Algorithms of the local nonlinear analysis

Nonlinear

Analysis,

Theory,

Methods

&Applications, Vol. 30, No. I. pp. 4683-4694, 1997 Proc. 2nd World Congress of Nonlinear Analysts

F’rinted

PII:SO362-546X(97)00194-6

ALGORITHMS Department of Mathematics,

0 1997 Elsevier Science Ltd in Great Britain. All rights reserved 0362-546X/97 $17.00 + 0.00

OF THE LOCAL NONLINEAR

ANALYSIS

ALEXANDER BRUNO Keldysh Institute of Applied Mathematics,

Moscow 125047, Russia

Iiey words and phmses: Algebraic equations, differential equations, asymptotics, first approximation, singular perturbation, power transformation, reduction of dimension, normal form, resolution of singularity, asymptotic expansions of solution, applications. 1. INTRODUCTION

All local (or asymptotic) first approximations of a polynomial, of a differential polynomial and of a system of such polynomials may be selected algorithmically [l-6]. It allows to find all first approximations of a system of equations (algebraic and ordinary differential and partial differential). Here the first approximation of a solution of the system of equations is a solution of the corresponding first approximation of the system of equations. The algorithm is based on the geometry of exponents and consists of computations for each equation: of the set D of the vector exponents Q E &!“, of faces I’p’ of a polyhedron M spanning D, of boundary subsets Df’ = I’l,“’ fl D in R” and of the normal cones Cry)of these faces Up) in the dual space Iw:. The corresponding computer program see in [7]. The power transformations [1,2,10] a 11ow to reduce the dimension of a first approximation of a system of equations. They induce linear transformations of vector exponents Q and commute with the operation of selecting first approximations. The algorithm consists of computation of a constant matrix of that linear transformation with prescribed properties [la]. If the first approximation is the linear system, then in many cases the system of equations can be transformed into the normal form by means of the formal change of coordinates. The normal form is reduced to the problem of less dimension by means of the power transformation. The algorithms of computation of the normal form are especially developed for ODE systems resolved with respect to derivatives [1,2,13,15]. Combining these algorithms, in rnauy problems we can resolve a singularity, find parameters determining properties of solutions and obtain the asymptotic expansions of solutions. Some applications to problems from Mathematics [l&19], Celestial Mechanics [20-271, Physics [28] and Hydrodynamics [29-301 were considered.

2.

FIRST

APPROXIMATIONS

Let in the space &!” with Cartesian coordinates ql,. . . , q,, we have a finite set D of points Qj = (Plj, . . ,4nj), j = 1,. . .T m. Let R: denote the dual space such that for P = (pl, . . . ,p,) E Ry and Q = (ql,... , qn) E Iw” there is the scalar product (P, Q) = plql + . . . + p,q,. For a fixed vector P # 0 let Dp denote such subset of the set D on which the scalar product (P, Q) has the maximal value c, that is (P, Qj) = c for Qj E D p and

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(I’, Qj) <

c

for Qj E D \ Dp.

(2.1)

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Now for the given set D and for each P # 0 we will find the corresponding set Dp. Let M denote the convex hull of the set D. The boundary 8M of the polyhedron M consists of faces rk(d) of different dimension d (0 5 d < n). Zerodimensional faces I’p’ are vertices of the polyhedron M, onedimensional faces rp’ are its edgesetc. According to (2.1) to each vector P E &!: there corresponds in R” a supporting to the set D hyperplane Lp = {Q : (P, Q) = c}. It intersects the polyhedron M along a face I’l,“‘. The set Up’ of all vectors P E WY such, that LP n M = I$), 1scalled the normal cone of the face I’r’. If dimM = n, the normal cone cIi”-‘) of the hyperface I‘p-l’ is a ray orthogonal to the face and directed out of the polyhedron M. The normal cone Ujn-‘) is a sector bounded by rays U/“-l’ and U!$‘-‘) if rp-2) = rjn-1) n I’:-‘), etc. The union of normal cones of all faces ia Ny \ (0). Let us consider the boundary subsets Df’ = I’f) n D of the set D. THEOREM 2.1 [I]. If P E Uid’, then Dp = Dp’. EXAMPLE 2.1. For n = 2 and (2.2)

D = {(3,0), (O,% (111))

the polygon M, faces I’?) and their normal cones are shown in Figure 1 [lS] and Figures 6, 7 [l]. Here I)(‘) = Q2D(O) = &I, D(O) = Q3, 4

h -

(01, &A

4

u(l) = -X(1 > 2) 7 1

fl) _ -

{Q~,Qd>

u(‘) = -X(2 2

,

I) 1

4

?v -

{Q:>Qd;

up = -X(1

7

1) (X > 0).

Let us consider a polynomial f(X)

= c

fsXQ

for

Q ED

(2.3)

where XQ = X9’ 1 . . . xr:, coefficients f~ E @ \ 0 and D is a set in R”. The set D is called the support of the polynomial. The convex hull M of the set D is called the Newton polyhedron of the polynomial f, and we can construct all accompanying objects as described above. Let Df’ be a boundary subset of the set D. The sum jr’(X)

= x,X*

for

Q E Dp’

(2.4)

is called the truncation of the polynomial f(X)[l, 21. It is a quasihomogeneousfunction with respect to the vector P. Now we consider a curve of the form xi

= biTY8 3 bi#O,

i=l,...,n,

where 7 + 00. On such a curve a monomial XQ = BQ~(W) 7 where B = (bl,. . . , b,) and P = (PI,. . . ,p,). f(X)

= jy’(

If P E Uf), then on the curve (2.5) B)F

+ Fo( 1)>

(2.5)

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where c is from (2.1). Thus, the truncation (2.4) is the first approximation of the polynomial f(X) on curves (2.5) with P E CJLd’. Let the equation f(X) X

= 0 have a solution of the form

,, = g(X’) = cgQ,X’Q’

for

Q’ E D’ c R”-l,

(2.6)

where X’ = (xl,. . . , z,-~). By letters with prime we shall denote the (n - l)-dimensional objects for the first n - 1 coordinates. To the expansion (2.6) th ere correspond the polyhedron M’ , faces ri(‘) of its boundary dM’ and the boundary subsets D{(6) = I’{(‘) n D’ in R”-’ and the normal cones U ‘(‘) 1 of the faces r iCa)in the dual space R:-‘. To each face I’j(‘) there corresponds the truncation j{6’(X’)

= CggtX”’

for

Q’ E DI(‘).

(2.7)

It is the first approximation of the function g(X) on curves xi = biTPa7 bi#Oy where P’ = (PI,. . . ,p,-1) E V/“‘.

i=l,...,n-l,

(2.8)

M oreover, OII these curves accordiug 2, = i$‘(B’)T

+ o(T’)

to (2.6) and (2.7)

7

where c > 0 and T = (P’,Q’)

for

Q’ E rj6).

Hence, on curves (2.8) the solution (2.6) has the form x,, = b,f + o(T’-‘). Now let us select from the sum (2.3) its truncation (2.4) corresponding to the vector P = (P’,r), i.e. p, = r and P E Ujd’. G THEOREM 2.2 [3]. Let 5, = g(X’) be a solution of the equation f(X) = 0, i.e. f(X’,g(X’)) 0: Let ~PP’(X’) be the truncation of the function g(X) corresponding to the vector P’ and [p(X) be the truncation of the sum (2.4) corresponding to the vector P = (P’,,). Then fP(cb(X’)) f 0, i.e. the first approximation of a solution is a solution of the corresponding first approximation of the equation. If we have found the first approximation (2.7) of the solution (2.6), then we can make the substitution x, = c$“(X’) + &, into the equation f(X) = 0 and look for the second approximation for 5, as the first approximation for &, and so on. EXAMPLE 2.2 (continuation of Example 2.1). Let f = S$t xi - 2x1~2, then the set D is (2.2) and among truncations fp’ there are A” = CC:,jj”’ = -22122, jp’ = xi, #’ = z; - 2x1x2, j(l) = xi - 2x122. Here n = 2 and (2.6) is an expansion of 52 in powers of x1. Its first aiproximation (2.7) is x2 = gqxi with gq # 0 and P = const (1, q). If (2.6) is a solution of the equation f = 0 with P # 0, then P can not lie in Up). Indeed if P E Uf’ then 3~ = fsXQ 1 and fp(xl,gpxT) = fQggp1xy’+qq2 s 0; but ‘t1 is impossible if gq # 0. If on the solution (2.6)

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51,~s --t 0 then P < 0, so P E U{‘) or Uj”. If P E I!J~“, i.e. 2pr = p2 < 0, then the first approximation 52 = 62x: satisfies the truncated equation xi - 2x1x2 = 0, i.e. x2 = x:/2. Similarly, if P E ,!I;“, then the first approximation of the solution (2.6) is xl = xz/2. That is two branches of the curve f = 0 go through the origin xl = x2 = 0 (see Figure 2 [16] or Figure 19 [l]). We can compute their asymptotic expansions of the form ccl x2

=

c m=l

gmx;m,

where r,,-1 < r,.

The considered construction can be transferred to ordinary differential equations in the following manner. We define a differential monomial s(X) as a product of powers of coordinates X and derivatives d”x,,/dx~-:_, , where integer m > 0. To the monomial u(X) we put in correspondence the point Q E R”: to the product const XQ there corresponds the point Q, to the derivative dmx,/dxc,“_, there corresponds Q = (0,. . . ,O, -m, I), and to the product of monomials there corresponds the sum of their vectors Q. To a differential polynomial f(X), th at is a sum of monomials, we put in correspondence in W” the set D(f) of points Q of its monomials. We call the set D(j) as the support of the differential polynomial f. Here one point Q can correspond to several different monomials. As above for the set D, we construct the polyhedron M, its faces I’p), boundary subsets IIf’ in IR” and their normal cones Up’ in the dual space IR:. Theorem 2.2 is true for a solution (2.6) of the equation f(X) = 0 151. Analogously for a partial differential equation. Now there is a set of independent variables where integer 11’2 = (x1+1,... , x,,-1) = X2 and derivatives have the form dK~z,/dXp, (h+*, ’ * . , k,-r) 2 0. To that derivative there corresponds the vector Q = (0,. . . ,O, -K2, 1) = The whole construction is repeated and Theorem 2.2 is true (0, * * * 10, -h+1,. * * , -k,,-r,l). again [5,6]. Now we want to generalize Theorem 2.2 for solutions x, = g(X’) of more general nature than (2.6). Let a function 4(r) be defined for r E (0, co). Numbers

s = limlog I~(T)I -

log 7

for

r -+ 0,

and 3 = KE’O: !(r)’ 0 7

for

r t

$00

are called orders of the function C$in zero and infinity respectively. If 4(r) is a polynomial then g and 3 are the smallest and largest exponents of its monomials, here 3 5 S. But it is possible that 2 > 3 for a function of more general nature. For instance, for c$(T) = l/(1 + r) we have g = 0, 3 = -1. For the power function 4(r) = constra we have 2 = s = cr. The asymptotic expansion by power functions is unique. We shall select a wider class of functions with this property. A function 4(r) is called the function of the power type, if d(r) = $(logr)?‘, where the function $(cr) is defined for 0 E (-co, +oo) and has finite orders for 0 + foe. A function h(X’) is called pseudohomogeneous with respect to the vector P, if along the curves (2.8) we have h(X’) = +(r, B’)rr, where C$is a function of the power type with zero order for any finite value of parameters B’ # 0 and r = const. Finally, the function ip~(X’) is a truncation of the function g(X’) with respect to the vector P’, if 6~’ is pseudohomogeneouswith respect to P’ and along the curves (2.8) g = ip, + o(r’-’ ). For such a definition of the asymptotical first

SecondWorld Congressof NonlinearAnalysts approximation of a Moreover, Theorem are functions of X’ allows us to expand

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solution, Theorem 2.2 is true for both algebraic and differential equations. 2.2 is true if coefficients of the algebraic or differeutial polynomial S(X) which are pseudohomogeneous with respect to P’. For a fixed vector 1j’ it a solution xn = g(X’) into an asymptotic series g =

where hk are pseudohomogeneous

GP’

functions

+

2 kc1

Ilk

and along curves (2.8) F(hk) > r(hk+r).

EXAMPLE 2.3. Let the polynomial f = .z2 - x2 - yz + $, where 1 contains terms with order greater than 2. We consider the equation f = 0 near the origin 2 = y F .a = 0. The truncation jp for P = -( 1, 1,l) is jr, = 2’ - 5’ - y’. The truncated equation fp = 0 has two solutions z = w = &v’-. After the substitution z = w + .zr we obtain f(s, y,w + zr) = 2wz, + h3(x, y, w) + . . . = fr(z, y, zr), where hs is a homogeneous polynomial on z, y, w of order 3. Now the truncation jr = 0 of the equation fr = 0, corresponding to the vector P = -(l, 1,3/2), is jr = 2wzr + h3 = 0. Its solution zr = -hs/(Lw gives the second term of the asymptotics. By these computations we can obtain tile expansion

i=2

of the solution z = g(s, y) of the equation f = 0. Here gm(z, y,w) are homogeneous polynomials of order na and Ici - Ii 2 i, lci - Ii 2 &-I - Ii-r. Thus to apply this approach we must have an algorithm giving all truncations of polynomial. According to the beginning of Section 2 the algorithm must give the boundary subsets Df’ and their normal cones Crid)for any set D E IR”. Only for n 5 3 it can be done by drawing some pictures. For n > 3 it demands some computations. Corresponding algorithm was developed in [2] and was written as a computer program in [7]. The Newton polygon, i.e. the polyhedron M for n = 2, was introduced by Puiseux [8]; Newton himself found one its edge only [9]. The polyhedron M (for any n) was introduced in [lo] for a system of ordinary differential equations. Name “Newton polyhedron” was given by Gindikin [ll]. See its history in [1,2]. 3. POWER TRANSFORMATIONS Let n-vector X be divided into three parts: parameters X1, independent variables X2 and dependent variables Xs with dimensions nr, n2 and ns respectively (X = (X1,X2,X3), n = nr + 112+ 123,R; > 0). We consider the power transformation 1ogxr = wrr log Yr, logX2 = w21 log K t log x3 = w31 log Yl t

w22

log

w32

log Y2 t

yz,

(3.1) w33

log Y3.

Here log Xr = (log ~1, . . . , log z,, )*, Wii are nondegenerate square matrices, Wij are rectangle matrices, W = (Wij) is a square block matrix, the star denotes transposition.

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Let the coordinate change (3.1) transform a differential We want to study the relationship of their supports D(f) THEOREM 3.1. Under the power transformation {Qj} are connected by the linear transformation

Analysts

polynomial f(X) into g(Y) = f(X). = {Qj} and D(g) = {Sj}.

(3.1), supports

D(g)

S=W*Q

= {Sj}

and D(f)

=

(3.2)

and vectors of the dual space &?: are transformed

as

R = W-‘P.

(3.3)

Here all sets in R” and in &!: are also changed by the corresponding linear transformation (3.2) and (3.3). H ence the selection of truncations commutes with any power transformation. Moreover, the power transformation can be used to simplify the truncated equation making its support parallel to a coordinate subspace. Let d = dimM(f) < 11. There exists such a matrix W that, after THEOREM 3.2. transformation (3.1), values of n - d coordinates sj are constants for all S = (sr, . . . ,s,) E the coordinate yj is absent, if j 5 nr, D(g), whereg(Y) = f(X). Let sj = const. In yr”‘g(Y) and is present only in the form logyj, if j > ni. Note that the multiplication of the polynomial g(Y) by the factor YT means the parallel < n translation of the set D(g) by th e vector T: D(gYT) = D(g) + T. Thus, if dimM(f) then by the power transformation (3.1) and a multiplication by YT we can put the support of the polynomial f into d-dimensional coordinate subspace. It allows to reduce the dimension of the truncated problem. The initial equation is the regular perturbation of its truncation in the corresponding domain of the X-space, where the truncation is the first approximation. This is true after the power transformation. But now the domain can be made a vicinity of a coordinate subspace in the Y-space. EXAMPLE 3.1 (continuation of Example 2.2). Let us make the edge I’?) parallel to the axis sr by means of the power transformation y1 = xfx,', { Yz = &2, i.e.

1 21

=

w-1

YlY2,

p f

2 -1 -1 11 ’ (

w=

22 = YIY;,

Here

=

:;. (

(3.4) 1

= x; - 221s2 = yfyz” - 2yfy23 = y;ygy1 =

x:

+

x;

-

221x2

=

$$(Yl

+

y1y;

-

2).

- 2),

(3.5)

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Reducing by yfy; we obtain the full equation y1

+

y,I/;

-

2 =

(3.6)

0,

and the truncated equation y1 - 2 = 0. Its root y1 = yy = 2 is a simple one. Applying Implicit Function Theorem to the full equation (3.6) we can obtain y1 - 2 as a power series in yz. Here it can be found in the explicit form y1 = 2/(l + yz). Substituting that expression into (3.4) we obtain the parametric representation of the branch F1: $1

=

2Y2/(1

+

y;,,

x2 =

2Y,2/(1 + Y,“,.

Now I recommend to the reader to draw sets D, M etc. for polynomials (3.5) and (3.6), and to compare them with Figure 1 [16] or Figure 6 (11. Generally speaking, if we have a system of equations then we must consider simultaneously several Newton polyhedrons: its own polyhedron for each equation [1,3,5]. But the system of ordinary differential equations dlogX/&

= (log X) = F(X)

= CFQ.U”

for

QE D

(3.7)

has one support D, one Newton polyhedron M and one set of corresponding objects. particular, to each P E R: \ (0) th ere corresponds the truncated system (loi X) = I;;(X).

In

(3.8)

The power transformation log X = W log Y transforms (3.7) into the system (logy)

= w-’ CFQYS,

where S = W’Q. If dimM(fip) = d < n, then there exists such a matrix W that (3.8) has the form (logy) = G(yl,. . . , yd)YT. After the substitution dt, = YTdt, we obtain the system dlog Y/&t, = G(yl,. . . , yd). To solve it, we must solve only the d-dimensional subsystem

There are two kinds of the power transformation (3.1): with arbitrary real matrix W and with unimodular matrix W (i.e. det W = fl) having integer elements. A power transformation of the second kind gives one-to-one correspondence between X and Y outside coordinate subspaces. To find such a matrix W one can use the continued fraction algorithm [l],ifn=2, and’tI s generalizations [12], if n > 2. Particular casesof the power transformation were used long ago, but in the general form it was introduced in [lo], see also [l]. 4. NORMAL

Let X E R” or c”. system

FORM

In a neighborhood of the stationary point X = 0 we consider the ODE dX/&

Y x = AX + G(X)

(4.1)

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where Cp is a vector power series, ip = O(llXll”). means of a formal transformation of coordinates X=BY+Z(Y),

W e want

to simplify

the system

detB#O.

(4.1) by

(4.2)

Let it bring the system (4.1) into a normal form

P = CY

+ Q(Y).

(4.3)

Let the matrix C = (cij) be in a normal form (for instance, in the Jordan one). So its diagonal consists of its eigenvalues (Xl,. . . , X,) = A, and Cij = 0 if X; # Xj. Let US write components of the vector q in (4.3) in the form ~i(Y)~fyigi(Y)~fyiCgi*YQ, where Q = (q,,...,

i= I,...,?&,

(4.4)

qn) E Z” and Q + f?‘i 2 0 and Ei is i-th unit vector.

The system (4.3), (4.4) is called the resonant normal such terms gi*YQ for which the scalar product

form,

if expansions (4.4) contain only

(4.5)

(Q, A) ef Q,Xl + . . . t q,x, = 0.

THEOREM 4.1 [1,13]. For each system (4.1) tl lere exists a formal transformation (4.2) which brings (4.1) into its resonant normal form (4.3). Let the equation (4.5) have exactly k integer solutions Q E z” independent over R. Then the system (4.1) has the k-fold resonance.

which are linearly

THEOREM 4.2 [1,13]. If the system (4.1) has the k-fold resonance, then there exists such a power transformation Y -+ 2 which brings the resonant normal form (4.3) into the system (log z;) = j*(Zl,. . . , Zk),

i = l,...,n.

(4.6)

Thus, to solve the system (4.6) one must solve its subsystem of order k: ii=Ziji(Zly...,Zk),

i = l,...,k.

(4.7)

The system (4.7) has not a linear part. So to simplify the system (4.7) we must find its truncated systems ii = ZifjJ’)( Zl , . ..,Zk 1, i= l,...,k, (4.8) using the Newton polyhedron of the system (4.7). The system (4.8) is quasihomogeneous,so we can again use a power transformation to simplify it. The question of analyticity of the normalizing transformation (4.2) for the analytic system (4.1) was solved in [13] in the following form. There are two conditions on the normal form (4.3): Condition w on eigenvalues A (the small divisor condition) and Condition A on its

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nonlinear part q. If both conditions are satisfied then (4.2) is analytic. If Condition A is violated then (4.2) can diverge. If Re A = 0 then Condition A is very restrictive and usually (4.2) diverges in a whole neighborhood of the origin. But there are sets of dimension less then n on which the system (4.1) is transformed into lhe normal form by an analytical or smooth change of coordinates. Description of such sets is given below for the case Re A = 0. By the normal form (4.3)-(4.5) we define the set A = {Y : $i = Xiyia, i = 1,. . . , n} where (Y is a free parameter. It is the set where Condition A is satisfied. Let 7” be an invariant I-dimensional torus of the system (4.3) and PI+), . . . , p,, be eigenvalues of its variational system. In the set ,4 we isolate two subsets A, D: A consists of all stationary and periodic solutions, 13 consists of all invariant tori with all pj = 0. THEOREM 4.3 [l]. On the set A, the system (4.1) is analytically and the set A is analytical in X.

reduced to the normal form

THEOREM 4.4 [l]. If A satisfies condition w then on the set B the system reduced to the normal form and the set f3 is analytical in X.

(4.1) is analytically

Bifurcations of periodic and conditionally periodic solutions are branching sets of these To analyze the structure of the sets A and B solutions over the subspace of parameters. one can use the method of a resolution of a singularity by means of Newton polyhedra and power transformations (see Sections 2 and 3). Analogously we can study a neighborhood of a periodic solution, of an invariant tori and of some families of such tori (11. Note that A = A, if all Xj are pairwise commensurable. EXAMPLE 4.1. Let system (4.1) h ave a small parameter ~1 = E (i.e. E = 0) and two variables ~2 and zs. Then E = x2 = 23 = 0 is its stationary point and Ai = 0. Let Xz = -X3 = i = &i. Then equation (4.5) is qzX2 + q&A3 = i(q2 - q3) = 0. Its integer solutions Q = (q1,q2,q3) have arbitrary ql and q2 = 43. So the normal form (4.3)(4.5) of our system (4.1) is

i = 0, The power transformation

Yj = Yjtxj + 9j(E

p = ysys, ya = ys transforms

i = 0,

b=

t

P(92h4

93th

j =

? ~2~3))r

2,s.

it to

P)),

?j3 =

Y393h

PI.

The set j=2,3}

d = (~7 YZ, ~3 : yjgj = Xjyja,

=diudzudg

where di = {E, ~2, ~3 : 92 t 9s = 0) and dj = {yj = 0}, j = 2,3 are coordinate axis. By Theorem 4.3 the set A is analytic for analytical our initial system (4.1). If gj = &E + cjp + ..*, j = 2,3, then the component set A1 is defined by the equation 92

t

93

=

(b2

t

63)E

t

(cz

t

c3)p

t

. . . =

0.

(4.9)

If b2 + b3 # 0 then by Implicit Function Theorem this equation has unique solution E = E(P), i.e. di is a manifold. For a real system (4.1), p = ~2~3 is the radius vector and di is a

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family of periodic solutions bifurcating from the origin. This is the Hopf bifurcation [14]. If 62+ 63 = c2 +cs = 0, then equation (4.9) has several branches of solutions. To find all branches one can use the Newton polygon for (4.9;. Here Condition A is g2 + g3 = 0. If not all ReXj = 0 then the system (4.1) has the invariant center manifold, corresponding to Xj with ReXj = 0. After a reduction of the system on the center manifold, we can apply the described approach. Some partial differential equations can be written as infinite system of ordinary differential equations, so called evolution equation. If the spectrum of its linear part has only finite number of points in the imaginary axis, then after reduction on the center manifold we obtain an ODE system, that can be transformed to the normal form and so on. If the spectrum has infinitely many points in the imaginary axis then after reduction on the center manifold we obtain an evolution equation again. It can be transformed to the resonant normal form. If the multiplicity of the resonance is finite then the normal fornl can be reduced to an ODE system by means of a power transformation. On the contrary, the power transformation can give new PDF, which is more simple than initial one. Thus in any case the normalizing transformations followed by an appropriate power transformation simplifies a problem. In a neighborhood of a stationary point of (4.1) the algorithm for computation of the normal form and the normalizing transformation demands only arithmetic operations over coefficients of the right side series of the initial system (4.1), see [l, Ch. III, Section 1.7; 151. In a neighborhood of a periodic solution or an invariant torus the algorithm requires computation of some integrals as well as for PDE. Tl rere are several such algorithms [15] and their computer realizations, especially for Ilamiltonian systems. 5.

APPLICATIONS

In a concrete local (or asymptotic) nonlinear problem we must combine truncations, power transformations and normalizations. Each of them diminish the dimension of the problem. So after several such steps, we obtain a problem which is simple enough to be solved. It can serve as an unperturbed problem for the transformed initial one. It allows to resolve a singularity, to study asymptotics of solutions, their branching, bifurcations, stability and so on. Below we list some recent applications of the algorithms to problems from Mathcinatics, Celestial Mechanics, Physics and Hydrodynamics. Other examples see in [l-3,5,6,16]. Algebraic equations. Isolation of all branches of any algebraic or analytic space curve, and their computation with any accuracy [2].

Ordinary diflerential equations. A complicated bifurcation up to 10 periodic solutions from the stationary one in a 4-dimensional system (4.1) with one small parameter [17]. A way for estimation of number of limit cycles of such a polynomial system on the plane which is close to integrable one [18]. Algorithm for finding H amiltonian truncations of a Hamiltonian system [19]. The equation of the satellite motion around its center mass moving along an elliptical orbit with eccentricity e has a singularity when e = 1. Families of periodic solutions were computed for e = 0.99 and e = 0.999 [20] and for e = 1 [21]. Some families of critical solutions were computed and a new such family with e > 0.98 was found [22]. A theory of singular perturbations using the normal form was created for the almost symmetrical satellite [23].

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The restricted 3-body problem describes motion of a massless body under the Newtonian attraction of two bodies with masses 1 and p. The planar circular problem assumes that orbits of two big bodies are circumferences and the massless body moves of the plane of these orbits. For small ~1 the problem has singular perturbations for orbits with close approach to the p-mass body. Such orbits were studied in [24] and computed in [25,26]. In particular, the existence of stable periodic orbits with close approach to Jupiter and to Earth was shown in [27]. Some asymptotic properties of solutions of a finite-dimensional approximation of the Schrodinger equation were studied in [28]. The problem of surface waves on water after reduction on the center manifold gives a $-dimensional system (4.1) with a degenerate linear part having 2 small parameters. We have found its new periodic solutions and conditionally periodic solutions [29]. Sometimes computations demand 3 steps of reductions: a truncation, a normalizing transformation and another truncation. Partial diflwential equations. Asymptotics of the stationary viscous fluid flow around a flat plate were found directly through the Newton polyhedron of the Helmholtz form of the Navier-Stokes equations [30]. Supported

by RFHR,

Grant

96-01-01411. REFERENCES

1. BRUNT A.D., Local Method of Nonlinear Analysis of Differential Equations. Nauka, SpringerMoscow (1979) (Russian) = Local Methods in Nonlinenr D#ercratial Equations. Verlag, Berlin (1989) (English). 2. BRUNO A.D. & SOLEEV A., Local uniformization of branches of a space curve, and Newton polyhedra. Algebra i Analis 3:1, 67-101 (1991) (R) = St. Petersburg Math. J. 3:1, 53-82 (1992) (E). 3. BRUNO A.D. & SOLEEV A., First approximations of algebraic equations. Doklady Akademii Nauk 335:3, 277-278 (1994) (R) = n ussian AC. Sci. Doklady. Mathem. 49~2, 291293 (1994) (E). 4. KHOVANSKII A.G., Newton polyhedra (resolution of singularities), in Itogi Nauki i Tekhniki: Sovretnennye Problemy Mat., 22, 207-239, VINITI, Moscow (1983) (R) = J. Sov. Math. 27, 2811-2830 (1984) (E). 5. BRUNO A.D., First approximations of differential equations, Dokfady Akademii Nuuk 335:4, 413-416 (1994) (Russian) = Russian Acad. S’ci. Doklady. Mathem. 49:2, 334-339 (1994) (E).

6. BRUNO

A.D.,

General approach to the asymptotic analysis of singular perturbations, in

Dynamical Systems and Chaos (Edited by N. Aoki, I<. Shiraiwa and Y. Takahashi), v.1, pp. 11-17. World Scientific, Sigapore (1995). 7. SOLEEV A. & ARANSON A.B., Computation of a polyhedron and of normal cones of its faces. Preprint 36, Inst. Appl. Math., Moscow (1994) (R). 8. PUISEUX V., Recherches sur les fonctions algebriques, J. de Math. pures et appf. 15, 365480 (1850). 9. NEWTON I., A treatise of the method of fluxions and infinite series, with its application to the geometry of curve lines, in The Mathematical Works of Isaac Newton (Edited by Harry Woolf), v.1, pp. 27-137, Johnson Reprint Corp., N.Y. and London (1964). 10. BRUNO A.D., The asymptotic behavior of solutions of nonlinear systems of differential equations. Dokf. Akad. Nauk SSSR 143:4, 763-766 (1962) (R) = Soviet Math. Dokl. 3,

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464-467 (1962) (E). 11. GIDIKIN S.G., Energy estimates and Newton polyhedra, Trans. Moscow Math. Sot. 31, 193-246 (1974). 12. BRUNO A.D. & PARUSNIKOV V.I., Comparison of different generalizations of continued fractions, Math. Notes 60:6 (1996). 13. BRUNO A.D., Analytical form of differential equations, Trans. Moscow Math. Sot. 25, 131-288 (1971); 26, 199-239 (1972). 14. MARSDEN J.E. & MCCRACKEN M., The Hopf Bifurcation and Its Applications. Springer-Verlag, New-York (1976). 15. BRYUNO A.D, Ways of computing a normal form, Russian AC. Sci. Dokfady Math. 52, 200-202 (1995). 16. BRUNO A.D., The Newton polyhedron in the nonlinear Analysis. Vestnik Moskovskogo Universiteta, ser. 1, no. 6, 45-51 (1995) (R) = Mathem. Bulletin 50 (1995) (E). 17. BRUNO A.D., Bifurcations of periodic solutions in the case of a multiple pair of purely imaginary eigenvlues with a symmetry, in Numerical Solution of Ordinary Differential Equations (Edited by S.S. Filippov), pp. 161-176, Inst. Appl. Math, Moscow (1988) (R) = Bifurcation of the periodic solutions in the symmetric case of a multiple pair of imaginary eigenvalues. Selecta Math. formerly Sovietica 12:1, 1-12 (1993) (E). 18. BRUNO A.D., The normal form of a system, close to a 1Iamiltonia.n system, Matem. Zametki 48:5, 35-46 (1990) (R) = Matem Notes 48:5/6, 1100-1108 (1991) (E). 19. BRUNO A.D. & SOLEEV A., The Hamiltonian truncations of a Hamiltonian system, Doklady Akademii Nauk 349:2 (1996) (R) = R ussian Acad. Sci. Doklady. Mathematics 54 (1996) W 20. BRUNO A.D. & PETROVICH V.YU., Computation of periodic oscillations of a satellite. Singular case. Preprint 44, Inst. Appl. Math., Moscow (1994) (R). 21. BRUNO A.& VARIN V., Singularities of oscillations of a satellite on highly eccentric elliptic orbit, Proceedings of the 2nd WCNA, Elsevier, Amsterdam (1997). 22. BRUNO A.D. & PETROVICH V.YU., Regularization of oscillations of a satellite on the very stretched orbit. Preprint 4, Inst. Appl. Math., Moscow (1994) (R). 23. SADOV S.YU., Singular asymptotic theory for oscillations of an almost symmetrical satellite on a very stretched orbit., Proceedings of the 2nd WCNA, Elsevier, Amsterdam (1997). 24. BRUNO A.D., The Restricted PBody Problem. Walter de Gruyter, Berlin (1994). 25. BRUNO A.D., Simple (double, multiple) periodic solutiocns of the restricted three-body problem in the Sun-Jupiter case. Preprints 60, 67, 68, Inst. Appl. Math., Moscow (1993) w 26. BRUNO A.D., Singular perturbations in Hamiltonian Mechanics, in Hamiltonian Mechanics (Edided by J. Seimenis), pp. 43-49, Plenum Press, New-York (1994). 27. BRUNO A.D., On periodic flights round moon. &print 91, Inst. Appl. Math., Moscow (1978) (R) = On periodic flybys of the moon, Celestial Mechanics 24:3, 255-268 (1981) (E). 28. SADOV S.YU., On a dynamic system arising from a finite-dimensional approximation of the Schrtidinger equation, Matem. Zametki 56:3, 118-133 (1994) (R) = Mathemalical Notes 56:3,960-971 (1994) (E). 29. BRUNO A.D. & SOLEEV A., Local analysis of a reversible ODE system and the Newton polyhedron, Proceedings of the ,?nd WCNA, Elsevier, Amsterdam (1997). 30. BRUNO A.D. & VASIL’EV M., Asymptotic analysis of the viscous fluid flow around a flat plate by the Newton polyhedra, Ibid.