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On Ordinary Differential Equations Admitting a Finite Linear Group of Symmetries
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
216, 180]196 Ž1997.
AY975668
On Ordinary Differential Equations Admitting a Finite Linear Group of Symmetries Michael K. KinyonU Department of Mathematics and Computer Science, Indiana Uni¨ ersity at South Bend, South Bend, Indiana 46634
and Sebastian Walcher † Mathematisches Institut, Technische Uni¨ ersitat 80290 Munchen, Germany ¨ Munchen, ¨ ¨ Submitted by Thanasis Fokas Received June 21, 1996
1. INTRODUCTION The importance of Žconnected, local. Lie groups of symmetries for the study of differential equations has been recognized since Lie’s fundamental work in the last century and has received renewed interest in recent years Žcf. Olver w14x for a contemporary account.. While Lie’s own primary goal may have been the reduction of differential equations to integration problems via reduction of dimension, the construction of a suitable ‘‘reduced space’’ for differential equations admitting a compact linear group of symmetries, and its application to the qualitative theory of such equations, has been the focus of much research in recent times; cf. the monographs by Golubitsky et al. w7, 8x. The purpose of this article is to isolate and discuss the algebraic aspect of reduction for differential equations admitting a linear group of symmetries or orbital symmetries which satisfies a Žrather weak. additional condition. While the result should be considered well known in the case of a symmetry group, reduction in the presence of a group of orbital symme* E-mail address: [email protected]. † E-mail address: [email protected]. 180 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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tries Žwhich preserve solution orbits, but not necessarily their parametrization. seems to have gone unnoticed Žor at least unexploited. so far. We discuss a number of examples with finite linear Žorbital. symmetry groups to illustrate how the procedure provides quite strong information in some cases, although there is no reduction of dimension. The examples will show that, unlike the connected Lie group case, the consideration of complex symmetry groups may yield useful information even if one is only interested in differential equations in real vector spaces. It will also turn out that orbital symmetries have quite strong consequences, at least when the group is a reflection group. It is worth mentioning that when dealing with a differential equation of low dimension which exhibits symmetry or orbital symmetry with respect to some group, an experienced observer will often notice Žafter some experimentation. an appropriate change of variables that renders the equation into a more tractable form. We do not see the reduction procedure discussed herein as competing with such methods; indeed, our view is that the underlying symmetry is the reason such methods work. Our purpose is to present an approach to the study of ordinary differential equations with finite Žorbital. symmetry groups which is systematic and thus can be used whether or not an ad hoc approach immediately suggests itself. We start with a few definitions and preliminary results. We will always consider an autonomous ordinary differential equation Ž*. ˙ x s f Ž x . on an open, connected neighborhood U of 0 in K n ŽK s R or C. with f analytic on U , and f / 0. If another differential equation ˙ x s g Ž x . Žwith analytic g . is given on m V : K , and there is a nonempty, open subset U˜ : U and an analytic map F : U˜ ª V that sends parametrized solutions of ˙ x s f Ž x . to parametrized Ž . solutions of ˙ x s g x , then we call F a solution-preser¨ ing map from ˙x s f Ž x . to ˙x s g Ž x .. If V s U , g s f, and F is Žlocally. invertible then F is called a symmetry of ˙ x s f Ž x .. If V s U , g s m f Žwith 0 / m : U ª K . Ž analytic , and F is locally. invertible then F is called an orbital symmetry of ˙ x s f Ž x .. If m is not identically 1 then we speak of a proper orbital symmetry. ŽNote that ˙ x s f Ž x . and ˙ x s m Ž x . f Ž x . have the same solution orbits near every point that is not a zero of m. Conversely, two equations that have locally the same solution orbits differ only by a scalar factor m that is necessarily the quotient of two analytic functions; cf. w19x.. There is a familiar ‘‘infinitesimal’’ criterion: With notations as above, F is solution-preserving from ˙ x s f Ž x . to ˙ x s g Ž x . if and only if DF Ž x . f Ž x . s g Ž F Ž x .. for all x g U˜; see, for instance, Olver w14x, or w19x. Now let G : GLŽ n, K. be a linear group. We call G an Ž orbital . symmetry group of ˙ x s f Ž x . if every element of G is an Žorbital. symmetry of ˙ x s f Ž x . Žresp. its complexification, if necessary .. Occasionally we speak
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of a proper orbital symmetry group if G contains at least one proper orbital symmetry. The criteria for orbital symmetries can be slightly sharpened for finite G. Recall that a character of G is a homomorphism x : G ª CU . PROPOSITION 1.1. The finite linear group G is an orbital symmetry group of ˙ x s f Ž x . if and only if there is a character x of G such that T ( f (Ty1 s x ŽT . f for all T g G. G is a symmetry group of ˙ x s f Ž x . if and only if T ( f (Ty1 s f for all T g G. Proof. In view of the discussion above, we only have to prove the necessity of the first condition. If T g G is an orbital symmetry of ˙x s f Ž x ., there is an analytic function m T so that ŽT ( f (Ty1 .Ž x . s m T Ž x . f Ž x . for all x. Because T has finite order and U is connected, m T is constant and its value is a < G < th root of unity, which we call x ŽT .. It is easily verified that x is a character of G. If w is analytic in U U : U with values in K, the Lie deri¨ ati¨ e L f Ž w . of w with respect to f is defined by L f Ž w . [ Dw Ž x . f Ž x .. ŽWe call w a semi-in¨ ariant, resp. first integral, of ˙ x s f Ž x . if L f Ž w . s mw for some analytic m : U ª K, resp. L f Ž w . s 0. The zero level set of a semi-invariant is an invariant set for ˙ x s f Ž x ., as is any level set of a first integral.. The following result may be seen as the technical basis for the reduction theorems. LEMMA 1.2. Let G be a linear group, and let x , h be characters of G such that T ( f (Ty1 s x ŽT . f and w (Ty1 s h ŽT . w for all T g G. Then L f Ž w .(Ty1 s x ŽT .h ŽT . L f Ž w . for all T g G. Proof. For x g U U , differentiate w ŽTy1 x . s h ŽT . w Ž x . to obtain Dw ŽTy1 x .Ty1 s h ŽT . Dw Ž x .. Now
x Ž T . h Ž T . L f Ž w . Ž x . s x Ž T . h Ž T . Dw Ž x . f Ž x . s Dw Ž Ty1 x . Ž Ty1x Ž T . f Ž x . . s Dw Ž Ty1 x . f Ž Ty1 x . s L f Ž w . Ž Ty1 x . . 2. REDUCTION IN THE CASE OF A SYMMETRY GROUP Let G be a subgroup of GLŽ n, K.. Recall that a polynomial w g Kw x 1 , . . . , x n x is called an in¨ ariant of G if w (Ty1 s w for all T g G. Moreover, a polynomial c is called a relati¨ e in¨ ariant of G if there is a character h of G so that c (Ty1 s hc for all T g G. The invariants of G form a subalgebra I Ž G ., while the relative invariants corresponding to a fixed character h form an I Ž G .-module we denote by Ih Ž G ..
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There are linear groups for which I Ž G . is not a finitely generated K-algebra, but I Ž G . is finitely generated for large classes of groups, including every group that acts completely reducibly on K n Žsee Springer w17x on these issues. and therefore every finite and every Žreal. compact group. We state the reduction theorem in a special case: THEOREM 2.1. Let G be a linear group such that I Ž G . has a finite system of generators w 1 , . . . , wr . If f is a polynomial and T ( f (Ty1 s f for all T g G then the map
¡K
F:
~
ª Kr w1 . x ¬ .. wr
¢
n
0
is solution-preser¨ ing from ˙ x s f Ž x . to a differential equation ˙ y s g Ž y . with g a polynomial. Proof. We have to verify the existence of a polynomial mapping g such that DF Ž x . f Ž x . s g Ž F Ž x .. for all x. Lemma 1.2 Žwith x s h s 1. implies L f Ž w . g I Ž G . for all j, hence there are polynomials g j such that L f Ž w . s g j Ž w 1 , . . . , wr ., 1 F j F r. Thus
g1 . g s .. gr
0
will suffice. Remark 2.2. Ža. If G acts completely reducibly on K n, then Theorem 2.1 continues to hold with ‘‘polynomial’’ replaced by ‘‘analytic.’’ This follows from a theorem of Luna w13x, which states that every analytic G-invariant K-valued function s defined near 0 can be represented as s s r Ž w 1 , . . . , wr ., with r a power series that has a nontrivial domain of convergence. ŽIt is this nontriviality that is the difficult part of the result.. For the C` case Žwith Žreal. compact G ., see w8, 16x. Žb. It is not necessary for there to be an algebraically independent set of generators for I Ž G .. The Žpolynomial. relations between w 1 , . . . , wr define a variety Y : K r which can be shown to be invariant for ˙ y s gŽ y. and contains the image of F. Y is sometimes called the discriminant variety of G; see Cox et al. w5x. If K s C and G is finite, then points of Y are in one-to-one correspondence with orbits of G. If K s R and G is compact then there is a semi-algebraic subset Y˜ of Y whose points are in
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one-to-one correspondence with orbits of G; see Schwarz w16x for the ˜ inequalities defining Y. Žc. For real compact groups, the structure and geometry of the aforementioned orbit space Žthe semialgebraic variety Y˜ . has been discussed both in general and for many specific groups; see Gaeta w6x, Abud and Sartori w1, 2x, Chossat w4x, and Jaric et al. w11x. It should be noted that another approach to characterize strata in the orbit space uses isotropy subgroups, cf. w6, 1, 2, 8x. Žd. Concerning the question of finding a system of generators for I Ž G ., G finite, see Cox et al. w5, Chap. 7x and Sturmfels w18x. Že. For a given differential equation with polynomial right hand side, the problem of finding linear Žorbital. symmetries leads to a system of polynomial equations for the matrix entries Žafter some coordinate system is fixed.. In principle, this is accessible with Grobner basis methods Žsee ¨ w x. Cox et al. 5 . The following examples will illustrate how Theorem 2.1 may help in analyzing or solving a differential equation. Note that most of them are taken from problems arising in physics or mathematical biology, and are therefore of some ‘‘practical’’ interest. EXAMPLE 2.3. Kasner’s equation Žcf. w12x.
˙x 1 s x 2 x 3 y x 12 ˙x 2 s x 3 x 1 y x 22
Ž briefly, ˙x s f Ž x . in R 3 .
˙x 3 s x 1 x 2 y x 32 describes Einstein’s gravitation equations in a special case. This equation has S3 Žthe group of linear maps defined by permutations of the standard basis. as a symmetry group. The invariant algebra I Ž S3 . is generated by the elementary symmetric polynomials w 1Ž x . s x 1 q x 2 q x 3 , w 2 Ž x . s x 1 x 2 q x 2 x 3 q x 3 x 1 , w 3 Ž x . s x 1 x 2 x 3 . Therefore
w1 F s w2 w3
0
is solution-preserving from ˙ x s f Ž x . to an equation ˙ y s g Ž y . in R 3 that turns out to be easily solved by elementary functions; see w12x. ŽIt should be noted that ˙ x s f Ž x . admits a two-dimensional abelian linear group of orbital symmetries, generated by dilations and rotations about 1 R? 1 . 1
ž/
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185
Therefore the solution of ˙ x s f Ž x . can be reduced to integration problems ´a la Lie. We include this example for historical reasons; it shows that for specific examples, the reduction technique has been around for some time.. EXAMPLE 2.4. The real equation
˙x 1 s a x 1 q b Ž x 14 , x 1 x 2 , x 24 . x 23 ˙x 2 s a x 2 y b Ž x 14 , x 1 x 2 , x 24 . x 13 Ž˙ x s f Ž x . in R 2 . }with a a real number and b : R 3 ª R a polynomial} admits the complex symmetry group G generated by T s diagŽ i, yi .. The algebra I Ž G . is generated by w 1Ž x . s x 14 , w 2 Ž x . s x 1 x 2 , and w 3 Ž x . s x 24 . Now 3
L f Ž w 1 . s 4a x 14 q 4b ? Ž x 1 x 2 . s 4aw 1 q 4bw 23 , L f Ž w 2 . s 2 a x 1 x 2 q b ? Ž x 24 y x 14 . s 2 aw 2 q b ? Ž w 3 y w 1 . , 3
L f Ž w 3 . s 4a x 24 y 4b ? Ž x 1 x 2 . s 4aw 3 y 4bw 23 . Therefore
w1 F s w2 w3
0
is solution-preserving from ˙ x s f Ž x . into
˙y 1 s 4a y 1 q 4b Ž y 1 , y 2 , y 3 . y 23 ˙y 2 s 2 a y 2 q b Ž y 1 , y 2 , y 3 . Ž y 3 y y 1 .
Ž ˙y s g Ž y . in R 3 .
˙y 3 s 4a y 3 y 4b Ž y 1 , y 2 , y 3 . y 23 . ŽThe variety Y : R 3 Žsee Remark 2.2Žb.. is defined as the zero set of y 1 y 3 y y 24 , since w 1 w 3 y w 24 s 0 is essentially the only relation between w 1 , w 2 , w 3 .. While ˙ x s g Ž x . does not appear to be any easier to solve than the original equation, we can still easily obtain nontrivial information from it: Obviously, Ž y 1 q y 3 .X s 4a Ž y 1 q y 3 ., and pulling this back yields L f Ž w 1 q w 3 . s 4a Ž w 1 q w 3 .. Thus s [ w 1 q w 3 : R 2 ª R is a solution-preserving map from ˙ x s f Ž x . into the one-dimensional equation ˙ y s 4a y, and this yields enough information to determine the qualitative behavior. As can be seen here, complex symmetries may provide interesting information about real systems.
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EXAMPLE 2.5. Equations of the type
˙x 1 s x 1 Ž m x 1 q n Ž x 2 q x 3 . . ˙x 2 s x 2 Ž m x 2 q n Ž x 3 q x 1 . .
Ž ˙x s f Ž x . in R 3 , with real
m, n .
˙x 3 s x 3 Ž m x 3 q n Ž x 1 q x 2 . . are of some importance in selection models of mathematical biology; see Hofbauer and Sigmund w10x. From S3-invariance it follows that
w1 F s w2 w3
0
Žwith the elementary symmetric polynomials. is solution-preserving to an equation ˙ y s g Ž y . in R 3. Computation shows
g Ž y. s
m y 12
Ž m q n . y1 y 2 Ž m q 2 n . y1 y 3
where
hŽ y . [
m y1
Ž m q n . y2 Ž m q 2n . y3
0 0
2 y2 q Ž n y m . 3 y 3 s y1Ž h Ž y . q k Ž y . . , 0
0
and
kŽ y. [
Ž n y m. y1
2 y2 3 y3 . 0
0
For the Lie bracket of these two vector fields, we compute
w h, k x Ž y . s Dk Ž y . h Ž y . y Dh Ž y . k Ž y . s Ž n y m . k Ž y . . From this commutator relation it follows that the general solution of
˙y s hŽ y . q k Ž y . can be expressed via the general solutions of ˙y s hŽ y . and ˙ y s k Ž y .: Denote by H Ž t, z . the solution of ˙ y s hŽ y ., y Ž0. s z, and by K Ž t, z . the solution of ˙ y s k Ž y ., y Ž0. s z. Then the solution of ˙ ys
hŽ y . q k Ž y ., y Ž0. s z, is given by H Ž t, K ŽŽ1rŽ n y m ...Ž e Ž ny m .t y 1., z ... ŽWe assume m / n ; the case m s n allows easy integration of ˙ y s g Ž y ... w x The proof of this formula follows as in 20 . Now
HŽ t, z. s
z1exp Ž m t . z 2 exp Ž Ž m q n . t . z 3 exp Ž Ž m q 2 n . t .
0
,
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while K Ž t, z . is found from the solution of 2 y2 ˙y s Ž n y m . 3 y 3 0
2
0
z1 q Ž n y m . tz 2 q 12 Ž n y m . t 2 z 3
which is
z 2 q Ž n y m . tz 3 z3
00
by replacing t with the solution t Ž t . to ˙ t s 1rŽ z1 q Ž n y m .t z 2 q 12 Ž n y 2 2 m . t z 3 ., t Ž0. s 0. ŽThus t can be explicitly determined from the equation z1t q 12 Ž n y m . z 2t 2 q 16 Ž n y m . 2 z 3t 3 s t using Cardano’s formula.. Putting pieces together, we see that the solution of ˙ y s Ž1ry 1 . g Ž y . can be determined explicitly, and so can the solution of ˙ x s Ž1rw 1 . f Ž x .. ŽNote that F can be explicitly inverted, using Cardano’s formula once more.. In the positive orthant Žwhich is the interesting region in applications. this equation and ˙ x s f Ž x . have the same solution orbits, and thus the analysis of ˙ x s f Ž x . can be carried out by elementary means. The reduction from ˙x s f Ž x . to ˙y s g Ž y . is a quite important step in this example, because it yields an easy-to-recognize decomposition of Ž1ry 1 . g Ž y . into a sum of vector fields with a well-behaved commutator relation. Moreover, it can be checked that there is no corresponding additive decomposition for ˙ x s f Ž x. itself. EXAMPLE 2.6. The equation
˙x 1 s x 2 x 3 ˙x 2 s x 3 x 1
Ž ˙x s f Ž x . in R 3 .
˙x 3 s yx 1 x 2 comes from the Euler equation for the Žasymmetric. spinning top in the absence of external forces by appropriate scaling of the variables. This equation admits the complex linear symmetries
T1 x [
x2 x1 , x3
0
ix 3 x2 , T2 x [ yix 1
0
and
yx 1 T3 x [ yx 2 , x3
0
which generate a group of order 12. ŽIt can be shown that no other linear symmetries exist for ˙ x s f Ž x ... G has no nontrivial linear invariant, and Žup to scalar multiples. one quadratic invariant c 1Ž x . s x 12 q x 22 y x 32 . Now L f Ž c 1 . s 6 c 2 , with c 2 Ž x . [ x 1 x 2 x 3 , L f Ž c 2 . s c 3 , with c 3 Ž x . [ yx 12 x 22 q x 22 x 32 q x 32 x 12 , and
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L f Ž c 3 .Ž x . s y4 x 1 x 2 x 3 Ž x 12 q x 22 y x 32 . s y4c 1 c 2 . Thus
c1 C s c2 c3
0
is solution-preserving from ˙ x s f Ž x . into
˙y 1 s 6 y 2 ˙y 2 s y 3
in R 3 ,
˙y 3 s y4 y 1 y 2
{
which is equivalent to the third order equation y s y4 yy. ˙ Integrating yields ¨ y s y2 y 2 q g Žg g R., which is a second-order equation for a Weierstrass `-function. Thus the reduction procedure provides a natural route to the Žwell-known. fact that Euler’s equations can be solved with the help of elliptic functions. The invariants c 1 , c 2 , and c 3 do indeed generate I Ž G . Žthis follows most easily from the fact that G is conjugate to the tetrahedral group in GLŽ3, C. and using Jaric et al. w11x.. It may be worth noting, however, that in this example we did not start with a system of generators for I Ž G ., but only with an element of smallest degree. Thus it is not necessary to have a complete system of generators at the beginning. To close this section we note another consequence of Lemma 1.2 that is occasionally useful in determining invariant sets. ŽFor sake of simplicity we do not consider the most general situation.. PROPOSITION 2.7. Let G : GLŽ n, K. and h a character of G such that the I Ž G .-module Ih Ž G . has a finite set of generators r 1 , . . . , r s . If f is a polynomial and G is a symmetry group for ˙ x s f Ž x . then the set Z of common zeros of r 1 , . . . , r s is an in¨ ariant set for ˙ x s f Ž x .. Proof. According to Lemma 1.2, L f Ž r i . g Ih Ž G . for 1 F i F s. Therefore there exist m i j g I Ž G . such that L f Ž r i . s Ý m i j r j Ž1 F i F s .. The invariance of Z follows from w19x. EXAMPLE 2.8. For S3 : GLŽ3, R. consider the character h that attains the value y1 on the reflections. It is known that the discriminant r Ž x . s Ž x 1 y x 2 .Ž x 2 y x 3 .Ž x 3 y x 1 . generates the I Ž S3 .-module Ih Ž S3 .; cf. w17x. Therefore the planes x i y x j s 0 Ž i / j . are invariant sets for every S3-symmetric equation. For the equations from Example 2.5, one has additional semi-invariants x 1 , x 2 , x 3 ; and the method developed in Grobner ¨ and Knapp w9x yields a number of first integrals, e.g., s Ž x . s Ž x 2 y n x 3 .yŽ m q2 n . Ž x 2 x 3 .Ž m q n . xy 1 .
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189
3. REDUCTION IN THE CASE OF AN ORBITAL SYMMETRY GROUP Intuitively, a Žproper. orbital symmetry group should be less restrictive for a differential equation than a symmetry group. Indeed, reduction has a more complicated look here. Once more we formulate the theorem only for the case of a polynomial differential equation. THEOREM 3.1. Let G be linear group and x : G ª CU a character of order s q 1 ) 1. Suppose that I Ž G . has a finite system w 1 , . . . , wr of generators and further that e¨ ery I Ž G .-module Ix k Ž G . Ž1 F k F s . is finitely generated by c k, 1 , . . . , c k, m k. Let pk s Ý kjs1 m k , 1 F k F s. If T ( f (Ty1 s x ŽT . f for e¨ ery T g G, then the map
¡K
Ž with R s r q m1 q ??? qm s . F :~ x ¬ Ž w 1 Ž x . , . . . , wr Ž x . , c 1, 1 Ž x . , . . . , c 1, m 1Ž x . , . . . ,
¢c
n
ª KR
s, 1
Ž x . , . . . , cs, m sŽ x . .
t
is solution-preser¨ ing from the polynomial differential equation ˙ x s f Ž x . in K n R to a polynomial differential equation ˙ y s g Ž y . in K which is of the following specific type, rqp 1
˙yi s Ý mŽ0. i j Ž y 1 , . . . , yr . y j
Ž1 F i F r .
jsrq1 rqp 2
˙yi s
Ý
jsrqp 1q1
mŽ1. i j Ž y 1 , . . . , yr . y j
Ž r q 1 F i F r q p1 .
.. . rqp s
˙yi s
Ý
jsrqp sy1 q1
mŽi sy1. Ž y 1 , . . . , yr . y j j
˙yi s n i Ž y 1 , . . . , yr .
Ž r q psy2 q 1 F i F r q psy1 .
Ž r q psy1 q 1 F i F r q ps .
with all the mŽi kj . and n i polynomials. Proof. By Lemma 1.2 we have L f Ž w i . g Ix Ž G . Ž1 F i F r ., and L f Ž c k, j . g Ix kq 1 Ž G . Ž1 F j F m k . for all k, where I1Ž G . [ I Ž G .. Remark 3.2. Ža. The image of F is contained in the subvariety Y of K R defined by all the relations between the w i and c k, j ; and again Y is invariant for ˙ y s g Ž y ..
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Žb. As there seems to be no counterpart of Luna’s theorem w13x available in the literature for this situation, generalization to the analytic case has to be investigated individually for every group. Žc. Theorem 3.1 turns out to be most useful in the case that every module Ix k Ž G . Ž1 F k F s . is generated by one element. The underlying reason is that the image equation ˙ y s g Ž y . starts with a ‘‘block’’
˙y 1 s yrq1 m 1 Ž y 1 , . . . , yr . .. .
˙yr s yrq1 m r Ž y 1 , . . . , yr . and information about solution orbits of the differential equation
˙y 1 s m 1 Ž y 1 , . . . , yr . .. .
˙yr s m r Ž y 1 , . . . , yr . in K r can be carried over to ˙ y s g Ž y .. ŽFor instance, a first integral of the latter will also be a first integral of ˙ y s g Ž y ... In the case of finite groups it is known that reflection groups have the desired property Žcf. Springer w17, Theorem 4.3.4x.. The following examples will therefore involve orbital symmetry groups that are generated by reflections. EXAMPLE 3.3. The equation
˙x 1 s x 1 Ž x 2 y x 3 . ˙x 2 s x 2 Ž x 3 y x 1 .
Ž ˙x s f Ž x . in R 3 .
˙x 3 s x 3 Ž x 1 y x 2 . is of some significance for a mathematical model in biology; cf. Hofbauer and Sigmund w10, Sect. 16.5x. ŽIt may also be viewed as describing the population dynamics of three species that cyclically prey upon each other.. For this equation we have S3 as a proper orbital symmetry group, with x the only nontrivial character of S3 , and Ix Ž S3 . generated by the discriminant: cf. Example 2.8. The image equation ˙ y s g Ž y . turns out to be
˙y 1 s 0 ˙y 2 s y4 ˙y 3 s 0 ˙y4 s y6 y 22 q y 12 y 2 q 9 y 1 y 3 .
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191
Here the first integrals y 1 and y 3 are immediate, and pulling these back yields the first integrals r 1Ž x . s x 1 q x 2 q x 3 and r 2 Ž x . s x 1 x 2 x 3 for ˙x s f Ž x .. These are already known from w10x Žand r 1 , at least, is obvious by inspection.. Returning to the image equation, for y 1 s g 1 , y 3 s g 3 constant we obtain the two-dimensional Hamiltonian system
˙y 2 s y4 ˙y4 s y6 y 22 q y 12 y 2 q 9 y 1 y 3 , with first integral 12 y42 q 2 y 23 y 12 g 12 y 2 y 9g 1 g 3 y 2 . Thus we have Ž ˙ y2 .2 s 3 2 2 Ž y4 y 2 q g 1 y 2 q 18g 1 g 3 y 2 q d with d determined by initial conditions., and this is essentially the differential equation of a Weierstrass `-function Žunless the right hand side has multiple roots, in which case the solution is elementary.. Thus in addition to the known first integrals, we add the new piece of information that ˙ x s f Ž x . can be solved by employing elementary and elliptic functions. EXAMPLE 3.4. Ža. Let G be the group Žof order 4. generated by T1 s diagŽ1, y1. and T2 s diagŽy1, 1., and let x be the character of G with value y1 on T1 and T2 . The algebra I Ž G . is generated by w 1Ž x . s x 12 and w 2 Ž x . s x 22 , and the module Ix Ž G . is generated by c Ž x . s x 1 x 2 . Consider the differential equation
˙x 1 s x 2 p Ž x 12 , x 22 . ˙x 2 s x 1 q Ž x 12 , x 22 . Ž˙ x s f Ž x . in R 2 . where p, q : R 2 ª R are orbital symmetry group for this equation, F s Ž w 1 , w 2 , c . t is solution-preserving from
polynomials. Then G is an with character x . The map ˙x s f Ž x . into
˙y 1 s 2 y 3 p Ž y 1 , y 2 . ˙y 2 s 2 y 3 q Ž y 1 , y 2 . ˙y 3 s y 2 p Ž y 1 , y 2 . q y 1 q Ž y 1 , y 2 . . ŽHere the variety Y is given by y 1 y 2 y y 32 s 0.. We are primarily interested in the first two lines. In particular, if p and q are such that determining a first integral s of the equation
˙y 1 s p Ž y 1 , y 2 . ˙y 2 s q Ž y 1 , y 2 . is elementary, then r Ž x . [ s Ž x 12 , x 22 . will be an elementary first integral of ˙ x s f Ž x .. Specific examples include the cases where p and q are given
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by a Hamiltonian function Ž pŽ x 1 , x 2 . s 2 hŽ x 1 , x 2 ., q Ž x 1 , x 2 . s y 1 hŽ x 1 , x 2 .., or where p and q are affine Ž pŽ x 1 , x 2 . s m q a x 1 q b x 2 , q Ž x 1 , x 2 . s n q g x 1 q d x 2 .; the latter case includes every polynomial differential equation of degree F 3 satisfying T ( f (Ty1 s x ŽT . f for all T g G. One may view the reduction from a slightly different perspective. For instance, in part Ža. one has d dt d dt
Ž x 12 . s 2 x 1 x 2 p Ž x 12 , x 22 . Ž x 22 . s 2 x 1 x 2 q Ž x 12 , x 22 . ,
and thus the mapping x ¬ Ž w 1 , w 2 . is locally orbit-preserving from ˙ xs f Ž x . to
˙y 1 s p Ž y 1 , y 2 . ˙y 2 s q Ž y 1 , y 2 . So it can be said that the reduction consists of a local coordinate transformation, together with a change in time scale. As noted in the Introduction, an experienced observer may, of course, notice this directly. The point we are making here, however, is that the symmetry group gives the underlying reason why this procedure works, and Theorem 3.1 provides a systematic approach. Žb. Let z s expŽ2p ir3., T s diagŽ z , 1., and G be the group generated by T. Denote by x the character that has value z on T. The algebra I Ž G . is generated by w 1Ž x . s x 13 and w 2 Ž x . s x 2 , while the modules Ix Ž G . and Ix 2 Ž G . are generated by c 1Ž x . s x 12 and c 2 Ž x . s x 1 , respectively. Consider the differential equation
˙x 1 s p Ž x 13 , x 2 . ˙x 2 s x 12 q Ž x 13 , x 2 . Ž˙ x s f Ž x . in R 2 . where p, q : R 2 ª R are polynomials. Then G is an orbital symmetry group for this equation, with character x . The map F s Ž w 1 , w 2 , c 1 , c 2 . t is solution-preserving from ˙ x s f Ž x . into
˙y 1 s 3 y 3 p Ž y 1 , y 2 . ˙y 2 s y 3 q Ž y 1 , y 2 . ˙y 3 s 2 y4 p Ž y 1 , y 2 . ˙y4 s p Ž y 1 , y 2 . .
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ŽThe variety Y is given by y 1 y y43 s 0, y 3 y y42 s 0.. Again, the first two lines are of principal interest. If p and q are such that determining a first integral s of the equation
˙y 1 s 3 p Ž y 1 , y 2 . ˙y 2 s q Ž y 1 , y 2 . is elementary, then r Ž x . [ s Ž x 13, x 2 . will be an elementary first integral of ˙ x s f Ž x .. Examples include the Hamiltonian case, and the case where pŽ x 1 , x 2 . s a x 1 q uŽ x 2 . for some polynomial u and q Ž x 1 , x 2 . s b q g x 2 , for this reduces to two one-dimensional linear equations. ŽThe latter case includes every polynomial differential equation of degree F 3 satisfying T ( f (Ty1 s x ŽT . f for all T g G.. This example provides another indication that complex orbital symmetry groups carry useful information about real equations. EXAMPLE 3.5. The existence of orbital symmetries forces certain qualitative properties on the differential equation, which in some instances can be recovered from the reduction procedure. We illustrate this assertion by considering the following class of ‘‘reversible’’ systems in K n, n G 2, O
˙x s f Ž x . [ O
O 0 b
a ? x q higher order terms, 0
with f satisfying T ( f (Ty1 s yf for T [ diagŽ1, . . . , 1, y1.. ŽNote that the linear part B s Df Ž0. satisfies T ( B(Ty1 s yB. Moreover, up to a suitable change of coordinates B is the most general linear map admissible.. In the following we will assume a / 0. The invariant algebra of the group G s T, id4 is generated by w 1Ž x . s x 1 , . . . , wny1Ž x . s x ny1 and wnŽ x . s x n2 , and the module Ix Ž G . is generated by c Ž x . s x n . ŽHere x denotes the character mapping T to y1.. By Theorem 3.1 the image equation ˙ y s g Ž y . in K nq 1 is of the type
˙y 1 s ynq1 m 1 Ž y 1 , . . . , yn . .. .
˙yn s ynq1 m n Ž y 1 , . . . , yn . ˙ynq 1 s n Ž y 1 , . . . , yn . . ŽAs follows from Remark 3.2Žc., this is also true for analytic f.. Since L f Ž wny1Ž x .. s a x n q h.o.t.s c Ž x .Ž a q h.o.t.. we have m ny1Ž0, . . . , 0. s a
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/ 0. Therefore 0 is not a stationary point of
˙y 1 s m 1 Ž y 1 , . . . , yn . .. .
Ž ˙y s h Ž y . . ,
˙yn s m n Ž y 1 , . . . , yn . and therefore ˙ y s hŽ y . has n y 1 independent first integrals s 1 , . . . , sny1 near 0. These Žconsidered as functions of y 1 , . . . , ynq1 . are also first integrals of ˙ y s g Ž y ., and pulling back yields the first integrals r i Ž x . s si Ž x 1 , . . . , x ny1 , x n2 . for ˙ x s f Ž x . Žwhich, again, are easily checked to be independent.. Therefore ˙ x s f Ž x . has n y 1 independent first integrals in a neighborhood of the stationary point 0, a highly unusual property. Let us look at two special cases. ŽBoth are reversible systems in R 2 .. 0
˙x s y1
ž
1 ? x q h.o.t.s f Ž x . 0
/
in U : R 2 .
Ž a.
The image equation here is
˙y 1 s y 3 Ž 1 q h.o.t.. ˙y 2 s y 3 Ž y2 y 1 q h.o.t..
Ž ˙y s g Ž y . . .
˙y 3 s ??? . We obtain a first integral s Ž y . s y 2 q y 12 q h.o.t. for ˙ y s g Ž y . and thus a first integral r Ž x . s x 12 q x 22 q h.o.t. for ˙ x s f Ž x .. Since r has a local minimum at 0 g R 2 , the Žlocal. level sets of r near 0 are closed curves by the classification of critical points in R 2 . Therefore the stationary point 0 is a center for ˙ x s f Ž x .. Of course it is well known that a reversible system with linear part as stipulated above has a center at 0 Žcf. Sansone and Conti w15x for another proof., but here we have an elementary ‘‘algebraic’’ proof of this fact. 0
˙x s 0
ž
b3 x1 x 2 1 ?xq q h.o.t. 0 b 1 x 12 q b 2 x 22
/
ž
/
in U : R 2 .
Ž b.
ŽThe quadratic term here is the most general compatible with reversibility. We will assume b 1 / 0.. The image equation ˙ y s g Ž y . is
˙y 1 s y 3 Ž 1 q b 3 y 1 q h.o.t.. ˙y 2 s y 3 Ž 2 b 2 y 2 q 2 b 1 y 12 q h.o.t. . ˙y 3 s ??? .
Ž ˙y s g Ž y . . .
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Here we get a first integral s Ž y . s y 2 y 2 b 2 y 1 y 2 y Ž2r3. b 1 y 13 q Žirrelevant terms. for ˙ y s g Ž y ., hence a first integral r Ž x . s x 22 y 2 b 2 x 1 x 22 y 3 Ž2r3. b 1 x 1 q Žirrelevant terms. for ˙ x s f Ž x .. Ž‘‘Irrelevance’’ here is with respect to the level set r Ž x . s 0; see Bruno w3, Chap. 1, Sect. 2x on the Newton polygon behind this argument.. For r Ž x . s 0 we have 0 s x 22 Ž 1 y 2 b 2 x 1 . y 23 b 1 x 13 q ??? , 0 s x 22 y 23 b 1 x 13 Ž 1 y 2 b 2 x 1 .
y1
q ???
s x 22 y 23 b 1 x 13 q ??? , and the stationary point turns out to be a cusp. Again it is worth noting that an a priori determination of the topological type of the stationary point 0 is not an easy problem. The algebraic reduction formalism, however, offers easy access to its solution. The specific results in the two-dimensional cases can also be obtained using normal forms, although the arguments given above use less machinery. ŽFor instance, in Ža. one has to use the nontrivial theorem that reversibility is preserved by a suitable transformation into normal form, and that Bruno’s ‘‘Condition A’’ Žsee w3x. guarantees convergence of such a transformation. In Žb., one must argue similarly using the Belitskii normal form associated with the given nilpotent linear part.. In higher dimensions it seems that the normal form machinery is less suitable than the approach chosen here.
REFERENCES 1. M. Abud and G. Sartori, The geometry of spontaneous symmetry breaking, Ann. Physics 150 Ž1983., 307]372. 2. M. Abud and G. Sartori, The geometry of orbit-space and natural minima of Higgs potentials, Phys. Lett. B 104 Ž1981., 147]152. 3. A. D. Bruno, ‘‘Local Methods in Nonlinear differential Equations,’’ Springer-Verlag, New YorkrBerlin, 1989. 4. P. Chossat, Forced reflectional symmetry breaking of an O Ž2.-symmetric homoclinic cycle, Nonlinearity 6 Ž1993., 723]731. 5. D. Cox, J. Little, and D. O’Shea, ‘‘Ideals, Varieties, and Algorithms,’’ Undergrad. Texts Math., Springer-Verlag, New YorkrBerlin, 1992. 6. G. Gaeta, ‘‘Nonlinear Symmetries and Nonlinear Equations,’’ Kluwer Academic, Dordrecht, 1994. 7. M. Golubitsky and D. G. Schaeffer, ‘‘Singularities and Groups in Bifurcation Theory,’’ Vol. 1, Applied Math. Sci. Ser., Vol. 51, Springer-Verlag, New YorkrBerlin, 1985. 8. M. Golubitsky, I. Stewart, and D. G. Schaelfer, ‘‘Singularities and Groups in Bifurcation Theory,’’ Vol. 2, Applied Math. Sci. Ser., Vol. 69, Springer-Verlag, New YorkrBerlin, 1988.
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9. W. Grobner and H. Knapp, ‘‘Contributions to the Method of Lie Series,’’ Bibliograph. ¨ Inst., Mannheim, 1967. 10. J. Hofbauer and K. Sigmund, ‘‘The Theory of Evolution and Dynamical Systems,’’ Cambridge Univ. Press, Cambridge, United Kingdom, 1988. 11. M. Jaric, L. Michel, and R. Sharp, Zeros of covariant vector fields for the point groups: Invariant formulation, J. Physique 45 Ž1984., 1]27. 12. E. Kasner, Solution of the Einstein equations involving functions of only one variable, Trans. Amer. Math. Soc. 27 Ž1925., 155]162. 13. D. Luna, Fonctions differentiables invariantes sons l’operation d’un group reductif, Ann. Inst. Fourier 26, No. 1 Ž1976., 33]49. 14. P. J. Olver, ‘‘Applications of Lie Groups to Differential Equations,’’ Graduate Texts in Math., Vol. 107, Springer-Verlag, New YorkrBerlin, 1986. 15. G. Sansone and R. Conti, ‘‘Nonlinear Differential Equations,’’ Pergamon, Elmsford, New York, 1964. 16. G. W. Schwarz, The topology of algebraic quotients, in ‘‘Topological Methods in Algebraic Transformation Groups’’ ŽH. Kraft et al., Eds.., pp. 135]151, Birkhauser, Basel, ¨ 1989. 17. T. A. Springer, ‘‘Invariant Theory,’’ Lecture Notes in Math., Vol. 585, Springer-Verlag, New YorkrBerlin, 1977. 18. B. Sturmfels, ‘‘Algorithms in Invariant Theory,’’ Springer-Verlag, New YorkrBerlin, 1993. 19. S. Walcher, Symmetries of ordinary differential equations, No¨ a J. Alg. Geom. 2, No. 3 Ž1993., 245]275. 20. S. Walcher, On sums of vector fields, Results in Math. 31, No. 1r2 Ž1997., 161]169.
216, 180]196 Ž1997.
AY975668
On Ordinary Differential Equations Admitting a Finite Linear Group of Symmetries Michael K. KinyonU Department of Mathematics and Computer Science, Indiana Uni¨ ersity at South Bend, South Bend, Indiana 46634
and Sebastian Walcher † Mathematisches Institut, Technische Uni¨ ersitat 80290 Munchen, Germany ¨ Munchen, ¨ ¨ Submitted by Thanasis Fokas Received June 21, 1996
1. INTRODUCTION The importance of Žconnected, local. Lie groups of symmetries for the study of differential equations has been recognized since Lie’s fundamental work in the last century and has received renewed interest in recent years Žcf. Olver w14x for a contemporary account.. While Lie’s own primary goal may have been the reduction of differential equations to integration problems via reduction of dimension, the construction of a suitable ‘‘reduced space’’ for differential equations admitting a compact linear group of symmetries, and its application to the qualitative theory of such equations, has been the focus of much research in recent times; cf. the monographs by Golubitsky et al. w7, 8x. The purpose of this article is to isolate and discuss the algebraic aspect of reduction for differential equations admitting a linear group of symmetries or orbital symmetries which satisfies a Žrather weak. additional condition. While the result should be considered well known in the case of a symmetry group, reduction in the presence of a group of orbital symme* E-mail address: [email protected]. † E-mail address: [email protected]. 180 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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tries Žwhich preserve solution orbits, but not necessarily their parametrization. seems to have gone unnoticed Žor at least unexploited. so far. We discuss a number of examples with finite linear Žorbital. symmetry groups to illustrate how the procedure provides quite strong information in some cases, although there is no reduction of dimension. The examples will show that, unlike the connected Lie group case, the consideration of complex symmetry groups may yield useful information even if one is only interested in differential equations in real vector spaces. It will also turn out that orbital symmetries have quite strong consequences, at least when the group is a reflection group. It is worth mentioning that when dealing with a differential equation of low dimension which exhibits symmetry or orbital symmetry with respect to some group, an experienced observer will often notice Žafter some experimentation. an appropriate change of variables that renders the equation into a more tractable form. We do not see the reduction procedure discussed herein as competing with such methods; indeed, our view is that the underlying symmetry is the reason such methods work. Our purpose is to present an approach to the study of ordinary differential equations with finite Žorbital. symmetry groups which is systematic and thus can be used whether or not an ad hoc approach immediately suggests itself. We start with a few definitions and preliminary results. We will always consider an autonomous ordinary differential equation Ž*. ˙ x s f Ž x . on an open, connected neighborhood U of 0 in K n ŽK s R or C. with f analytic on U , and f / 0. If another differential equation ˙ x s g Ž x . Žwith analytic g . is given on m V : K , and there is a nonempty, open subset U˜ : U and an analytic map F : U˜ ª V that sends parametrized solutions of ˙ x s f Ž x . to parametrized Ž . solutions of ˙ x s g x , then we call F a solution-preser¨ ing map from ˙x s f Ž x . to ˙x s g Ž x .. If V s U , g s f, and F is Žlocally. invertible then F is called a symmetry of ˙ x s f Ž x .. If V s U , g s m f Žwith 0 / m : U ª K . Ž analytic , and F is locally. invertible then F is called an orbital symmetry of ˙ x s f Ž x .. If m is not identically 1 then we speak of a proper orbital symmetry. ŽNote that ˙ x s f Ž x . and ˙ x s m Ž x . f Ž x . have the same solution orbits near every point that is not a zero of m. Conversely, two equations that have locally the same solution orbits differ only by a scalar factor m that is necessarily the quotient of two analytic functions; cf. w19x.. There is a familiar ‘‘infinitesimal’’ criterion: With notations as above, F is solution-preserving from ˙ x s f Ž x . to ˙ x s g Ž x . if and only if DF Ž x . f Ž x . s g Ž F Ž x .. for all x g U˜; see, for instance, Olver w14x, or w19x. Now let G : GLŽ n, K. be a linear group. We call G an Ž orbital . symmetry group of ˙ x s f Ž x . if every element of G is an Žorbital. symmetry of ˙ x s f Ž x . Žresp. its complexification, if necessary .. Occasionally we speak
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of a proper orbital symmetry group if G contains at least one proper orbital symmetry. The criteria for orbital symmetries can be slightly sharpened for finite G. Recall that a character of G is a homomorphism x : G ª CU . PROPOSITION 1.1. The finite linear group G is an orbital symmetry group of ˙ x s f Ž x . if and only if there is a character x of G such that T ( f (Ty1 s x ŽT . f for all T g G. G is a symmetry group of ˙ x s f Ž x . if and only if T ( f (Ty1 s f for all T g G. Proof. In view of the discussion above, we only have to prove the necessity of the first condition. If T g G is an orbital symmetry of ˙x s f Ž x ., there is an analytic function m T so that ŽT ( f (Ty1 .Ž x . s m T Ž x . f Ž x . for all x. Because T has finite order and U is connected, m T is constant and its value is a < G < th root of unity, which we call x ŽT .. It is easily verified that x is a character of G. If w is analytic in U U : U with values in K, the Lie deri¨ ati¨ e L f Ž w . of w with respect to f is defined by L f Ž w . [ Dw Ž x . f Ž x .. ŽWe call w a semi-in¨ ariant, resp. first integral, of ˙ x s f Ž x . if L f Ž w . s mw for some analytic m : U ª K, resp. L f Ž w . s 0. The zero level set of a semi-invariant is an invariant set for ˙ x s f Ž x ., as is any level set of a first integral.. The following result may be seen as the technical basis for the reduction theorems. LEMMA 1.2. Let G be a linear group, and let x , h be characters of G such that T ( f (Ty1 s x ŽT . f and w (Ty1 s h ŽT . w for all T g G. Then L f Ž w .(Ty1 s x ŽT .h ŽT . L f Ž w . for all T g G. Proof. For x g U U , differentiate w ŽTy1 x . s h ŽT . w Ž x . to obtain Dw ŽTy1 x .Ty1 s h ŽT . Dw Ž x .. Now
x Ž T . h Ž T . L f Ž w . Ž x . s x Ž T . h Ž T . Dw Ž x . f Ž x . s Dw Ž Ty1 x . Ž Ty1x Ž T . f Ž x . . s Dw Ž Ty1 x . f Ž Ty1 x . s L f Ž w . Ž Ty1 x . . 2. REDUCTION IN THE CASE OF A SYMMETRY GROUP Let G be a subgroup of GLŽ n, K.. Recall that a polynomial w g Kw x 1 , . . . , x n x is called an in¨ ariant of G if w (Ty1 s w for all T g G. Moreover, a polynomial c is called a relati¨ e in¨ ariant of G if there is a character h of G so that c (Ty1 s hc for all T g G. The invariants of G form a subalgebra I Ž G ., while the relative invariants corresponding to a fixed character h form an I Ž G .-module we denote by Ih Ž G ..
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There are linear groups for which I Ž G . is not a finitely generated K-algebra, but I Ž G . is finitely generated for large classes of groups, including every group that acts completely reducibly on K n Žsee Springer w17x on these issues. and therefore every finite and every Žreal. compact group. We state the reduction theorem in a special case: THEOREM 2.1. Let G be a linear group such that I Ž G . has a finite system of generators w 1 , . . . , wr . If f is a polynomial and T ( f (Ty1 s f for all T g G then the map
¡K
F:
~
ª Kr w1 . x ¬ .. wr
¢
n
0
is solution-preser¨ ing from ˙ x s f Ž x . to a differential equation ˙ y s g Ž y . with g a polynomial. Proof. We have to verify the existence of a polynomial mapping g such that DF Ž x . f Ž x . s g Ž F Ž x .. for all x. Lemma 1.2 Žwith x s h s 1. implies L f Ž w . g I Ž G . for all j, hence there are polynomials g j such that L f Ž w . s g j Ž w 1 , . . . , wr ., 1 F j F r. Thus
g1 . g s .. gr
0
will suffice. Remark 2.2. Ža. If G acts completely reducibly on K n, then Theorem 2.1 continues to hold with ‘‘polynomial’’ replaced by ‘‘analytic.’’ This follows from a theorem of Luna w13x, which states that every analytic G-invariant K-valued function s defined near 0 can be represented as s s r Ž w 1 , . . . , wr ., with r a power series that has a nontrivial domain of convergence. ŽIt is this nontriviality that is the difficult part of the result.. For the C` case Žwith Žreal. compact G ., see w8, 16x. Žb. It is not necessary for there to be an algebraically independent set of generators for I Ž G .. The Žpolynomial. relations between w 1 , . . . , wr define a variety Y : K r which can be shown to be invariant for ˙ y s gŽ y. and contains the image of F. Y is sometimes called the discriminant variety of G; see Cox et al. w5x. If K s C and G is finite, then points of Y are in one-to-one correspondence with orbits of G. If K s R and G is compact then there is a semi-algebraic subset Y˜ of Y whose points are in
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one-to-one correspondence with orbits of G; see Schwarz w16x for the ˜ inequalities defining Y. Žc. For real compact groups, the structure and geometry of the aforementioned orbit space Žthe semialgebraic variety Y˜ . has been discussed both in general and for many specific groups; see Gaeta w6x, Abud and Sartori w1, 2x, Chossat w4x, and Jaric et al. w11x. It should be noted that another approach to characterize strata in the orbit space uses isotropy subgroups, cf. w6, 1, 2, 8x. Žd. Concerning the question of finding a system of generators for I Ž G ., G finite, see Cox et al. w5, Chap. 7x and Sturmfels w18x. Že. For a given differential equation with polynomial right hand side, the problem of finding linear Žorbital. symmetries leads to a system of polynomial equations for the matrix entries Žafter some coordinate system is fixed.. In principle, this is accessible with Grobner basis methods Žsee ¨ w x. Cox et al. 5 . The following examples will illustrate how Theorem 2.1 may help in analyzing or solving a differential equation. Note that most of them are taken from problems arising in physics or mathematical biology, and are therefore of some ‘‘practical’’ interest. EXAMPLE 2.3. Kasner’s equation Žcf. w12x.
˙x 1 s x 2 x 3 y x 12 ˙x 2 s x 3 x 1 y x 22
Ž briefly, ˙x s f Ž x . in R 3 .
˙x 3 s x 1 x 2 y x 32 describes Einstein’s gravitation equations in a special case. This equation has S3 Žthe group of linear maps defined by permutations of the standard basis. as a symmetry group. The invariant algebra I Ž S3 . is generated by the elementary symmetric polynomials w 1Ž x . s x 1 q x 2 q x 3 , w 2 Ž x . s x 1 x 2 q x 2 x 3 q x 3 x 1 , w 3 Ž x . s x 1 x 2 x 3 . Therefore
w1 F s w2 w3
0
is solution-preserving from ˙ x s f Ž x . to an equation ˙ y s g Ž y . in R 3 that turns out to be easily solved by elementary functions; see w12x. ŽIt should be noted that ˙ x s f Ž x . admits a two-dimensional abelian linear group of orbital symmetries, generated by dilations and rotations about 1 R? 1 . 1
ž/
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185
Therefore the solution of ˙ x s f Ž x . can be reduced to integration problems ´a la Lie. We include this example for historical reasons; it shows that for specific examples, the reduction technique has been around for some time.. EXAMPLE 2.4. The real equation
˙x 1 s a x 1 q b Ž x 14 , x 1 x 2 , x 24 . x 23 ˙x 2 s a x 2 y b Ž x 14 , x 1 x 2 , x 24 . x 13 Ž˙ x s f Ž x . in R 2 . }with a a real number and b : R 3 ª R a polynomial} admits the complex symmetry group G generated by T s diagŽ i, yi .. The algebra I Ž G . is generated by w 1Ž x . s x 14 , w 2 Ž x . s x 1 x 2 , and w 3 Ž x . s x 24 . Now 3
L f Ž w 1 . s 4a x 14 q 4b ? Ž x 1 x 2 . s 4aw 1 q 4bw 23 , L f Ž w 2 . s 2 a x 1 x 2 q b ? Ž x 24 y x 14 . s 2 aw 2 q b ? Ž w 3 y w 1 . , 3
L f Ž w 3 . s 4a x 24 y 4b ? Ž x 1 x 2 . s 4aw 3 y 4bw 23 . Therefore
w1 F s w2 w3
0
is solution-preserving from ˙ x s f Ž x . into
˙y 1 s 4a y 1 q 4b Ž y 1 , y 2 , y 3 . y 23 ˙y 2 s 2 a y 2 q b Ž y 1 , y 2 , y 3 . Ž y 3 y y 1 .
Ž ˙y s g Ž y . in R 3 .
˙y 3 s 4a y 3 y 4b Ž y 1 , y 2 , y 3 . y 23 . ŽThe variety Y : R 3 Žsee Remark 2.2Žb.. is defined as the zero set of y 1 y 3 y y 24 , since w 1 w 3 y w 24 s 0 is essentially the only relation between w 1 , w 2 , w 3 .. While ˙ x s g Ž x . does not appear to be any easier to solve than the original equation, we can still easily obtain nontrivial information from it: Obviously, Ž y 1 q y 3 .X s 4a Ž y 1 q y 3 ., and pulling this back yields L f Ž w 1 q w 3 . s 4a Ž w 1 q w 3 .. Thus s [ w 1 q w 3 : R 2 ª R is a solution-preserving map from ˙ x s f Ž x . into the one-dimensional equation ˙ y s 4a y, and this yields enough information to determine the qualitative behavior. As can be seen here, complex symmetries may provide interesting information about real systems.
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EXAMPLE 2.5. Equations of the type
˙x 1 s x 1 Ž m x 1 q n Ž x 2 q x 3 . . ˙x 2 s x 2 Ž m x 2 q n Ž x 3 q x 1 . .
Ž ˙x s f Ž x . in R 3 , with real
m, n .
˙x 3 s x 3 Ž m x 3 q n Ž x 1 q x 2 . . are of some importance in selection models of mathematical biology; see Hofbauer and Sigmund w10x. From S3-invariance it follows that
w1 F s w2 w3
0
Žwith the elementary symmetric polynomials. is solution-preserving to an equation ˙ y s g Ž y . in R 3. Computation shows
g Ž y. s
m y 12
Ž m q n . y1 y 2 Ž m q 2 n . y1 y 3
where
hŽ y . [
m y1
Ž m q n . y2 Ž m q 2n . y3
0 0
2 y2 q Ž n y m . 3 y 3 s y1Ž h Ž y . q k Ž y . . , 0
0
and
kŽ y. [
Ž n y m. y1
2 y2 3 y3 . 0
0
For the Lie bracket of these two vector fields, we compute
w h, k x Ž y . s Dk Ž y . h Ž y . y Dh Ž y . k Ž y . s Ž n y m . k Ž y . . From this commutator relation it follows that the general solution of
˙y s hŽ y . q k Ž y . can be expressed via the general solutions of ˙y s hŽ y . and ˙ y s k Ž y .: Denote by H Ž t, z . the solution of ˙ y s hŽ y ., y Ž0. s z, and by K Ž t, z . the solution of ˙ y s k Ž y ., y Ž0. s z. Then the solution of ˙ ys
hŽ y . q k Ž y ., y Ž0. s z, is given by H Ž t, K ŽŽ1rŽ n y m ...Ž e Ž ny m .t y 1., z ... ŽWe assume m / n ; the case m s n allows easy integration of ˙ y s g Ž y ... w x The proof of this formula follows as in 20 . Now
HŽ t, z. s
z1exp Ž m t . z 2 exp Ž Ž m q n . t . z 3 exp Ž Ž m q 2 n . t .
0
,
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EQUATIONS WITH FINITE SYMMETRY GROUPS
while K Ž t, z . is found from the solution of 2 y2 ˙y s Ž n y m . 3 y 3 0
2
0
z1 q Ž n y m . tz 2 q 12 Ž n y m . t 2 z 3
which is
z 2 q Ž n y m . tz 3 z3
00
by replacing t with the solution t Ž t . to ˙ t s 1rŽ z1 q Ž n y m .t z 2 q 12 Ž n y 2 2 m . t z 3 ., t Ž0. s 0. ŽThus t can be explicitly determined from the equation z1t q 12 Ž n y m . z 2t 2 q 16 Ž n y m . 2 z 3t 3 s t using Cardano’s formula.. Putting pieces together, we see that the solution of ˙ y s Ž1ry 1 . g Ž y . can be determined explicitly, and so can the solution of ˙ x s Ž1rw 1 . f Ž x .. ŽNote that F can be explicitly inverted, using Cardano’s formula once more.. In the positive orthant Žwhich is the interesting region in applications. this equation and ˙ x s f Ž x . have the same solution orbits, and thus the analysis of ˙ x s f Ž x . can be carried out by elementary means. The reduction from ˙x s f Ž x . to ˙y s g Ž y . is a quite important step in this example, because it yields an easy-to-recognize decomposition of Ž1ry 1 . g Ž y . into a sum of vector fields with a well-behaved commutator relation. Moreover, it can be checked that there is no corresponding additive decomposition for ˙ x s f Ž x. itself. EXAMPLE 2.6. The equation
˙x 1 s x 2 x 3 ˙x 2 s x 3 x 1
Ž ˙x s f Ž x . in R 3 .
˙x 3 s yx 1 x 2 comes from the Euler equation for the Žasymmetric. spinning top in the absence of external forces by appropriate scaling of the variables. This equation admits the complex linear symmetries
T1 x [
x2 x1 , x3
0
ix 3 x2 , T2 x [ yix 1
0
and
yx 1 T3 x [ yx 2 , x3
0
which generate a group of order 12. ŽIt can be shown that no other linear symmetries exist for ˙ x s f Ž x ... G has no nontrivial linear invariant, and Žup to scalar multiples. one quadratic invariant c 1Ž x . s x 12 q x 22 y x 32 . Now L f Ž c 1 . s 6 c 2 , with c 2 Ž x . [ x 1 x 2 x 3 , L f Ž c 2 . s c 3 , with c 3 Ž x . [ yx 12 x 22 q x 22 x 32 q x 32 x 12 , and
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KINYON AND WALCHER
L f Ž c 3 .Ž x . s y4 x 1 x 2 x 3 Ž x 12 q x 22 y x 32 . s y4c 1 c 2 . Thus
c1 C s c2 c3
0
is solution-preserving from ˙ x s f Ž x . into
˙y 1 s 6 y 2 ˙y 2 s y 3
in R 3 ,
˙y 3 s y4 y 1 y 2
{
which is equivalent to the third order equation y s y4 yy. ˙ Integrating yields ¨ y s y2 y 2 q g Žg g R., which is a second-order equation for a Weierstrass `-function. Thus the reduction procedure provides a natural route to the Žwell-known. fact that Euler’s equations can be solved with the help of elliptic functions. The invariants c 1 , c 2 , and c 3 do indeed generate I Ž G . Žthis follows most easily from the fact that G is conjugate to the tetrahedral group in GLŽ3, C. and using Jaric et al. w11x.. It may be worth noting, however, that in this example we did not start with a system of generators for I Ž G ., but only with an element of smallest degree. Thus it is not necessary to have a complete system of generators at the beginning. To close this section we note another consequence of Lemma 1.2 that is occasionally useful in determining invariant sets. ŽFor sake of simplicity we do not consider the most general situation.. PROPOSITION 2.7. Let G : GLŽ n, K. and h a character of G such that the I Ž G .-module Ih Ž G . has a finite set of generators r 1 , . . . , r s . If f is a polynomial and G is a symmetry group for ˙ x s f Ž x . then the set Z of common zeros of r 1 , . . . , r s is an in¨ ariant set for ˙ x s f Ž x .. Proof. According to Lemma 1.2, L f Ž r i . g Ih Ž G . for 1 F i F s. Therefore there exist m i j g I Ž G . such that L f Ž r i . s Ý m i j r j Ž1 F i F s .. The invariance of Z follows from w19x. EXAMPLE 2.8. For S3 : GLŽ3, R. consider the character h that attains the value y1 on the reflections. It is known that the discriminant r Ž x . s Ž x 1 y x 2 .Ž x 2 y x 3 .Ž x 3 y x 1 . generates the I Ž S3 .-module Ih Ž S3 .; cf. w17x. Therefore the planes x i y x j s 0 Ž i / j . are invariant sets for every S3-symmetric equation. For the equations from Example 2.5, one has additional semi-invariants x 1 , x 2 , x 3 ; and the method developed in Grobner ¨ and Knapp w9x yields a number of first integrals, e.g., s Ž x . s Ž x 2 y n x 3 .yŽ m q2 n . Ž x 2 x 3 .Ž m q n . xy 1 .
EQUATIONS WITH FINITE SYMMETRY GROUPS
189
3. REDUCTION IN THE CASE OF AN ORBITAL SYMMETRY GROUP Intuitively, a Žproper. orbital symmetry group should be less restrictive for a differential equation than a symmetry group. Indeed, reduction has a more complicated look here. Once more we formulate the theorem only for the case of a polynomial differential equation. THEOREM 3.1. Let G be linear group and x : G ª CU a character of order s q 1 ) 1. Suppose that I Ž G . has a finite system w 1 , . . . , wr of generators and further that e¨ ery I Ž G .-module Ix k Ž G . Ž1 F k F s . is finitely generated by c k, 1 , . . . , c k, m k. Let pk s Ý kjs1 m k , 1 F k F s. If T ( f (Ty1 s x ŽT . f for e¨ ery T g G, then the map
¡K
Ž with R s r q m1 q ??? qm s . F :~ x ¬ Ž w 1 Ž x . , . . . , wr Ž x . , c 1, 1 Ž x . , . . . , c 1, m 1Ž x . , . . . ,
¢c
n
ª KR
s, 1
Ž x . , . . . , cs, m sŽ x . .
t
is solution-preser¨ ing from the polynomial differential equation ˙ x s f Ž x . in K n R to a polynomial differential equation ˙ y s g Ž y . in K which is of the following specific type, rqp 1
˙yi s Ý mŽ0. i j Ž y 1 , . . . , yr . y j
Ž1 F i F r .
jsrq1 rqp 2
˙yi s
Ý
jsrqp 1q1
mŽ1. i j Ž y 1 , . . . , yr . y j
Ž r q 1 F i F r q p1 .
.. . rqp s
˙yi s
Ý
jsrqp sy1 q1
mŽi sy1. Ž y 1 , . . . , yr . y j j
˙yi s n i Ž y 1 , . . . , yr .
Ž r q psy2 q 1 F i F r q psy1 .
Ž r q psy1 q 1 F i F r q ps .
with all the mŽi kj . and n i polynomials. Proof. By Lemma 1.2 we have L f Ž w i . g Ix Ž G . Ž1 F i F r ., and L f Ž c k, j . g Ix kq 1 Ž G . Ž1 F j F m k . for all k, where I1Ž G . [ I Ž G .. Remark 3.2. Ža. The image of F is contained in the subvariety Y of K R defined by all the relations between the w i and c k, j ; and again Y is invariant for ˙ y s g Ž y ..
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Žb. As there seems to be no counterpart of Luna’s theorem w13x available in the literature for this situation, generalization to the analytic case has to be investigated individually for every group. Žc. Theorem 3.1 turns out to be most useful in the case that every module Ix k Ž G . Ž1 F k F s . is generated by one element. The underlying reason is that the image equation ˙ y s g Ž y . starts with a ‘‘block’’
˙y 1 s yrq1 m 1 Ž y 1 , . . . , yr . .. .
˙yr s yrq1 m r Ž y 1 , . . . , yr . and information about solution orbits of the differential equation
˙y 1 s m 1 Ž y 1 , . . . , yr . .. .
˙yr s m r Ž y 1 , . . . , yr . in K r can be carried over to ˙ y s g Ž y .. ŽFor instance, a first integral of the latter will also be a first integral of ˙ y s g Ž y ... In the case of finite groups it is known that reflection groups have the desired property Žcf. Springer w17, Theorem 4.3.4x.. The following examples will therefore involve orbital symmetry groups that are generated by reflections. EXAMPLE 3.3. The equation
˙x 1 s x 1 Ž x 2 y x 3 . ˙x 2 s x 2 Ž x 3 y x 1 .
Ž ˙x s f Ž x . in R 3 .
˙x 3 s x 3 Ž x 1 y x 2 . is of some significance for a mathematical model in biology; cf. Hofbauer and Sigmund w10, Sect. 16.5x. ŽIt may also be viewed as describing the population dynamics of three species that cyclically prey upon each other.. For this equation we have S3 as a proper orbital symmetry group, with x the only nontrivial character of S3 , and Ix Ž S3 . generated by the discriminant: cf. Example 2.8. The image equation ˙ y s g Ž y . turns out to be
˙y 1 s 0 ˙y 2 s y4 ˙y 3 s 0 ˙y4 s y6 y 22 q y 12 y 2 q 9 y 1 y 3 .
EQUATIONS WITH FINITE SYMMETRY GROUPS
191
Here the first integrals y 1 and y 3 are immediate, and pulling these back yields the first integrals r 1Ž x . s x 1 q x 2 q x 3 and r 2 Ž x . s x 1 x 2 x 3 for ˙x s f Ž x .. These are already known from w10x Žand r 1 , at least, is obvious by inspection.. Returning to the image equation, for y 1 s g 1 , y 3 s g 3 constant we obtain the two-dimensional Hamiltonian system
˙y 2 s y4 ˙y4 s y6 y 22 q y 12 y 2 q 9 y 1 y 3 , with first integral 12 y42 q 2 y 23 y 12 g 12 y 2 y 9g 1 g 3 y 2 . Thus we have Ž ˙ y2 .2 s 3 2 2 Ž y4 y 2 q g 1 y 2 q 18g 1 g 3 y 2 q d with d determined by initial conditions., and this is essentially the differential equation of a Weierstrass `-function Žunless the right hand side has multiple roots, in which case the solution is elementary.. Thus in addition to the known first integrals, we add the new piece of information that ˙ x s f Ž x . can be solved by employing elementary and elliptic functions. EXAMPLE 3.4. Ža. Let G be the group Žof order 4. generated by T1 s diagŽ1, y1. and T2 s diagŽy1, 1., and let x be the character of G with value y1 on T1 and T2 . The algebra I Ž G . is generated by w 1Ž x . s x 12 and w 2 Ž x . s x 22 , and the module Ix Ž G . is generated by c Ž x . s x 1 x 2 . Consider the differential equation
˙x 1 s x 2 p Ž x 12 , x 22 . ˙x 2 s x 1 q Ž x 12 , x 22 . Ž˙ x s f Ž x . in R 2 . where p, q : R 2 ª R are orbital symmetry group for this equation, F s Ž w 1 , w 2 , c . t is solution-preserving from
polynomials. Then G is an with character x . The map ˙x s f Ž x . into
˙y 1 s 2 y 3 p Ž y 1 , y 2 . ˙y 2 s 2 y 3 q Ž y 1 , y 2 . ˙y 3 s y 2 p Ž y 1 , y 2 . q y 1 q Ž y 1 , y 2 . . ŽHere the variety Y is given by y 1 y 2 y y 32 s 0.. We are primarily interested in the first two lines. In particular, if p and q are such that determining a first integral s of the equation
˙y 1 s p Ž y 1 , y 2 . ˙y 2 s q Ž y 1 , y 2 . is elementary, then r Ž x . [ s Ž x 12 , x 22 . will be an elementary first integral of ˙ x s f Ž x .. Specific examples include the cases where p and q are given
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by a Hamiltonian function Ž pŽ x 1 , x 2 . s 2 hŽ x 1 , x 2 ., q Ž x 1 , x 2 . s y 1 hŽ x 1 , x 2 .., or where p and q are affine Ž pŽ x 1 , x 2 . s m q a x 1 q b x 2 , q Ž x 1 , x 2 . s n q g x 1 q d x 2 .; the latter case includes every polynomial differential equation of degree F 3 satisfying T ( f (Ty1 s x ŽT . f for all T g G. One may view the reduction from a slightly different perspective. For instance, in part Ža. one has d dt d dt
Ž x 12 . s 2 x 1 x 2 p Ž x 12 , x 22 . Ž x 22 . s 2 x 1 x 2 q Ž x 12 , x 22 . ,
and thus the mapping x ¬ Ž w 1 , w 2 . is locally orbit-preserving from ˙ xs f Ž x . to
˙y 1 s p Ž y 1 , y 2 . ˙y 2 s q Ž y 1 , y 2 . So it can be said that the reduction consists of a local coordinate transformation, together with a change in time scale. As noted in the Introduction, an experienced observer may, of course, notice this directly. The point we are making here, however, is that the symmetry group gives the underlying reason why this procedure works, and Theorem 3.1 provides a systematic approach. Žb. Let z s expŽ2p ir3., T s diagŽ z , 1., and G be the group generated by T. Denote by x the character that has value z on T. The algebra I Ž G . is generated by w 1Ž x . s x 13 and w 2 Ž x . s x 2 , while the modules Ix Ž G . and Ix 2 Ž G . are generated by c 1Ž x . s x 12 and c 2 Ž x . s x 1 , respectively. Consider the differential equation
˙x 1 s p Ž x 13 , x 2 . ˙x 2 s x 12 q Ž x 13 , x 2 . Ž˙ x s f Ž x . in R 2 . where p, q : R 2 ª R are polynomials. Then G is an orbital symmetry group for this equation, with character x . The map F s Ž w 1 , w 2 , c 1 , c 2 . t is solution-preserving from ˙ x s f Ž x . into
˙y 1 s 3 y 3 p Ž y 1 , y 2 . ˙y 2 s y 3 q Ž y 1 , y 2 . ˙y 3 s 2 y4 p Ž y 1 , y 2 . ˙y4 s p Ž y 1 , y 2 . .
EQUATIONS WITH FINITE SYMMETRY GROUPS
193
ŽThe variety Y is given by y 1 y y43 s 0, y 3 y y42 s 0.. Again, the first two lines are of principal interest. If p and q are such that determining a first integral s of the equation
˙y 1 s 3 p Ž y 1 , y 2 . ˙y 2 s q Ž y 1 , y 2 . is elementary, then r Ž x . [ s Ž x 13, x 2 . will be an elementary first integral of ˙ x s f Ž x .. Examples include the Hamiltonian case, and the case where pŽ x 1 , x 2 . s a x 1 q uŽ x 2 . for some polynomial u and q Ž x 1 , x 2 . s b q g x 2 , for this reduces to two one-dimensional linear equations. ŽThe latter case includes every polynomial differential equation of degree F 3 satisfying T ( f (Ty1 s x ŽT . f for all T g G.. This example provides another indication that complex orbital symmetry groups carry useful information about real equations. EXAMPLE 3.5. The existence of orbital symmetries forces certain qualitative properties on the differential equation, which in some instances can be recovered from the reduction procedure. We illustrate this assertion by considering the following class of ‘‘reversible’’ systems in K n, n G 2, O
˙x s f Ž x . [ O
O 0 b
a ? x q higher order terms, 0
with f satisfying T ( f (Ty1 s yf for T [ diagŽ1, . . . , 1, y1.. ŽNote that the linear part B s Df Ž0. satisfies T ( B(Ty1 s yB. Moreover, up to a suitable change of coordinates B is the most general linear map admissible.. In the following we will assume a / 0. The invariant algebra of the group G s T, id4 is generated by w 1Ž x . s x 1 , . . . , wny1Ž x . s x ny1 and wnŽ x . s x n2 , and the module Ix Ž G . is generated by c Ž x . s x n . ŽHere x denotes the character mapping T to y1.. By Theorem 3.1 the image equation ˙ y s g Ž y . in K nq 1 is of the type
˙y 1 s ynq1 m 1 Ž y 1 , . . . , yn . .. .
˙yn s ynq1 m n Ž y 1 , . . . , yn . ˙ynq 1 s n Ž y 1 , . . . , yn . . ŽAs follows from Remark 3.2Žc., this is also true for analytic f.. Since L f Ž wny1Ž x .. s a x n q h.o.t.s c Ž x .Ž a q h.o.t.. we have m ny1Ž0, . . . , 0. s a
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KINYON AND WALCHER
/ 0. Therefore 0 is not a stationary point of
˙y 1 s m 1 Ž y 1 , . . . , yn . .. .
Ž ˙y s h Ž y . . ,
˙yn s m n Ž y 1 , . . . , yn . and therefore ˙ y s hŽ y . has n y 1 independent first integrals s 1 , . . . , sny1 near 0. These Žconsidered as functions of y 1 , . . . , ynq1 . are also first integrals of ˙ y s g Ž y ., and pulling back yields the first integrals r i Ž x . s si Ž x 1 , . . . , x ny1 , x n2 . for ˙ x s f Ž x . Žwhich, again, are easily checked to be independent.. Therefore ˙ x s f Ž x . has n y 1 independent first integrals in a neighborhood of the stationary point 0, a highly unusual property. Let us look at two special cases. ŽBoth are reversible systems in R 2 .. 0
˙x s y1
ž
1 ? x q h.o.t.s f Ž x . 0
/
in U : R 2 .
Ž a.
The image equation here is
˙y 1 s y 3 Ž 1 q h.o.t.. ˙y 2 s y 3 Ž y2 y 1 q h.o.t..
Ž ˙y s g Ž y . . .
˙y 3 s ??? . We obtain a first integral s Ž y . s y 2 q y 12 q h.o.t. for ˙ y s g Ž y . and thus a first integral r Ž x . s x 12 q x 22 q h.o.t. for ˙ x s f Ž x .. Since r has a local minimum at 0 g R 2 , the Žlocal. level sets of r near 0 are closed curves by the classification of critical points in R 2 . Therefore the stationary point 0 is a center for ˙ x s f Ž x .. Of course it is well known that a reversible system with linear part as stipulated above has a center at 0 Žcf. Sansone and Conti w15x for another proof., but here we have an elementary ‘‘algebraic’’ proof of this fact. 0
˙x s 0
ž
b3 x1 x 2 1 ?xq q h.o.t. 0 b 1 x 12 q b 2 x 22
/
ž
/
in U : R 2 .
Ž b.
ŽThe quadratic term here is the most general compatible with reversibility. We will assume b 1 / 0.. The image equation ˙ y s g Ž y . is
˙y 1 s y 3 Ž 1 q b 3 y 1 q h.o.t.. ˙y 2 s y 3 Ž 2 b 2 y 2 q 2 b 1 y 12 q h.o.t. . ˙y 3 s ??? .
Ž ˙y s g Ž y . . .
EQUATIONS WITH FINITE SYMMETRY GROUPS
195
Here we get a first integral s Ž y . s y 2 y 2 b 2 y 1 y 2 y Ž2r3. b 1 y 13 q Žirrelevant terms. for ˙ y s g Ž y ., hence a first integral r Ž x . s x 22 y 2 b 2 x 1 x 22 y 3 Ž2r3. b 1 x 1 q Žirrelevant terms. for ˙ x s f Ž x .. Ž‘‘Irrelevance’’ here is with respect to the level set r Ž x . s 0; see Bruno w3, Chap. 1, Sect. 2x on the Newton polygon behind this argument.. For r Ž x . s 0 we have 0 s x 22 Ž 1 y 2 b 2 x 1 . y 23 b 1 x 13 q ??? , 0 s x 22 y 23 b 1 x 13 Ž 1 y 2 b 2 x 1 .
y1
q ???
s x 22 y 23 b 1 x 13 q ??? , and the stationary point turns out to be a cusp. Again it is worth noting that an a priori determination of the topological type of the stationary point 0 is not an easy problem. The algebraic reduction formalism, however, offers easy access to its solution. The specific results in the two-dimensional cases can also be obtained using normal forms, although the arguments given above use less machinery. ŽFor instance, in Ža. one has to use the nontrivial theorem that reversibility is preserved by a suitable transformation into normal form, and that Bruno’s ‘‘Condition A’’ Žsee w3x. guarantees convergence of such a transformation. In Žb., one must argue similarly using the Belitskii normal form associated with the given nilpotent linear part.. In higher dimensions it seems that the normal form machinery is less suitable than the approach chosen here.
REFERENCES 1. M. Abud and G. Sartori, The geometry of spontaneous symmetry breaking, Ann. Physics 150 Ž1983., 307]372. 2. M. Abud and G. Sartori, The geometry of orbit-space and natural minima of Higgs potentials, Phys. Lett. B 104 Ž1981., 147]152. 3. A. D. Bruno, ‘‘Local Methods in Nonlinear differential Equations,’’ Springer-Verlag, New YorkrBerlin, 1989. 4. P. Chossat, Forced reflectional symmetry breaking of an O Ž2.-symmetric homoclinic cycle, Nonlinearity 6 Ž1993., 723]731. 5. D. Cox, J. Little, and D. O’Shea, ‘‘Ideals, Varieties, and Algorithms,’’ Undergrad. Texts Math., Springer-Verlag, New YorkrBerlin, 1992. 6. G. Gaeta, ‘‘Nonlinear Symmetries and Nonlinear Equations,’’ Kluwer Academic, Dordrecht, 1994. 7. M. Golubitsky and D. G. Schaeffer, ‘‘Singularities and Groups in Bifurcation Theory,’’ Vol. 1, Applied Math. Sci. Ser., Vol. 51, Springer-Verlag, New YorkrBerlin, 1985. 8. M. Golubitsky, I. Stewart, and D. G. Schaelfer, ‘‘Singularities and Groups in Bifurcation Theory,’’ Vol. 2, Applied Math. Sci. Ser., Vol. 69, Springer-Verlag, New YorkrBerlin, 1988.
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9. W. Grobner and H. Knapp, ‘‘Contributions to the Method of Lie Series,’’ Bibliograph. ¨ Inst., Mannheim, 1967. 10. J. Hofbauer and K. Sigmund, ‘‘The Theory of Evolution and Dynamical Systems,’’ Cambridge Univ. Press, Cambridge, United Kingdom, 1988. 11. M. Jaric, L. Michel, and R. Sharp, Zeros of covariant vector fields for the point groups: Invariant formulation, J. Physique 45 Ž1984., 1]27. 12. E. Kasner, Solution of the Einstein equations involving functions of only one variable, Trans. Amer. Math. Soc. 27 Ž1925., 155]162. 13. D. Luna, Fonctions differentiables invariantes sons l’operation d’un group reductif, Ann. Inst. Fourier 26, No. 1 Ž1976., 33]49. 14. P. J. Olver, ‘‘Applications of Lie Groups to Differential Equations,’’ Graduate Texts in Math., Vol. 107, Springer-Verlag, New YorkrBerlin, 1986. 15. G. Sansone and R. Conti, ‘‘Nonlinear Differential Equations,’’ Pergamon, Elmsford, New York, 1964. 16. G. W. Schwarz, The topology of algebraic quotients, in ‘‘Topological Methods in Algebraic Transformation Groups’’ ŽH. Kraft et al., Eds.., pp. 135]151, Birkhauser, Basel, ¨ 1989. 17. T. A. Springer, ‘‘Invariant Theory,’’ Lecture Notes in Math., Vol. 585, Springer-Verlag, New YorkrBerlin, 1977. 18. B. Sturmfels, ‘‘Algorithms in Invariant Theory,’’ Springer-Verlag, New YorkrBerlin, 1993. 19. S. Walcher, Symmetries of ordinary differential equations, No¨ a J. Alg. Geom. 2, No. 3 Ž1993., 245]275. 20. S. Walcher, On sums of vector fields, Results in Math. 31, No. 1r2 Ž1997., 161]169.