A NECESSARY AND SUFFICIENT CONDITION OF SMOOTH LINEARIZATION OF AUTONOMOUS PLANAR SYSTEMS IN A NEIGHBORHOOD OF A CRITICAL POINT V. S. Samovol We consider the problem of smooth linearization in a neighborhood of a critical point of a system of C ~ ordinary differential equations. The analytical case was studied in [1-4]. In [5] a condition on the support of the normal form, which is sufficient for its ck-linear - ization k _> i, was formulated (the S(k) condition). Moreover, in [5, 6] it is shown that the condition S(k) is nonvacuous. In the present work we give a full proof of that fact that in the case of two-dimensional systems the condition above is not only sufficient, but also ne- cessary for ck-linearization. An infinitely smooth system of ordinary differential equations in a neighborhood of a nondegenerate critical point has the form ~----A~ +H(~), [IH(~)N =o(H ~l]). (1) Here ~, //(~)~R n, H (~)~ C a, the spectrum A = {Ai} of the matrix A does not intersect the imaginary axis. Necessity and sufficiency of condition S(k) for the ck-linearization of system (i) for n = 2, Re Al, Re A2 > 0 follows from the results of [5].. It only remains for us to consider a critical point of the saddle type. In a neighborhood of such a point system (i) has the form ~._ = ~: + h~ (~), r = (~, ~); h, (g) ~ C% I h, (r I = o (ll r 11 ) (, = i, 2), )u > 0 > ).2. The normal form of system (2) depends on whether resonance relations )'i = kli•l. -~ k2i}~2 (i = l, 2), where kij are nonnegative integers, kli + k2i > i, are present. (2) (3) If the number ~i/~ 2 is irrational, then relation (3) cannot hold, and the normal form is linear. In this case, due to a theorem of Sternberg [7, pp. 307-308, Theorem 12.1] there exists a C~ transformation that reduces system (2) to a linear system. Let Az/l 2 = p/q be an irreducible fraction. Then relation (3) holds if and only if klz = qn + i, k2i = pn, kz2 = qm, k22 = pm + i, where n, m are natural numbers. In this case the normal form of system (2) is '= For a system of form (4) we shall use the notation (4) a,~ = max(qn + 1, pn). ~.~ = max(qm, pro+ t), a= rain a,,, ]3= rain ~,~, 7=rain(a, 13). Let k > 0 be an integer. Then condition S(k), which is a sufficient condition for Ck - linearization, reduces in the case of system (2) to the inequality ~ > k. Let us consider two finite sets of natural numbers 31: (k) = {n: cz,, < k, a,, =/s= 0}, -]12 (k) ~- {m: j3,,z ~ 1% bm #= 0}. International Institute of Economical Problems of the World Socialist System. Translated from Matematicheskie Zametki, Vol. 46, No. i, pp. 67-77, July, 1989. Original article submit- ted October 6, 1987. 0001-4346/89/4678- 0543 $12.50 9 1990 Plenum Publishing Corporation 543 THEOREM i. Let k > 0. Then, if either of the sets M1(k), M2(k) is nonempty, system (2) cannot be reduced by a nondegenerate C k transformation to a linear system, that is, the in- equality y > k is a necessary condition for ck-linearization of system (2). The following theorem is a direct corollary of Theorem 1 and of the results of [6]. THEOREM 2. A necessary and sufficient condition for ck-linearization of system (2) is the inequality Y > k. Before we prove Theorem i, let us consider the polynomial normal form of the general system (I), which determines the possibility of ck-linearization of this system: 9~ = ~xi-1 + ~.~xi + a i (x) (t < i < n), X,H= 2 a~2:s , 8 = (81, 9 9 S,,) (5) x ~ C , e~ = 0 i = 0 asif or ~, e > 0, a ~ (X) = is a collection of nonnegative integers, Moreover, we shall 2p=lSpXp:~.i, Xs=Xl '. \"'\" .xnn , ]3[=~~ pn lSp' L=L(k), limL(k)=~. =assume that Re Xi+ z > Re h i > 0 for i < i <_ nl - 1 and that Re Xi+z < Re ~• < 0 for nz + 1 _< i _< nl + n 2 -- i, nl + n2 ---- n. Condition S(k), which is sufficient for Ck-linearization of system (i) is as follows: for every collections hold: s~P,P= U~=1{s: a~ =~0} ~'j=lsjRekj~kRe).m nJ n one of the following n inequalities should (i ~<~ m ~ nl); ~\" n,-~-I sj[Rekjl>k]Bek,,~[ j=(nl-i- l
1 transformation for system (i) = B~ 1 + H (~1), Net B ~ 0, [1H (~)[] = o (ll ~1 []), which reduces that system to the linear system %1 = .4 ~. (9) (8) Then, clearly, the equality AB = BA holds, from which it follows that the linear transforma- tion :~1 - B-I~_ (10) does not change the form of system (9). Hence we see that the transformation which is the superposition of the transformations (8) and (10) is a near identity ck-linearizing trans- formation for system (1), which is what had to be proved. Let us consider two cases: 1) (z -- ~: 544 In the first case let us take for definiteness ~ < ~, and therefore y = ~. Then from Chen's theorem [7, Theorem 12.2, p. 309] and Theorem 3 it follows that system (2) is CT- equivalent to the following one: 9~ ' = XsX + axqr~+ly p~, a =/= O, ~) = where max (qn + i, pn) = u Cu transformation, ;',.,y, (li) In [6] it is shown that system (ii) cannot be linearized by a If ~ = ~ then again from Chen's theorem and from Theorem 3 it follows that system (2) is CY-equivalent to the system , = klx + axqn+lypiT, a =/= 0; fJ = ~'iY + bxqmg p'~+I, b ::/= 0, ( 12 ) where max(qfi + 1, pn) = max(qm, pm + 1) = 'y, from which it easily follows that p = q. Then ~ = X > 0, X2 = - X and from the equality a = $ we obtain that m = n. Thus system (12) can be written down as 9 = ~x + ax~+ay ~, a =/= 0; b :/= 0, (13) 0 = --~u + bx~y ~+~, where n = y - i. Let us now demonstrate that for system (13) there does not exist a C ~+~ diffeomorephism of the form x = / (~, ~]), / (L q) = ~ + o (II (~, ~) II), y = g (L % g (L ~]) = n + o (11 (L ~])II), (1A) which reduces it to the form = ~,~, = --~. (15) We shall argue from the contrary. t) a+b~O; 2) a-+-b=O. Let us consider two cases: In the first case solutions of system (13) x(t, xo, Yo), y(t, xo, Yo), satisfying ini- tial conditions x(O, xo, Yo) = xo, y(O, xo, Yo) = Yo have the form x (t, x o, Yo) -- Xo exp (~.t). (i -- cn)xo~\"Y o~ t J~ , b y (t, x o, Yo) =- Yo exp (-- ~.t). (t -- cn\"x o~Y o'~t ) cry, where c = a + b. Solutions of system (15) have the form (t, ~o) = ~oexp (%t), ~ (t, Bo) = 1]o exp (--%t). Let us fix $o = 6, 0 < 6 < E, where r is arbitrarily For a number ~, 0 < p < e let us define that, T-- small. q(T, qo) = P. It is clear a number T such that 1, ~ln-~o , ~ (T, 6) -- bt~xl o Following a trajectory of system (13) for time T, the point (f (6, q0), g(6, q0)) is mapped into the pint (f, g), where f = / (6, rio) exp (kT). [P -- c~ (/(6, ~lo) g (8, 11o))~ T]-a'c~,: g = g (8, 00) exp (-- %T). [1 -- cn (/(6, ~1o) g (6, I1o))~ T] -b/c~. On the other hand, the point (5, g) must coincide with the point ( 6qo 545 Hence we obtain = / (6, no) exp ().T). [t -- c~ (/(6, ~lo) g (6, ~1o))a T] -'.:r g / &l,~ xp (-- LT). [i -- c5 (/(6, qo) g (6, rb))'~T]-b/c'L k, \"W' ~') -- g (6, ~1o)eMultiplying the right- and left-hand sides of the two last equalities, and substituting T = (-i/%) in (~/q0), we have , = + \"-Z-/(h (6, ~lo)) In qo J ' ( 16 ) where h(~, D) = f(~, ~)g(~, q). Let us note that the function h($, D) is of class C 6+~, as are the functions f($, q), g(g, q). From the fact that invariant manifolds Mz = {~ = 0}, N~ = {q = 0} are mapped under trans- formation (14) into, respectively, the manifolds M= = {x = 0} and N= = {y = 0}, it follows that ] (0, n) ~ O, g (~, O) ~_ O. Moreover, from (14) we clearly have that (17) /(~,0)=~+o(1~1)4=0 for ~:#:0, g(O,~l) =q +o( I~l I)# =0 for q:~O, O'f(~'rl) -- I .-~0('1_) for ll(%,n)ll~O, (18) Og (~aq, ~1) = I 4'- 0 (t) for II (~- , ~) II -~ O. Hence, in particular it follows that h(~,O) = h(O, rl)_----O, oh n(~).,a ~ ~.=o = n + o(I n t), Let us also note that ~ lan ~=o =a~+o(l~.l). (6, rio))'~I n + § o((h (6, ~1o))'~I n-~o ). (19) [l'--r \"-f-era (h (6, Ilo))'~ In ~]-''~J 'Io = t--+(h (20) From (19) we have that h(6, D0) = (6 + o(6))No + o(n0), whence it follows that o (h(6,~lo))~ln-~7 = o ~lo ln-~a 9 (21) Expanding the left hand side of Eq. (16) in a Taylor series at the point (0, ~), and the function h(6, q0) in the right hand side at the point (6, 0), we obtain, after taking into account equalities (19)-(21), ~no: * h?+.(o,..F/j6n,,~+~ + o((6.o?+'1= h{ 1~(o, p)-7 ~ ... § 0,- ~ I): \\-V) \\,, ~ ,' e . ,~+I, l I ~)~ \"(',~, +~)(6, O) ~l',~+ ~ ~~ o trio ~j 9 = /,~? (6, o) no = ..- , ~ ._,~(~wL): an Here h(~i ) (0, p) = o% a(--7 ~, ~) I %=0' ~lo -roolu - ~ o~,qo In )j In '~no qT/J t~.(~> (a, O) = a~h (6, ~]) 1 O~li tq=o\" Let us now select the term c (I) -~+- fi+ll -L'- (hn (5, ~u-))' * ~l,, ,u I]o in the right-hand side of the above equality and note that all the other summands on both sides of this equality are either of higher or of lower order in Do than this term. Hence 546 it follows that h~i)(6,~- 0) : 0. This, however, contradicts condition (19). The contradic- tion we have obtained shows that it is impossible to perform a cn+z-lienarization of system (13) in the case when a + b # 0. Let us now consider the case ~ + b = 0, that is, b = - a. In this case solutions of system (13) have the form x (t, Xo, Yo) = Xo exp ((k + a (xoYo)~ ) t): Y (t, Xo, Yo) = yoexp ((--~--a(xoYo)~)t) 9 Let us define the numbers 5, r ~, T as in the case a + b # 0. Following a trajectory of system (13) for time T, the point (f(5, no), g(6, no)) is mapped into the point (f, g), where f = f (6, ~o) exp ((~ + a (h (6, no))~ ) T), = g (~, ~o) exp ((--~ -- a (h (~, ~o))':) T). Here h(~, q) = f(~, q)g(~, n). On the other hand, the point (f, g) must coincide with the point (/(~ (6, r), ~ 0~o, T)), g (~ (~, T), ~101o, T))) = [I [~-,, ), g k-~-' Hence we obtain / /( --&-~lo- -, ,,// = / (6, ~lo) exp (0' + a (h (6, ~1o))'~)[T ). Substituting T = (l/X) in (P/q0) and taking logarithms of both sides of the last equality, we have that In/(-~-~t, ~t)= ln/(& ~o)+ +[k + a (h (6, ,,o))~] ln-~ . (22) Now we expand the function f(dq0/~, p)in a Taylor series at the point (0, p); the func- tions f(6, q0), h(6, q0) we expand in Taylor series at the point (6, 0). = ~)~-+-.- i (~+])!:t ~,~)\" 9 \\-~-/ +o =C~qo+...+c~+~qo +O(qo ), /(6,~o)=/(6,O)+m4(1w)/R, O)~o~~ --.+ (~+] ~)! :*(-n +l) /w;~ , 0)\" 9~ 1o+ ~+oU. lo~ +1\\) =d~+d~lo~-.: ..~-d~+~qo +~+oU9 lon +l~) ; h (~, ~o) = h (~, o) + hi? (~, 0) ,~o + (~. ~)~ hi~+ \" (6,o). 9 ~]2 +~ + o 0- ]o? ~+I\\) = ep]o v-..- + e,~+lqo +~ + o 0]o+l). Here ]i ~ (o, ~) = ~i.~e( Lii' ' ) ~=o ' Ill) (6, o) = s a(~~]T, ,) ln =o\" Moreover, note that from (19) it follows that e I = -/Z(l1i ) (~, O) = ~ + 0 (6) =# 0 for 6 =~ O. Using the Taylor expansions given above, we have ln/\\ ~ ,~,=ln~lo+eo+elqo+....+C~qo +o(qo~); ln/(f,~o)=~o+~ll]o+...+d~+xlto :-OUlo ), (h(S, no)) ~ = e~o + o01~), ~0. Substituting these equalities in (22), we obtain ln~lo+~o+ePio+...+c,~l]o n +O0lon )=go+~llo+. . 9 =9- d~, ~+~~l o~+ i ~-- O(qon +l ) In + ~ elB o tn T + o ~-\" 547 Let us now pick the term (a/~,) ~1~1~~ ln. qo in the right-hand side of the above equality and note that all the other summands on both sides of this equality are either of higher or of lower order in ~0 than this term. Hence we must have the equality (~/%)e~ = 0, which is impossible. The contradiction we have ob- tained shows that it is impossible to perform a cn+~-linearization of system (13) in the case when a + b = 0. Theorem i is proved. Proof of Theorem3. For each i (i <_ i < n) let us order the set Q~= {s: ~j=~sjk~=k~] of collections of resonances in the following way. We shall say that s~ , ~Q~,s~>-s ~ (s ~, follows s =) if either {s~{ > {s={ or if {s~{ = {s2] and there exists an m (i~m~z)such that s m~ > s m2 and sj~ = sj2 for i <_ j < m. Let us renumber the collections of the set Q~ so that QV= {s~,/~N} and s~,~-s ~s if j~ > J2\" Let us note that if sij ~ Qi (I ~ j ~ s then in view of system (7), derivatives of the functions y s~ (I <]. j(i), where s iy(0.= max {s}. s~P~nQ i Considering Y = {Yij} in (23) as independent variables, using Theorem 1 of [98], we have that ior any integer N 0 there is a number M0 = M0(N, A) such that if M(s > M0, then system (23) can be reduced by a CN transformation y = z+Ql(z,Z), Z---- {z~j}, r = Z + •e (z, Z),IlQi (z, Z) N = o (H (z, Z)]{ N) (i = i, 2) (24) to the form Since system (23) has an invariant manifold r=h@,h(v)={S}, system (25) will also have an invariant manifold Z = ~ (z) ~ C N, ~ (z) = {~j (z)}. Here from the form of the mapping (24) it follows that II ~ (z) - h (z)1{ = o (11 z II~'). The tranformation (24) induces a nondegenerate change of variables y = z + g~ (z, 7/(z)) ~ C~', after which system (7) assumes the form -~i = eizi-1 + ~iz~ + ~2 (z) (1 ~< i ~< n), ( 26 ) 548 where ~'2 (z) ~ C ,~'. Clearly, the following estimates hold: ttai2(z)--a~2(z)tl=o(llzll \"v) (l~.<-n)~ (27) From (25) it is seen that in view of (26) the derivatives (hij(z))' of the functions hij(z) satisfy the equalities Let us now pick numbers N and ~ so large, that N > N(k, A) and M(Z) > Mo(N, A), where N(k, A) is the number from Theorem 1 of [8]. We shall show that under these conditions sys- tems (26) and (5) are ck-equivalent which clearly proves the assertion of the theorem. We seek the desired change of variables in the form z,=z~§ I/~(z) l=o(llzll)(l biTu~J where P is some set of collections of resonances, that satisfy condition S(k), the terms of the polynomial Fi~(w, g) are also resonant, Fis(w) e C N, IFi3(w)l = o(IIwlIN). Our goal is to show for system (31) existence of an invariant C k manifold of the form g = G(w), IIG(w)[I = o([[wlI). The proof of this fact is essentially identical to the corresponding part of the proof of Theorem 3 of [5], so we give only its outline. First of all, we construct for the collections s e p auxiliary functions t~(w), which have the following properties: 549 - by force of system (31), the deriavive of the function t~(w) is equal to one; - for any integer q > 0 the functions wS(t~(w))q is in C k. Using the auxiliary functions t~(w) we construct the mapping g = gl ._~ G1 (W) ~ C ~, II G ~ (w)II -- o (11 w II), that reduces system (31) to the form ~v = ~pwv + Y p