semi-simple holomorphic singular vector field X =

∑nj=1 aj(y)

∂∂yj

, aj(0) = 0

(j = 1, . . . , n) defined in some neighborhood of the origin of Cn. The following equationsgive the change of the coordinates y = x+ v(x) which linearizes X

(H) ε−1Lvj = λjvj +Rj(x+ v(x)), L =n∑

j=1

λjxj∂

∂xj, j = 1, . . . , n,

where R = (R1, . . . , Rn) is the nonlinear part of X , R = O(|x|2) when |x| → 0.

We can uniquely construct a singular perturbative solution (SP-solution in short) of (H)v(x, ε) in the form

v(x, ε) =∞∑

ν=0

ε−νvν(x) = v0(x) + ε−1v1(x) + · · · ,

where the sum with respect to ε is a formal sum, and vν(x) are vector-valued functionswhich are holomorphic in some neighborhood of the origin independent of ν.

2. Borel-Laplace resummation. We use the Borel-Laplace transform with respect toε in order to construct an exact solution of (H) although (H) is a semilinear equation.Let the direction ξ (0 ≤ ξ < 2π) and the opening θ > 0 are given. We define Sξ,θ bySξ,θ =

{ε ∈ C; |arg ε− ξ| < θ

2

}. Then we have

Theorem 2. (Resummation) . Assume that the Poincare condition or the followingcondition is satisfied.

∃ t, 0 ≤ t ≤ 2π, eitλj ∈ R \ {0} (j = 1, 2, . . . , n).(0.1)

Then there exist a ξ, θ > 0, a neighborhood U of the origin x = 0 and V (x, ε) such thatV (x, ε) is holomorphic in (x, ε) ∈ U × Sξ,θ and it gives a solution of (H). The SP-solutionv(x, ε) is the G2-asymptotic expansion of V (x, ε) in U × Sξ,θ, when ε → ∞. Namely, foreach N ≥ 1 and R > 0, there exist C > 0 and K > 0 such that

∣∣∣∣∣V (x, ε)−N∑

ν=0

ε−νvν(x)

∣∣∣∣∣ ≤ CKNN !|ε|−N−1,(0.2)

for all (x, ε) ∈ U × Sξ,θ, |ε| ≥ R.

3. Analytic continuation of SP-solution. We now study the analytic continuationwith respect to ε of the resummed SP-solution, and we apply it to the study of asymptoticproperty of divergent series which appears in solving (H) in a small denominator case.

First we study the analytic continuation in the Poincare case. We recall that thereexist an infinite number of ε’s on the right half-plane � ε > 0 for which the resonaceoccurs. For the sake of simplicity we call these values of ε ε-resonances. These valuesof ε accumulate only at infinity in the Poincare case, and may accumulate on the realline in other cases by the assumption (0.1). The resummed SP-solution V (x, ε) may besingular with respect to ε at these values in view of the proof of Theorem 2. Our first

1

2

observation is that the analytic continuation (with respect to ε) of the SP-solution to theright half-plane is identical with the classical Poincare series solution. More precisely wehave

Theorem 3. Suppose that the Poincare condition is verified. Then V (x, ε) is ana-lytically continued as a single-valued function of ε to a non ε-resonant point along anybounded path which does not meet with the ε-resonances . If ε = 1 is nonresonant, thenthe analytic continuation of the resummed SP-solution to ε = 1 coincides with the clas-sical Poincare series solution of (H).

Now we study the solvability of (H) without assuming a Diophantine condition. Let σbe a nonnegative integer and Γ be an open connected neighborhood of the origin, andlet 0 ≤ c < π/2. We define Hσ ≡ Hσ,c,Γ as the set of holomorphic (vector) funtionsv(ζ) = (v1(ζ), . . . , vN(ζ)) of ζ = η + iξ ∈ Γ + iRn such that

v σ,c,Γ := supη∈Γ

∫

�n

〈ζ〉σec|ξ||v(ζ)|dξ <∞,(0.3)

where 〈ζ〉 = 1 +∑n

j=1 |ζj|, |ξ| = |ξ1| + · · · + |ξn|, and |v(ζ)| = (∑N

j=1 |vj(ζ)|2)1/2. The

space Hσ,c,Γ is a Banach space with the norm (0.3). We define the multi sector Sc by

Sc := (S0)n, S0 = {z ∈ C; z = reiθ, |θ| < c, r > 0}.(0.4)

Let f(x) be an integrable N - vector function on Rn+, R+ := {t ∈ R; t ≥ 0} and let f(ζ)

be the Mellin transform of f

f(ζ) ≡ M(f)(ζ) =

∫

�n+

f(x)xζ−edx, e = (1, . . . , 1), ζ = η + iξ, η ∈ Γ, ξ ∈ Rn,(0.5)

where xζ−e = xζ1−11 · · ·xζn−1

n , ζ = (ζ1, . . . , ζn). It is easy to see that f (ζ) is analytic if theintegral (0.5) absolutely converges. The inverse Mellin transform is given by

f(x) = M−1(f )(x) = (2πi)−n

∫

�n

f(η + iξ)x−η−iξdξ,(0.6)

where η is so taken that the integral converges. We note that if f ∈ Hσ,c,Γ, then theintegral (0.6) is a holomorphic function of x in Sc. We note that these formulas followfrom the corresponding ones of the Fourier-Laplace transform by the change of variableseiθj → xj.

We define Hσ,c,Γ as the inverse Mellin transform of Hσ,c,Γ. We note that the Mellintransform gives the one to one correspondence between the spaces Hσ,c,Γ and Hσ,c,Γ. Foru ∈ Hσ,c,Γ we define the norm ‖u‖σ,c,Γ of u by

‖u‖σ,c,Γ := M(u) σ,c,Γ.

For an integer k ≥ 1 we denote by (Hσ,c,Γ)k the product of k copies of Hσ,c,Γ. Thenorm in (Hσ,c,Γ)k is defined as the sum of the norms of each component. For simplicity,we denote the norm in (Hσ,c,Γ)k by ‖ · ‖σ,c,Γ if there is no fear of confusion.

Let Γ0 be a connected neighborhood of the origin in Rn. Let Γ1 ⊂ −Rn+ be an open

connected convex cone with vertex at the origin such that 〈λ, η〉 �= 0 for every η ∈ Γ1,

η �= 0. Let �Γ(ζ) := Γ(ζ1) · · ·Γ(ζn), where Γ(z) is the Gamma function. Let R(ζ) be the

3

Mellin transform of R(ζ). We assume that the nonlinear term Rj(x) (j = 1, 2, . . . , n) hasthe form

Rj(ζ) =∑

α∈�n+

rjα(ζ)�Γ(ζ + α) (j = 1, 2, . . . , n).(0.7)

Here rjα(ζ) is an entire function such that there exists K ≥ 1 satisfying that |rj

α(ζ)| ≤e−K|ξ|/|α||α| when ξ = �ζ → ∞.

Example. Let α = (α1, . . . , αn), αk > 0 and let cj be given constants. If Rj(x) (j =1, 2, . . . , n) are given by Rj(x) =

∑α,finite cj,αx

α exp(−x1 − · · · − xn), then the condition(0.7) holds.

Let 0 < θ0 < π/2 be a given constant, and let τ0 = ±π/2. We define the sector Στ0,θ0

with the vertex at ε = 1, the direction τ0 and opening θ0 by

Sτ0,θ0 = {ε ∈ C; |arg(ε− 1) − τ0| < θ0/2} .(0.8)

Let V (x, ε) be the analytic continuation of the resummed SP-solution given in Theorem2. Let � ε �= 0 and let us, for the moment, assume that we can expand

V (x, ε) =∑

α

Vα(ε)xα(0.9)

in the convergent series of x at x = 0. We note that the radius of convergence of the seriesmay tend to zero when ε → 1 in the case where all λj’s are real numbers. Then we have

Theorem 5. Suppose that the λj ’s in (H) are real numbers and nonresonant, λj−〈λ, α〉 �=0 for all α ∈ Z

n+, |α| ≥ 2, j = 1, 2, . . . , n. Assume that (0.7) is satisfied. Then there exist

K0 > 0 and an integer k0 > 0 such that if ‖R‖k0,c,Γ0 < K0 and ‖∇R‖k0,c,Γ0 < K0, thenthere exists u(x, ε) holomorphic in (x, ε) ∈ Sc×Sτ0,θ0 such that, for every N = 0, 1, 2, . . . ,there exists RN (x, ε) being holomorphic in (x, ε) ∈ Sc × Sτ0,θ0 and satisfying RN(x, ε) =O(|x|N+1) as |x| → 0, (x, ε) ∈ Sc × Sτ0,θ0 such that

u(x, ε)−∑

|α|≤N

Vα(ε)xα = RN (x, ε), (x, ε) ∈ Sc × Sτ0,θ0 .(0.10)

Borel-Laplace transform of a difference equation associated with Henon

maps

Chihiro Matsuoka and Koichi Hiraide1

Department of Physics, Graduate school of Science and Technology, Ehime University,1Department of Mathematics, Graduate school of Science and Technology , Ehime University,

Matsuyama, Ehime, 790-8577, Japan

A polynomial diffeomorphism f : C2 → C2

f :(

xy

)7→(

1 + y − ax2

bx

)(1)

is called the (complex) Henon map , where a = 0 and b = 0 are complex numbers, and fixes the following twopoints: (

b − 1 ±√

(b − 1)2 + 4a

2a,

b − 1 ±√

(b − 1)2 + 4a

2ab

).

Let P = (xf , yf ) be one of them, and let α = α1 be an eigenvalue of the derivative DfP at P . Then we have thequadratic equation

α2 − λα − b = 0,(2)

where λ = −2axf , and another eigenvalue α2 of DfP coinsides with λ − α = −b/α. As usual, define the stableand unstable manifolds at P by

W s(P ) = {Q ∈ C2|fn(Q) → P as n → ∞},Wu(P ) = {Q ∈ C2|fn(Q) → P as n → −∞}

respectively. It is well-known that if P is a saddle point, i.e. 0 < |α1| < 1 and |α2| > 1, then W s(P ) is actually ananalytic submanifold of C2 which is an injective immersion of the complex plane C, and tangent to the eigenspacefor α1 in the tagent space TP C2 at P , and the similar fact for Wu(P ) holds.

We construct a novel function in order to describe the stable and unstable manifolds of the Henon map whichis constructed by the use of Borel-Laplace transform.

Let α be one of eigenvalues of the derivative DfP of the Henon map f at a fixed point P = (xf , yf ). Weconsider an f -invariant curve at P = (xf , yf ) parameterized by the complex variable t ∈ C as follows:

t 7→(

x(t) + xf

y(t) + yf

)=(

X(t)Y (t)

),

such that

f :(

X(t)Y (t)

)7→(

X(t + 1)Y (t + 1)

)=(

1 + Y (t) − aX(t)2

bX(t)

).

Replacing X(t) and Y (t) as x(t) and y(t) again, the following difference equation of the second kind is obtained

x(t + 1) − λx(t) − bx(t − 1) = −a{x(t)}2,(3)

together with y(t) = bx(t − 1).In order to solve (3), we express x(t) with the Laplace integral on some Riemann surface X;

x(t) = L[X](t) =∫

γ

e−ζtX(ζ)dζ(4)

1

where the contour γ is chosen depending on the positions and the form of branch points of X. Substituting (4)into (3), we obtain an integral equation for X(ζ):

AX = −aX ∗ X + C,(5)

where C(ζ) is an entire function of exponential type, A(ζ) = e−ζ −λ− beζ , and ∗ denotes the convolution definedby

F ∗ G =∫ ζ

0

F (ζ − ζ ′)G(ζ ′)dζ ′.

Setting

X(ζ) = a0 + X(ζ),(6)

and substituting (6) into (5), we have

AX + 2aa0 ∗ X = W.(7)

where W = W0 − aX ∗ X and W0 = −aa20ζ − a0A + C.

We expand X(ζ) and W (ζ) with a formal parameter σ as

X(ζ) =∞∑

n=1

σnXn(ζ), W (ζ) =∞∑

n=0

σn+1Wn(ζ).

Substituting these into (7), we have the solution Xn (n = 1, 2, · · · )

Xn = A−1F0

∫ ζ

0

W ′n−1

F0dζ ′ (n = 1, 2, . . . ),(8)

where W ′n−1 = dWn−1

dζ .Let α = 0 be one of eigenvalues of the derivative DfP at P . We define the lattice Γα generated by −log|α| as

follows. For k ∈ Z, let ζk = ρ + (2kπ + θ)i ≡ ζk1, where

ρ = −log|α|, −π < θ = arg α ≤ π,

and let

Γα = {ζ ∈ C | ζ =N∑

l=1

ζkl, ζkl = lρ + (2kπ + θ)i, N = 1, 2, · · · }.(9)

It is easy to see that Γα is on the right half plane of C if 0 < |α| < 1, on the left half plane if |α| > 1, and on theimaginary axis if |α| = 1. Note that Γα is dense in the imaginary axis in the case of |α| = 1 (see Figure 1).

Lemma 1 Let |α| = 1. For ζ ∈ C \ Γα and a path ω from the origin to ζ in C \ Γα, there is a smooth path δ fromthe origin to ζ homotopic to ω in C \ Γα such that ζ/2 ∈ δ and δ is symmetrical with respect to ζ/2.

Theorem 1 Let ζ = ζkN + ξ (|ξ| < ρ, k ∈ Z) and δ be a path from the origin to ζ in C \ Γα. Then the solutionX(ζ) = X(ζ, δ) to Eq. (7) is given as the limit of a Riemann suface X(N)(ζ):

X(ζ) = limN→∞

X(N)(ζ), X(N)(ζ) =∞∑

n=1

X(N)n (ζ).

The convolution W(N)n−1(ζ) and the solution X

(N)n (ζ) (N ≥ 2) are given by

W(N)n−1(ζkN + ξ) =

∑∞

m=0 v(N)n−1,m+n−1ξ

m+n−1(logξ)n + reg(n−1)(ξ), (1 ≤ n ≤ N − 1)

∑∞m=0 v

(N)n−1,m+n−1ξ

m+n−1(logξ)N + reg(N−1)(ξ), (n ≥ N)

X(N)n (ζkN + ξ) =

∑∞

m=0 b(N)n,m+n−1ξ

m+n−1(logξ)n + reg(n−1)(ξ), (1 ≤ n ≤ N − 1)

∑∞m=0 b

(N)n,m+n−1ξ

m+n−1(logξ)N + reg(N−1)(ξ), (n ≥ N)

(10)

2

ζ11

ζ21

ζ01

ζ−11

ζ−21

ζ32

ζ22

ζ12

0 ζ02

ζ03

ζ04

ζ13

ζζ23

ζ44

4πi

2πi

−2πi

−4πi

Figure 1: Lattice Γα and path δ (thick solid curve), where we set 0 < |α| < 1 and θ = 0 in ζk1.

where v(N)n−1,m+n−1 and b

(N)n,m+n−1 are complex coefficents which do not depend on the choice of the vertical index k

in the N -th singularity. The notation reg(n−1) is given by

reg(n−1)(ξ) =n−1∑m=0

Rm(ξ)(logξ)m,

where Rm(ξ) = ∗ + ∗ξ + ∗ξ2 + · · · (m = 0, 1, 2, · · · ) is a regular function with complex coefficients ∗’s.

From now on, we drop the index k and rewrite the N -th singularity ζkN as ζN . Thus, we obtain the solutionX(N)(ζ) for the N -th singularity

X(N)(ζ) =∞∑

n=1

X(N)n (ζ)

=∞∑

n=N

∞∑m=0

b(N)n,m+n−1(ζ − ζN )m+n−1 [log(ζ − ζN )]N + reg(N−1)(ζ − ζN ).(11)

For the coefficient b(N)n,m+n−1 in the solution X

(N)n for the N -th singularity ζ = ζN (N ≥ 1), the following lemma

holds.

Lemma 2 There exist constants C > 0 and K > 0 which depend on the eigenvalue α, and the first coefficientb(N)n,n−1 and the higher order coefficient b

(N)n,m+n−1 (m = 0, 1, 2, · · · ) of the n-th convolution for n ≥ N in the N -th

solution X(N)n (ζN + ξ) are estimated as

|b(N)n,n−1| ≤ Cn |α|N logN

n!,

|b(N)n,m+n−1| ≤ Kn|b(N)

n,n−1|.

for all n ≥ N (N ≥ 1).

3

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