Spontaneous linearization and periodic solutions in Hopf and symmetric bifurcations

Volume 116, number 7

PHYSICS LETTERS A

30 June 1986

S P O N T A N E O U S L I N E A R I Z A T I O N AND P E R I O D I C S O L U T I O N S IN H O P F AND S Y M M E T R I C B I F U R C A T I O N S Giampaolo C I C O G N A Dipartimento di Fisica, Universit~ di Pisa, 56100 Pisa, Italy and

Giuseppe G A E T A 1 Department of Physics, New York University, New York, NY 10003, USA Received 7 February 1986; revised manuscript received 21 April 1986; accepted for publication 29 April 1986

We show that for a rather large class of symmetries,symmetric polynomial evolution equations - and therefore symmetric bifurcation equations - exhibit spontaneous finearization; i.e. although the equations are nonlinear, their asymptotic solutions are governed by linear equations. The same mechanismleads to periodicity of such solutions. This is exemplified in the cases of SO(2) and SU(2) symmetries,corresponding to standard Hopf and quaternionic bifurcations. The purpose of the present letter is to stress the role played in bifurcations of a fixed point to a periodic orbit, described by the mechanism called "spontaneous linearization", and its relation with the symmetry and the geometry of the problem. W e shall look at the bifurcation theory from the point of view of symmetry considerations, so we start by briefly recalling the relevant features of symmetric bifurcation theory [1,2], thereby always assuming sufficient regularity (e.g. analyticity) of all the operators entering into the discussion. Let the original problem be

2= c(x, u)=L(X)u+N(X, u),

(1)

where u G H, G: R × H ---, H, h is a real parameter, L()~) is the linear part (depending on )~) of the operator G, and N its nonlinear part, which we assume to be at least quadratic; H is some space, whose detailed nature is not important (e.g., a Hilbert space). We assume also that u = 0 is a solution for any ~,: G ( h , O) = O, V~.

If h 0 is a bifurcation point, we can consider the bifurcation equation (obtained e.g. by the L y a p u n o v - - S c h m i d t reduction [1]) :~ = F(•,

x).

(2)

In this letter we assume that the space (t in which the bifurcation takes place, i.e. the space corresponding to the critical eigenvalues of oi(h) of L ( ~ ) (those satisfying Re o i ( h o ) = 0 and Re o i ' ( h 0 ) > 0), is V = R u, so that in eq. (2) x R u, F: R x R " ~ R u.

Now, one knows from bifurcation theory [1] that F can be written as F ( X , x ) = Y'.Bm(X, x ) ,

(3)

m

where each Bm is m-linear in x. If the original problem (1) has a symmetry described by the representation ]" of the group G, namely G(X, T s u ) = TsG(X, u),

Vg~G,

(4)

the bifurcation equation shares this, namely 1 Present address: Dipartimento di Fisica, Universith di Roma, 00185 Rome, Italy

F ( X , Tgx) = T g F ( X , x ) ,

0375-9601/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Vg ~ G,

(5) 303

V o l u m e 116, n u m b e r 7

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where Tz: R u ~ R u and Tg = T~ I ,~For linear representations now follows that each Bm in eq. (3) is covariant, B,,()~, Tgx)= T~Bm(X, x), and, given T, the most general form of F can be determined by building up m-linear covariants; this also means that here we just have to consider covariant polynomial equations in R N, and conversely our result will be valid for any such equation. When one considers real irreducible linear representations, these can be classified according to Schur's lemma [3] as real, complex and quaternionic type, names corresponding to the nature of the center C(T) of the representation T under consideration (i.e. the set of linear operators commuting with the whole representation). Real-type representations give raise to stationary bifurcation, complex-type to Hopf bifurcation and quaternionic-type to quaternionic bifurcation [4]. As remarked elsewhere [4], stationary and Hopf bifurcations are special cases of quaternionic bifurcation, and any bifurcation is a "combination" of these three basic types. We recall now a known result: let T: R u ~ R u be linear, isometric, and transitive on S u-1 (the unit sphere in RN), with moreover D ( T ) = d , where d = dim{y ~ a N [G~ ___G~}, G~ being the isotropy group of x under T, G~ = { g I Tgx = x ), and D(T) = dim C(T), which is 1, 2, or 4 if T is of real, complex, or quaternionic-type respectively [3]. This implies ~V2, = 0 ,

#V2,+, = D ( T ) ,

~$2,=1,

-~S2,+l=0,

under these hypotheses, on a sphere of given (arbi-

trary) radius, any couariant is equal to a linear one. This leads to spontaneous linearization: if the dynamics are such that the attracting manifold of the motion is a sphere (or a union of spheres), the asymptotic motion and the motion of stable solutions will be described by linear evolution equations, even if the general motion is described by nonlinear evolution equations. It also brings immediately theorems on the correspondence between full and truncated bifurcation equations (and their solutions). Indeed, since one is only interested in the long term (or stable) solutions, both truncated and full equation are of the form k =ED~)a,~i(x), with a i ~ R, ~/i ~ V1 and x restricted to satisfy ix] = pl/2: therefore, considering the full equation brings merely a rescaling in the a,'s and in the P of the truncated equation. Eq. (7) is also at the basis of variational properties of bifurcation equations [5]. Due to symmetry assumptions it is clear that the most general form of the attracting manifold can be a union of spheres (including possibly the cases of radius 0 and oo), which differs from the case of a single sphere only in a trivial way. Let's now look in greater detail and in a less qualitative way at the mechanism sketched above. First note that, when eqs. (6) hold, eq. (3) becomes F(X, x ) = E B b ( X , x) = E B 2 s + , ( X , x ) m

s

D(T)

(6)

where # Vm (:~ S,,) is the dimension of the space Vm (Sin) of m-linear covariant (invariant) functions, q~/q~(Tgx) = Tsq~(x ) (o/o(Tgx) = o(x)). We also recall that the fundamental real representations of classical orthogonal groups (SO(N), SU(N), O(N), U ( N ) ) satisfy the above hypotheses [5]. Now, the point is that, when eqs. (6) are true,

SN-', 3,1 v , / ¢ ( x ) =,7(x),

(7) and due to covariance and transitivity of T, the equality extends to the whole sphere SN-1; that is, 304

30 J u n e 1986

= E ( x , x) s E < ( x ) n i ( x ) , s

(8)

k=l

where a, ~ R, and ~i ~ V1, i.e. ~i(x) = K,x, with {K,} a basis of C(T). One of the K,'s can always be taken to be the identity, of course, so that F can also be decomposed as F(X, x ) = P(X, x ) + Q(X, x ) P(X, x)=(F(X,

x), 2),

Q(X, x ) = F(;k, x ) -

with

P(X, x).

5 c = x / I x I, (9)

Now, the motion in the radial direction is controlled by P alone, which in turn depends on x only through Ix l, d(x, x)/dt=2P(X,

x ) - O ( lxl).

(10)

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Therefore, the motion is driven toward spheres specified by 0( I x [) = 0, 8'( [ x 1) < 0, say (x, x) = p with 0 < p < o o (this can be guaranteed by classical hypotheses), on which the asymptotic motion takes place. This, of course, unless the initial datum x(0) satisfies 8(Ix I ) = 0, regardless of 0 ' ( I x ( 0 ) I): in this case the motion takes place entirely on the sphere of radius Ix(0) l and the following reasoning applies as well, being based merely on the condition that the asymptotic motion takes place on a sphere satisfying 0( Ix I) --- 0. Then, the equation of motion is

equations obtained via Lyapunov-Schmidt reduction: the SO(2) symmetry has then to be interpreted as a "temporal" symmetry, arising from the periodicity requirement [7,4]). Now, the SO(2) covariants are of the form

:t = F(X, x ) [(x,x)=o = O ( h , x ) [
P(X, x)= E(x,

(11)

B2m+l(~., x) = (x, X)m[am(~k)I-[-/3m(~k)J]x, where a, /3 ~ R and

,=(1 o),

-1);

P and Q are given simply by

x)",~.(X)x,

n

but this is, see eq. (8),

O(X, x ) = E ( x , x)"fl~(X)Jx. X = EosOlis(~k ) g i x ,

n

(12)

s,i

where the orthogonal C(T). Now, p, asymptotic

sum over i is extended to operators to the identity in a basis spanning X, a are constants, so we can write the equation of motion as

k = to( X )K( h )x,

(13)

with to ~ R, K ~ C ( T ) and (x, Kx) = O, Vx. This is precisely the spontaneous linearization. Moreover, due to general properties [3], K 2 is proportional to - 1 , so take, e.g., K 2 = - 1 ; now, if one considers the solutions of eq. (13),

x ( t ) = e'~Xtx(O),

(14)

and the series expansion for the exponential, this results in

x( t ) = cos[to(X)t] Ix(O) + sin[to(~,)t] K( X )x(O)

(15)

Substituting (x, x) = 0 solution of P(X, x) = 0 in .~ = F(~, x) one gets ~ = toJx, which describes the motion along a circle with angular velocity to. In the standard quaternionic case [4], i.e. when the bifurcation equation is covariant with respect to the fundamental real representation of SU(2), the covariants are of the form

B~,.+,(X, x) = (x, x)" g . ( x ) / +

i

x,

i=1

with or, /3 ~ R, I the four-dimensional identity matrix, and the three matrices K,'s (corresponding to the possible symplectic forms "t(u, v ) = (u, K,v) on R 4) given by

KI=

0 -1 0 0

Therefore, this same mechanism generates periodic solutions. In particular, this is the mechanism at work in standard Hopf and quaternionic bifurcations: in fact, as remarked above, this mechanism applies, among the others, to the fundamental real representations of SO(2) and SU(2). In the standard (i.e. two-dimensional) Hopf case, one can write the equations in normal form, obtaining an SO(2) symmetric system [6] (we note of course that the same holds - in this case - for

E

K2=

K3=

1

0

0 0 0

0 0 -1

-1

0 0 -1 0

0 0 -1 0

0 0 0 1

0 0 0

0 1 0 0 1 0 0 0

0 0 1

'

0 1

0 0 ' 0 0 -1 0 0

In this case, P()~, x ) = E . ( x , x)"a.(X)x, and 305

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s u b s t i t u t i n g the solution ( x , x ) = p of P ( ) , , x ) = 0 one gets for ~ = F(X, x ) the e q u a t i o n ~ = 09(X)A(k,)x, with

A()~)A 2 = -I

1

3

n i E EPfl~()~)K,,

09(X) ,=t

and

.

092

3 E i=1

2. P ~nt(

T h e s o l u t i o n is x(t) = e ' ° ( a ) A ( X ) t x ( 0 ) = (cos[09(?Qt]I+sin[09(h)t]A(h))x(O), which describes the m o t i o n on a circle whose p l a n e can vary as a function of ?~. T o c o n c l u d e this note, we w o u l d like to a d d s o m e remarks, T h e first is j u s t that, as was said before, these s a m e c o n s i d e r a t i o n s a p p l y to a n y c o v a r i a n t p o l y n o m i a l e q u a t i o n a n d can therefore b e of use also out of the b i f u r c a t i o n theoretic c o n t e x t we are interested in. W e w o u l d also stress again that the s a m e m e c h a n i s m comes into p l a y for a rather large class of symmetries, a n d therefore n o t o n l y for the s t a n d a r d H o p f a n d q u a t e r nionic bifurcations; in p a r t i c u l a r [5], the f u n d a m e n t a l real r e p r e s e n t a t i o n s of S U ( N ) , N >/3, give o t h e r e x a m p l e s of this m e c h a n i s m in c o m p l e x type ( n o n - s t a n d a r d H o p f ) bifurcations, with "v" = R 2N, D(T) = d = 2; a n d p e r i o d i c b i f u r c a t i n g solutions. T h e last r e m a r k we wish to m a k e is that in o r d e r to have s p o n t a n e o u s linearization the crucial

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30 June 1986

p r o p e r t y is D(T)=d, V x ~ S N-~, while the transitivity on S N-a (which is a n y w a y g r a n t e d for the f u n d a m e n t a l real r e p r e s e n t a t i o n s of classical o r t h o g o n a l groups) plays a s e c o n d a r y role; that is, it g u a r a n t e e s the p a r t i c u l a r l y simple structure of eq. (6). The f u n d a m e n t a l p r o p e r t y :¢V,, < ~ V 1 is a n y w a y g u a r a n t e e d even b y D(T)= d alone, a n d so is the equivalence of linear a n d m-linear cov a r i a n t s on a n y given orbit, a n d therefore the m e c h a n i s m of s p o n t a n e o u s linearization.

References [1] D.H. Sattinger, Group theoretical methods in bifurcation theory, Lecture Notes in Mathematics, no. 762 (Springer, Berlin, 1979). [2] D.H. Sattinger, Branching in the presence of symmetry (SIAM, Philadelphia, 1983). [3] A.A. Kirillov, Elrments de la throrie des reprrsentations (MIR, Moscow, 1974). [4] G. Cicogna and G. Gaeta, Quaternionic bifurcation, preprint Dipartimento di Fisica Univ. di Pisa, IFUP TH 85/43 (1985). [5] G. Gaeta and P. Rossi, Gradient property of standard representations for classical orthogonal groups, preprint Cornell Univ. (1985), to be published in Nuovo Cimento. [6] J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields (Springer, Berlin, 1983). [7] M. Golubitsky and I. Stewart, Arch. Ration. Mech. Anal. 87 (1985) 107.