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Degenerate Hopf bifurcation in the presence of symmetry
Physica 27D (1987) 338-356 North-Holland, Amsterdam
D E G E N E R A T E H O P F B I F U R C A T I O N IN T H E P R E S E N C E OF S Y M M E T R Y * K.-H. H O C K and D. H A C K E N B R A C H T Institut fftr Festkbrperphysik, Technische Hochschule Darmstadt, Hochschulstr. 2, D-6100 Darmstadt, Fed. Rep. Germany P. J O R D A N and H. T H O M A S lnstitut fftr Physik, Universitilt Basel, Klingelbergstr. 82, CH-4056 Basel, Switzerland Received 15 September 1986 Revised manuscript received 20 February 1987
We investigate the bifurcation of time-periodic states from a stationary state destabilized by the undamping of a set of modes associated with a degenerate pair of complex-conjugate frequencies. This problem is of particular interest for bifurcations in driven systems with symmetry whose order-parameter dimension n is even and n > 4. For this case of a degenerate Hopf bifurcation a star of symmetry-equivalentlimit cyclesbifurcates in analogy to the star of symmetry-related domains arising at a symmetry-breakingphase transition in equilibrium systems. We illustrate this fact by analyzinga concrete example with n = 4. Within the framework of an amplitude expansion, we explicitly construct the time-periodic states and discuss their stability. In particular, it is shown that fairly general conclusions for the bifurcation behaviour can be drawn on the sole basis of the knowledgeof the order-parameter symmetry.
1. Introduction In a recent paper [1] we have investigated the role of symmetry in autonomous, nonlinear dynamical systems in which soft-mode or hard-mode instabilities give rise to bifurcations of new states from a stationary reference state. It has been demonstrated that the application of group-theoretical methods to bifurcation problems provides important pieces of information on phase transitions in driven systems. For example, instabilities associated with the undamping of normal modes of a given stationary state may be classified by the type of representatio n associated with the order parameter of the system. As a consequence, symmetry criteria can be formulated for phase transitions to new stationary states or to permanently time-dependent structures. In particular, symmetry considerations lead to distinguish two types of hard-mode stabilities (coupling-induced or symmetry-induced hard-mode instabilities) which we briefly recall in section 2. Each of these instabilities gives rise to a degenerate Hopf bifurcation for even values of the order-parameter dimension n with n >__4. In that case, the well-known Hopf theorem [2] cannot be applied to assure the existence of stable limit cycles bifurcating from a stationary reference state. Very recently, Golubitsky and Stewart [3] succeeded in generalizing the Hopf theorem to the case of the coupling-induced, degenerate Hopf bifurcation. For the case of the symmetry-induced, degenerate H o p f bifurcation, on the other hand, the knowledge on the bifurcation behavior is still the subject of current research.
*Work supported by the Sonderforschungsbereich65 Frankfurt-Darmstadt and by the Swiss National Science Foundation. 0167-2789/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
K.-H. HSek et aL/ Degenerate Hopf bifurcation
339
In this paper, we investigate a representative example for this type of bifurcation based on a four-component order parameter. Unfortunately, such a case does not occur in systems described by spatially uniform tensorial quantities in three-dimensional space. Four-dimensional physically irreducible representations may occur in the case of wave instabilities of translationally invariant systems (order parameter components ~ exp (ikx), ~ exp ( - i k x ) [4], or as space-group representations of crystals or other spatially periodic structures. In the present paper, we consider specifically an order-parameter representation with matrix group Q A C2 (Q = quaternion group), which is realized by several space-group representations (see section 6). In section 2, we define the class of systems to be studied and construct the equation of motion for the order parameter on the basis of symmetry considerations. To investigate the bifurcation of time-periodic states, we derive the bifurcation equations by power series expansion and solve them explicitly. The results of the calculations are discussed in section 4 from the symmetry point of view. Section 5 is devoted to the stability analysis of the bifurcating solutions and section 6 concludes with a brief discussion of our results. 2. Construction of the equation of motion
We investigate an autonomous, non-linear dynamical system in which a hard-mode instability gives rise to bifurcations of time-periodic states from a given stationary state with symmetry group G. Within a Landan-type analysis, the bifurcation behaviour of the system is described in terms of an n-component order parameter (OP) ¢ which represents deviations from the stationary reference state q, = 0. The dynamics of the OP ~ is governed by an equation of motion of the form q~= B(O; #).
(1)
The real velocity field B(¢; #) is non-linear, analytic function on the OP-space ~2 satisfying B(0; #) = 0 for all values of the real parameter #, which can be externally controlled. Taylor expansion around ¢ = 0 yields 6 = L(/~) "q~ Y'~ o(k)(o; #).
(2)
k>l
Here, L(#)--VB]¢= o is the Jacobian whose eigenvalues X(#) determine the type of bifurcation. The higher-order terms O(k)(rk; ix), on the other hand, determine the position of the bifurcating solutions in OP-space $2, the variation of their amplitude with the control parameter/~, and their stability. Symmetry requires B(qD; #) to be equivariant with respect to the (real) representation F = ( F ( g ) J g ~ G) of the symmetry group G acting in OP-space/2, i.e. r(g)
•
= n(r(g)
• q,;
Vgea.
(3)
In particular, the linear operator I.(#) in (2) satisfies [L(/z), F ( g ) ] = 0
Vg~G.
(4)
From representation theory it then follows that the type of F uniquely determines the structure of L(#)
[51.
K.-H. Hgck et al./ Degenerate Hopf bifurcation
340
We are interested in the case that the stationary state ~ = 0 looses stability at the critical point/~c = 0 by' virtue of a degenerate pair of complex-conjugate eigenvalues of I.(/z) crossing the imaginary axis with nonvanishing imaginary part. As discussed in ref. 1, such a situation may occur in OP-dimensions n - 4, 6, 8 . . . . for three different types of OP-representation lr: Either F is irreducible over R but decomposes over C into two complex-conjugated irreducible representations 3', 3' *, i.e. F = V • 3' *, with (a) 3' equivalent to 3'* (type 2) or (b) F inequivalent to 3' * (type 3), o r / ' is reducible and decomposes into two equivalent representations (c) /" = 27 (type 1.2) with 3' irreducible over R and C, For cases (a) and (b), their instability is called a symmetry-induced hard-mode instability, while (c) is called a coupling-induced hard-mode instability. The latter has been analyzed recently in detail by Golubitsky and Stewart [3] and thus will not be discussed further in this paper. For types 2 and 3, the linear operation L(/~) has the following structure [1]: Type 2 L(/~) = a~°)(/x)l + a[1)(/~)l + a[2)(F)m + at3)(/~)n.
(5a)
Type 3 (5b)
L ( # ) = a[°)(#)l + a[1)(#)l, with at(°) real, and the matrices 1, I, m,
n
generate the quaternion algebra,
12= m 2 = n 2 = - 1 , I. m = - m . I = n
(6a) and cyclic.
(6b)
As pointed out in ref. 1, the structure of the matrices I.(/Q for type 2 and type 3 are closely related, i.e. (5a) can always be brought into a form identical to (5b) by a suitable choice of coordinates. As a representative example of a symmetry-induced hard-mode instability, we thus construct a specific model in OP-dimension n = 4 by considering an OP-representation lr of type 3 of the symmetry group G = Q/x C2 which is the semi-direct product of the quaternion group Q and the cyclic group of order 2. It is worth pointing out that this model comprises a whole class of systems with different symmetry groups G' which are determined by theorem 1 of ref. 1. The group G = Q/x C2 consists of 16 elements
G = ( +_e, +i, +j, + k , +l, +il, +jl, + k l }
(7a)
with multiplication rules defined by 1) i 2 = j 2 = k 2 = I 2 = _ e ;
(7b)
2) il= li,
k l = lk;
(7c)
and cyclic.
(7d)
j l = lj,
3) ij = - j i = k
It is convenient to choose a basis { e i } in OP-space ~2 such that the (real) OP-representation F = { F(g)lg
341
K.-H. HOck et al./Degenerate Hopf bifurcation
G } is generated by the three 4 × 4 matrices:
0
-1', '
1 J x
1
0',I
. . . . . . .
-4- . . . . . . . . I
0
i I-1
0
0
/°___1i.........
0
/
0
r
0
/
°/
',a
"=
-1/
1 0
°i
Ol -ll
0
o/ -1
/
.
(8)
/ /
Since F is of type 3, there exists exactly one nontrivial matrix commuting with all matrices of F [5]. Therefore the linear operator k(/~) has the form (5b) with I given in (8). Next we determine the form of the leading higher-order term O (k) in the Taylor expansion (2). The total number N ( k ) of basic equivariants 0 (k'j), j = 1. . . . . N(k), of given order k contributing to O (~) may be evaluated from a well-known formula in group theory [6] (see Appendix A). For the group G = Q A C2 under consideration, one obtains N(1) = 2,
N(2) = 0,
N(3) = 10,
N(4) = 0 . . . . .
However, their form has to be found by inspection, since no algorithm for a systematic construction is yet available. This task is greatly simplified by the observation that if O (k'j) is an equivariant of order k, the same is true for I • 0 (k'j). For the group G = Q A C2, we therefore have to determine five basic equivariants of order three. These can be derived as gradient terms from suitably chosen invariants of order four, i.e. 0 (3,j)(+) ___~gr+i(4,j)(+),
(9)
with 1 +14 "~ +42 "1=+34 + +4), 1 ( 4 ' 1 ) - 4( 1 +1+3 2 2 + +i+ 2 ), 1(4'3) = ~(
1 +1+2 2 2 ={-+2+2), •(4,2) : ~(
2 2 "q- +2+3 2 2 ), i(4,4) ~__.1 ~( +1+~
0o)
1(4'5) = +1+2+3+4"
Thus, up to third order the equation of motion takes the form 5
q~: l o t ) "+ :
E [a~2+-l)(~) 1 + a~2+)(~) I] • 0(3'+)(+) + o(1015) •
(11a)
j=l
For simplicity, we take the coefficient a[°)(/z) as control parameter:
a~0)(~) = ~, which satisfies the Hopf transversality condition d al(°)/d# ~ O. Further we assume a}t)(/~) = 0~0, a~2J-1)(#) = c2j-1,
a~ZJ)(l~) = c2j
to be independent of the bifurcation parameter/z.
(alb)
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K.-H. Hi~cket al./ Degenerate Hopf bifurcation
3. Calculation of the bifurcating solutions 3.1. Destabilization of the stationary state The normal modes of the stationary state ~ = 0, ~o(k)(t) = A e -i~'k',
(12)
are determined by the eigenvalue problem [t-(t0 + io~11 "I = 0.
(13)
Since L(/~) is specified by (5b) and (llb), one finds the degenerate pair of eigenfrequencies tox,2 = + O~o+ i/*, ~3,4 = - °~o + i~t
(14a)
with eigenvectors
1 [x=~-
-i 1 00'
1 [2=~
-
0 0 1i' .
/3:/1",
f4=[2 ""
(14b)
Thus on approaching the critical point/~: = 0 from below, the normal modes become undamped, and for ~o ~ 0 there occurs a hard-mode instability of the stationary state q~= 0 against critical perturbations of the form - A sin(o~0t + a) ]
A cos( ot + ¢3¢b(t) = R
[
B c o s ( 6 % t + fl) l ,
(15)
- B sin (~Oot +/3) ] where A 2 + B 2 = 1, such that ISq,(t)l = R is the mode amplitude. For later convenience, we set a = 0 + %/3 = 0 - q0 and choose 0 = O. Furthermore, we define the parameter p = ( A 2 - B 2 ) / ( A 2 + Bz), such that A=~(1+0)/2,
B=((1"0)/2.
(16a)
Eq. (15) describes a circular motion 6 ~ ( t ) = R [ u l ( p , ~ ) cos o~0t + u2(p, ~) sin O~ot]
(16b)
with radius R in a plane through the origin of the OP-space 12 spanned by the unit vectors u l ( p , ~ ) = - A ( e 1 sin~ - e2cos ~) + B(e3 cos ~ + e 4 sin ~),
u2(P, cp) = - A ( e 1 cos ~0+ e 2 sin 9~) + B ( e 3 sing~ - ¢4 COS 99).
(16C)
K.-H. Hi~cket aL/ DegenerateHopf bifurcation
343
Its orientation is determined by the two parameters p and tp. Thus, at # = 0 the stationary state becomes unstable against circular motions in a two-parameter family of planes through the origin of the OPspace ~2. Based on these results, we expect the bifurcating solutions to emerge at # = 0 + as a circular motion of frequency too in a plane given by (16c). With increasing g, the radius R will increase from its value R = 0 at # = 0, the frequency to will shift away from too, and higher harmonics will appear. The set of parameters (P, q0, R, to) will be determined by the higher-order terms in the equation of motion (11). 3.2. Bifurcation of time-periodic states We assume that the bifurcating solution is a periodic function, `#p(t) = `#p(t + T) with period T = 2~r/to, represented by the Fourier series
`#p(t, #) = )-~`#. (#)
(17a)
e -inwt
n
with the reality condition
We adopt the method of power series expansion [8] to construct explicitly the bifurcating solutions. The invariance of the equations of motion (1) against time translations leads to an indeterminateness of the phases of the Fourier coefficients ¢,(g): The transformation (17b)
,#,,(It) ~ `#,,(g) exp ( - i m T )
generates a solution differing from (17a) by a time shift • = ,//to. It is a common technique in bifurcation theory not to use the control parameter # as the expansion parameter, but the amplitude of the solution bifurcating from the stationary state `# = 0. Thus, a small real parameter e is introduced as a measure of this amplitude. Correspondingly, one then expands the Fourier amplitudes `#n, the frequency to, and the bifurcation parameter # into power series in e. Because of symmetry, the solution `# has to change sign with e, whereas # and to have to be invariant under e ~ - e. Therefore, we write
`#. = E
,
(lSa)
k>O
#=
2~ #~2k)eZk,
(18b)
k>l to = 620 +
E tot2k)e2k" k>l
(18C)
The expansion parameter e is identified with the modulus R=[`#+I[ of the first Fourier amplitude. Therefore, the expansion coefficients `#~k) satisfy the equations
I`# 1)12 -4- I`#O 1 =
---
1,
(19a)
k
E `#(12k-2j+l)'(`#(2j+l)) * = 0 j~O
( k = 1,2 . . . . ).
344
K.-H. Htck et al. / Degenerate Hopf bifurcation
Further, since the phase 71 in (17b) may depend in an arbitrary way on e, each expansion coefficient q,]k) is indeterminate with respect to an additive change by iaq~(1a) (a real). We remove this freedom by requiring Im[*{2k+*)'(*(11))* ] = 0
(k= 1,2,...).
(19b)
By substituting the expansions (17a) and (18) into the equation of motion (11) and equating the coefficients of the powers e k to zero, we obtain the perturbation equations listed in Appendix B. There it is shown that this set of equations represents a hierarchy which may be solved successively to arbitrary order in e. In lowest order, the branching solution is of the same form as the instability mode, eq. (16b), with amplitude e ~p(t, e) = e[ul( O, cp) cos 0aot + u2(p, ~) sin ,a0t ] ,
(20)
where (u 1, U2) are given by (16c), and the parameters p and ~o are yet unspecified. From solvability condition of the perturbation equation in e 3 we find that the two parameters P and cp determining the orientation of the plane as well as the second-order contributions oa(2) and/,(2) of (18) are given by the two coupled equations A [i:~o(2) + Ix(2) + C1A 2 + C2 B2 + C 3 B 2 e 4i~°] = 0,
(21a)
I
B [i,~a(2) +/,(2) + C1B 2 + C2A 2 + C3A2e-4i~] _ 0,
(21b)
where the remaining phase is fixed by a = - f l = ~. The three complex constants, Cj = C / + iCj", are defined by C 1 = ¼[(3q + c 3 ) - i ( 3 c 2 + c4)], (21c)
C 2 = ½[(c s + c 7 ) - i ( c 6+ c8)], C3 = ¼ [ ( - c 5 + c7 + c9) - i ( - c 6 + c8 + c10)],
where the c k, k = 1,..., 10, are the coefficients of the third-order equivariants defined in (11). For a given set of coupling constants (Cj }, a number of different limit cycle solutions (20) is allowed by (21). Each of these solutions may be appointed to one of the four classes listed in table I according to its symmetry properties discussed in section 4. The limit cycles of classes II-IV, which bifurcate on planes whose orientation in OP-space f2 is independent of the choice of the coupling constants, exists for all values { Cj } for which/,(2) >_ 0 is satisfied. In those cases, the solutions ~(t) are confined to the respective planes to any order of the perturbation expansion, i.e. the out-of-plane components are always zero. For the solutions of class I, on the other hand, the values of the parameters (p, ~) explicitly depend on the values of the coupling constants { Cj ), i.e. p2 =
(C3'2 + C~"2) 2 sin24~° [c ; , ( c ; c;(c;
cos4q0 =
- c;( c;,-
(22a) 2'
+
(C3, 2 + C3,,2)
(22b)
The existence region of these solutions is limited by the conditions 0 < p 2 < 1 (16a) and - 1 < cos 4~ < 1,
K.-H. Hrck et at~Degenerate Hopf bifurcation
345
i.e. the variation o f the coupling constants { C/, C/'} is restricted by c ; 2 + c ; "~ < ( c ; - c ~ ) ~ + ( c ; ' - c ~ ' ) 2
(22c)
[C3' :g ½ ( C [ - C~)] 2 + [C3" T- ½(C{'- C[')]2 > ¼ [ ( C / - C~) 2 + (C{'- C(')2].
(22d)
and
Each class may be divided into two subclasses (a) and (b). Let a bifurcation plane of (a) be spanned by the vectors u~(#,, %), i = 1,2. Then a bifurcation plane of (b) is spanned by the vectors u~(#u, ~b), i = 1,2, simply related to those of (a) by ili(Pb, ~b) -~ U,(--Pa, ~0a-t- '/1"//2).
(23a)
The set of orthonormal vectors
u,(o, ~p), u,(-O, ~P+ ~r/2), i = 1,2
(23b)
represents a convenient basis of the OP-space for the discussion of symmetry and stability of the bifurcation solutions. Finally, it can be shown that the higher-order terms q~tk) in the perturbation expansion (18) do not alter the symmetry properties of the branching solutions (20).
4. Symmetry properties of the bifurcating solutions We now discuss what happens on passing through the instability from the symmetry point of view. It is a basic feature of hard-mode instabilities in driven systems that the resulting phase transition may not only be associated with a symmetry change in OP-space 12 but also with a spontaneous breaking of time-translation symmetry. As explained in detail in ref. 1, for the discussion of the symmetry properties of the bifurcating solutions it is advantageous to augment the group G acting in OP-space 12 by the group .Y" of time translations, 0r: t ~ t + z, yielding c . = c × 3-,
(24)
the high-symmetry group (H-group) of the system. We use the convenient Seitz notation (gl'r) [7] to represent the symmetry operations of augmented groups, but with multiplication rule simply given by (gx[¢x)(g21~'2) = (gxg2l~'l + ¢2).
(25)
The group action of G H in 12 is defined by [(gl¢)~] (t) = gtk(t- ¢) = r ( g ) . q~(t- ~-), with I'(g) given in (8).
(26)
K.-H. HSck et a l l Degenerate Hopf bifurcation
346
Clearly, the stationary state q~= 0 is invariant under G and the continuous group oq"of time-translations. The bifurcating solutions (20), on the other hand, are invariant only under a subgroup G L of GH, the low-symmetry group (L-group) of the system. We now discuss in detail the various L-groups, i.e. all possible solutions, which are predicted by symmetry provided that the bifurcating solutions are of the form of the critical mode (16b), i.e. q~(t) = R [ux(p, cp) cos to0t + u2(p, rp) sin to0t] .
(27)
It is evident from the transformation law (26) that these solutions are only invariant under the subgroup J-r c J - of discrete time-translations, t ~ t + nT, i.e. = ( o.
1. = o, + 1 . . . . },
(28)
where T = 2~r/to is the period of the bifurcating limit cycle. Thus on passing through the hard-mode instability, a discrete "time lattice" is generated, spanned by the "basis vector" T. In this sense, the L-groups of time-periodic systems have the structure of a "space group". Correspondingly, a "reciprocal lattice '" is generated by the "basis vector'" to. Thus, we may introduce the concept of the Brillouin zone of the space group, which turns out to be helpful, for example, in the discussion of bifurcations from given time-periodic states. In order to derive the L-groups, we proceed to investigate the transformation behaviour of the limit cycles (27) under the group action of G. To find all possible L-groups, we first determine those elements of G which leave invariant a bifurcation plane with arbitrary parameters (O, qo). These elements are given by
g,,i( p, ) = Er, j( g ),,j( p, ),
(29)
with i = 1, 2 and comprise a subgroup G 1 of G. In general, there are specific planes in 12, i.e. specific sets of (p, ~), which are left invariant by a symmetry group Gi c G of higher order than G v In order to find these planes, we have to consider all (maximal) subgroups G i for which eq. (29) holds. For a given group G i c G, the elements g ~ G - G~ (i.e. g ~ G, g ~ G~) rotate the plane (that is left invariant by G~) in 12. If we factorize G into left cosets with respect to the subgroup G~, i.e. q
G = Y'~ g~G,,
q = [GI/IGi{,
(30)
a=l
the coset representatives g, generate a set of distinct bifurcation planes spanned by the basis vectors
g..,(o,
= .i,.(o,
(31)
For any fixed a, the subspace $2, c 12, spanned by the vectors (31), is invariant under the subgroup G}") c G, conjugate to G i, i.e. G}") = g . G , g ; 1.
(32)
An dement g of G: "), which leaves the subspace ~2, invariant, may change the associated limit cycle solution by shifting its phase. However, partial time translations ~o = n T / m (n, m integer) in combination
K.-H. H~ck et aL~Degenerate Hopf bifurcation
347
Table I The parameters p and q0, determining the orientation of a bifurcation plane, as well as the second-order corrections to the control p a r a m e t e r , / t (2), and to the frequency, (o(2), of the time-periodic state for the various classes of solutions. Class
p
(p
I
(a)
0
0 < qo < 3~r//4
(b)
O-" -O
qo --* qo + ~r//2
II (a)
w(2)
- C { + ½(1 - p2)(C~ - C~ - C~ cos4~o)
G'-
-c~
c;'
½(1 - p 2 ) ( C : ' - G ' -
G ' cos4qo)
1 arbitrary
(b)
/L(2)
-
1
ni (a)
o o
-G + flG - G - G)
c ; ' - ' ~(G" - G" - G')
- C ; + ~1 ( G t- G + G )
G ' - ~~(G" - G ' + G')
1
(b)
•
t
I
,~/2
~/4
IV (a) 0
(b)
3~r/4
with the elements g ~ G~(") can be found such that the augmented elements (g],rp) leave the limit cycle invariant, i.e. [(glvp)O](t)=O(t)
(33)
•
Thus, in general, the L-group GLi of a time-periodic state exhibits the structure of a "non-symmorphic space group", i.e.
(34a)
GL,= E
g~G, °
with elements of the form (gl% + n T ) . For given Gi, the conjugate groups G }") give rise to conjugate L-groups
G ~ )=
E
(glzp)-q'r.
(34b)
g ~ G('~)
Thus the left coset representatives of G i in G generate a star of symmetry-equivalent limit cycles. If we apply the procedure described above to the group G --- Q A C2, we find the following results: (1) For arbitrary (p, (p), a bifurcation plane is invariant under the subgroup
G, = {_+e, ±t} = C.,
(35)
which is isomorphic to the cyclic group of order )C4I = 4. With q = IG]/IC4] = 4, we obtain for a given limit cycle a star of four symmetry-equivalent solutions. The stars generated from the solutions with (p, q0) and ( - p , cp + ,r/2) are distinct.
348
K.-H. HOck et al./ Degenerate Hopf bifurcation
Table II The coset representatives'enteringthe low-symmetrygroup GLi associatedwith G,. Group G~
Coset-representatives of o~:v in GLi
61
((e#), (- el{ T), (11{T), (- 11¼T)}
G2
{(el0), ( - el½ T), (i{¼ T), ( - i[¼ T), (ll¼ T), ( - 11¼T), (il[0), (-
G3
{(e~0), (-el{T),
(kl¼ T), (- kl¼T),
(q¼T), (-II3T), (kll½T),
ill½T)} (- kl#)}
{(el0)' (_el{T), ( j i l T ) ' ( - j I.3z T), ( lhl T), (-/[~T), 3 • 1 T), (_jll0)} ( fl[~
(2) There are three pairs of special planes described by parameters (p, ¢p), i.e. {(1, arbitrary), ( - 1, arbitrary)},
{ (0,0), (0, ~r/2)) and {(0, rr/4), (0, 3~'/4) }. Each of the corresponding planes is invariant under a subgroup G i c G (i = 2, 3, 4) which is isomorphic to Caw Since IC4hl = 8, we find in all three cases a star of two symmetry-equivalent limit cycle solutions. A star generated from a solution with parameters (p, ¢p) already comprises the solution with ( - p, q~+ ~r/2). F r o m table II all possible L-groups for G = Q A C2 can be constructed, since we give the coset representatives (g]~-) that enter G[~) in (34) for one member of the set G~(~) of conjugate groups, i-- 1,2,3,4. The groups Gi may serve to define the various classes of solutions. For G = Q A C2, the solutions can therefore be appointed to one of the four classes listed in table I. Let us finally discuss the reduction of the OP-representation/" if we restrict the elements of G to G;. For i = 1,/" decomposes into two (over R) irreducible E-representations of C4, i.e. £ = E • E,
(36a)
while we find in all other cases that F decomposes into two inequivalent representations of C4h, i.e. F= Eg~E
u.
(36b)
This block-diagonal decomposition is achieved in the (parameter-dependent) basis given in (23b).
5. Stability of the bifurcating solutions In this section we analyze the stability of the (small amplitude) limit cycles q~p(t, ~) = q~p(t + T ( e ) , ~) found in section 3, because only stable solutions have physical significance. For this purpose, we study the dynamics of small deviations from this state, ~ ( t , e) -----cbp(t , e) = ~ ( t , e),
(37)
K.-H. Hi~ck et al. / Degenerate Hopf bifurcation
349
and linearize the equation of motion (11) about ~p(t, e), which yields 8q~(t) = L(~p(t, e))" 8q~(t).
(38)
The (real) linear operator L = WB]#~is periodic in time with period T(e) = 2~r/o:(e) and o:(e) given in (21) *. We therefore apply the Floquet theory [8] to solve (38). With the ansatz 8~(t)=~(t)e
°',
~(t)=~(t+T),
(39)
eq. (38) is converted into an eigenvalue problem for the Floquet exponent o = Re o + i Im e,
(
d
_ol).~(t)
=0,
(40)
where I m a is restricted to the first "Brillouin zone", i.e. - lo: _< I m o < ½o:. Since we deal with an autonomous system, the "Goldstone" mode 8~(aM)(t) = ~p(t) = ~(~M)
(41)
is always a solution to (8) associated with the Floquet exponent o~ra = 0. The stability of a limit cycle Op(t) is then established if all other exponents determined by (40) have negative real part. Since L(t) and ~(t) are periodic in time, they may be represented by a Fourier series L(t) = )"~Lne - i ~ ' ,
~ ( t ) = E ~ , e -i~°'',
(42)
with o: = o:(e). Inserting the expansion (42) into (40) yields an algebraic equations for the Fourier components ~,,
E L,-~,-r - (a -
ino:)~, = 0.
(43)
r
The perturbation expansion (18) of the periodic solution ~p(t, e) induces a corresponding one for the linear operator L, i.e.
L.= E
ekL(nk),
(44)
k>_O
and from the results of section 3 we derive the general structure of L up to e2: L =
L~ ) + e2(L~ ) +
L(22)e -2i°:t + --2'-/(2) , 2 i ~ , }.
(45)
With L(o°)= Oo|, L~) real and L~½= (L~)) *, L~) and L(~) depend on the specific limit cycle under consideration. *Since ~k (t + T/2) = -q~p(t) and L is even in ~v,Lp actually has period T/2. It is more convenient, however, not to take this fact explicitly into acount, We only have to note that the Fourier expansion of L in (43) below contains only even terms (see eq. (45)),
350
K.-H. HSck et aL/ Degenerate Hopf bifurcation
Consequently, we also expand ~ and o into power series of e, ~,=
Z ~k~}k),
0 = Y'~ e%(k),
k>_O
(46)
k>_O
where the unperturbed Floquet exponent has to be taken as zero, o(°) = 0, since the eigenfrequency o~o of the stationary state is at a reciprocal lattice vector of the time lattice defined by the limit cycle. The calculation, whose details are given in Appendix C, then yields in the lowest order t(t)
= t t 0) e-i~0t.-~ 4(._01ei~0 ',
(47)
where the Fourier components ~o) are found to be linear combinations of the orthonormal vectors (ui } given in (23b): I~°) = bl(u , + iuz) + b3(u 3 + iu4),
(48)
~0~ = b2(u 1 - iu2) + b 4 ( u 3 - iu4). (For convenience, we have defined u3(p, ~) = u l ( - p , ~o+ ~r/2) and u4(o, ~) = u z ( - p , ep + ~r/2).) Furthermore, we find that the first nonvanishing contribution to the Floquet exponents is of the order e2. It is determined together with the coefficients b i from the solubility condition for the inhomogeneous second-order perturbation equation,
L~). ~ o ) + L~). ~ = (0(2)_ i~(2)) ~o), L~). t(_o) + L~), . ~0) = (0 (2) + i0~(2))t(ol.
(49)
These coupled equations can be cast into an eigenvalue problem M" + = a(z)qJ
(50a)
b2
(50b)
(hi/
with
~b=-
b3
b4 and the matrix IVl given in Appendix C. As expected from symmetry (see end of section 4), the matrix IVI becomes block-diagonal in the basis ui(o, q~) for classes II-IV. For class I, on the other hand, such a decomposition is not expected, but it turns out that the matrix elements M31, M32, M41, M42 vanish. For each class of solutions we solve the eigenvalue problem. The results are listed in table III. We now discuss the conclusions we can draw from the explicit expressions for the Floquet exponents oj: (1) For bt(2) > 0, all solutions are (at least marginally) stable against small perturbations lying in the plane of the limit cycle under consideration. These "in-plane"-deviations are the phase mode (Goldstone mode) and the amplitude mode (Hopf mode), associated with the real Floquet exponents o 1 = O~M = 0 and (I2 = O H =
--2~ (2) <
0.
K.-H. H6ck et al./ Degenerate Hopf bifurcation
351
Table III The non-trivial Floquet exponents and the border of stability for the stable solutions. Class
o3,4 (2)
u
- ( c [ - c~) _+[-(C('
-
C~') 2 +
C; 2
02(2)
0 (2) = 0 b
2c(
c~ 2 + G '2 = (c~ - c~) 2 + ( c ; ' - c d ' ) :
+ C;'21 ~/2
}[(c/- c~) - 3c~1
III
[ c ; - ½ ( c ~ - c~)] 2 + [ c ; ' - ½ ( c ; ' - c ~ ' ) ] 2
~+~+~ +_ ~ [ ( c ;
IV
l[(c;
- c~ + c ; ) + 8 c ; ' ( c 7
- c~) + 3c;1
+_~ [ ( c ;
- c~ - c;)
= ¼[(c; - c~) 2 + (Cf' - cf) 2]
- c ~ ' - c ~ ' ) ] ~/~ It;
- 8c~'(c;'
- Q'
- c;')U
2
+ l(c;
- c~)] 2 + [ c ; ' + ½ ( c ; ' - c~')] 2
1 t = c~t + ~1 = ~[(C1 - c-;) 2 + ( c 7
c~') 2 ]
aa2(2) o H -- 2/~(2). bBorder of stability. =
=
(2) The stability is therefore determined by the (real or complex) Floquet exponents 03 and o4 associated with out-of-plane deviations. In particular, we find for class I, that the real part of one of these exponents is always positive. Therefore these solutions are unstable. It turns out, that the solutions of classes II-IV are stable within a limited region in the parameter space
(q}. We now briefly describe the types of instabilities which occur when the coupling constants { C/, C/' } are changed in such a way that the boundary of the stability region is crossed. If a real Floquet exponent changes sign, then the original limit cycle becomes unstable against a pitchfork bifurcation of a pair (generally: a multiple) of new limit cycles with the same period. If a pair of complex conjugate Floquet exponents crosses the imaginary axis with nonvanishing imaginary parts, then the original limit cycle becomes unstable against the formation of a torus. It should be noted, however, that the validity of the perturbation theory is limited to Floquet exponents of order e2. Therefore, the two frequencies of the motion on the torus differ only by an infinitesimal amount. For the same reason, an instability against period-doubling requiring Ime = -T- ½~00 cannot occur within the region of validity of the perturbation expansion.
6. C o n c l u s i o n s
In this paper we have analysed an example of a degenerate Hopf bifurcation associated with a four-component order parameter which is supposed to transform according to a single, physically irreducible representation of the symmetry group of the system. We have shown that fairly general conclusions can be drawn from symmetry considerations. The transformation behaviour of the order parameter completely specifies the structure of the linear operator which determines the form of the destabilizing mode. The solutions bifurcating at the critical point form stars of symmetry-equivalent limit cycles, each star being characterized by a distinct low-symmetry group. The existence of these solutions for the particular group G = Q A C2 is proved by explicitly solving the
K.-H. Htck et aL / Degenerate Hopf bifurcation
352
bifurcation equations. The stability analysis yields the result that the solutions with the lowest low-symmetry group are unstable, while all others (with low-symmetry groups of higher order) are stable. It is worth pointing out that the orientation of the bifurcation planes in OP-space ~2 associated with the stable solutions are determined by symmetry alone, while for the unstable solutions the orientation depends on the coupfing constants of the leading higher-order terms in the equation of motion. Though we do not known of any experimental data, such a bifurcation behaviour is expected to occur for the particular case G = Q A Cz, for example, in crystals with orthorhombic, tetragonal or cubic symmetry at certain high-symmetry points of the Brillouin zone. A simple example is represented by the (symmorphic) tetragonal space group I 4 / m (C~h) [7], where the complete space-group representation resulting from the P-point, k = ¼ (1,1,1), of the Brillouin zone with small representation of type E has Q A Cz as matrix group. Other representative space groups where either a small or a complete space-group representation leads to the matrix group Q A C2 are listed below: I) Orthorhombic systems 1212121 Pcca Pbcm Pbca
(D9), D8 (Zh), ~D n~ \ 2hi, is (D2h), (D 2h 27~J,
Ibca
Pnna Pccn Pbcn Pnma Imma
(D 26h ) ' (D10~ ~ 2hj, /DX4"~ \ 2h J, (D16-~ ~ 2h), (D~).
P4/ncc
(DSh),
F4132
(04).
II) Tetragonal systems 14122 P42/ncm
(Dam),
(D~°).
III) Cubic systems Pa3
(T6),
Acknowledgement We are indebted to Dr. J.W. Swift for letting us use his Ph.D. thesis [9] and a paper [10] prior to publication.
Appendix A Number of equivariants
The number of equivariants of order k is given by 1 N ( k ) = ~-~ E X t r p ( g ) x ( g ) ,
(A.1)
g~G
where Xtrl~ denotes the character of the symmetrized kth power of the representation /" and x ( g ) =
K.-H. HOck et aL/ Degenerate Hopf bifurcation
353
Tr F(g). X[rl~(g) is computed by X [ F ] k ( g ) --
E xi l(g--~)"-'xi'~(+k) it,J2 ..... ik 1'ti1!2'2i2! ... k'kik!
(A.2)
with the constraint Z~=tl. ik = k. Thus, for the first few k we have
Xtrl3(g) = {x3(g) + ½x(g)x(g 2) + ½x(g3),
(A.3)
X[r]'(g) = ~ x 4 ( g ) + ¼xZ(g)x(g z) + ½x(g)x(g 3) + {x2(g z) + ¼X(g4) •
Appendix B
Perturbation expansion for limit-cycle bifurcation Substituting the expansions (17a) and (18) into the equations of motion (11) and equating the coefficients of successive powers e k to zero yields the following set of perturbation equations: El:
- ~0[I + in1] "~,~1) --
=0,
•
lit
n
k k e2k+l: -- OOO[|"4-in1] °l/brn(2k+l)-~E fig{2j) q-inO0(2j)] °Cb(rgk-2j+l)-{- E R(2k+l'2j+l)' j=l j=l
(B.1)
Here R~,k' 1) denotes the contribution of the lth order equivariant O (o(~,) to the order e k of the equation of motion for the n th Fourier coe~cient. We now show that eqs. (B.1) may be solved successively to arbitrary order. For n = +1, the operators occurring on the left-hand sides are singular in the directions of the eigenvectors fl, 12 and I3, 14, respectively. We therefore decompose the vector space 9 into invariant subspaces 12(+) spanned b y / t , I2, and 12(-) spanned b y / 3 , I4, I2 =/2 (+) • 12(-),
(B.2)
with corresponding projection operators P(-+ ), stich that I + in1 =
i[(n - 1)P {+) + (n + 1)P(-)].
(B.3)
354
K.-H. HOck et al./ Degenerate Hopf bifurcation
In order e~, one finds
0(?= 1__[ v~- X(1)1"1+
Y(l>h] E
(B.4)
,/,.(~)=0 (n4= +1), with complex amplitudes X ~1)= A exp ( - ia), Y(]) = B exp ( - ifl). In all higher orders, the Fredholm alternative yields the solvability condition
P(+)"
r(12k+1) = 0,
(B.5)
where ¥(2k+1) denotes the fight-hand sides of (B.1). If (B.5) is satisfied, the solution is obtained in the form
= 1
[
+ r
+ % ] + "i p ( _ ) . r(2k+l),
(B.6)
n
tO0
with complex amplitudes X (2k+l), y(2k+ l) It remains to show that eq. (B.5) together with eqs. (20a) and (20b) determine successively all coefficients x ( 2 k + l ) , y(2k+l), a n d ~t(2k), 03(2k).
In order e 3, the solvability condition (B.5) takes the form [, (2) + iw(2))[ X(X)fl + yO)fz ] + p(+), R~3,3)= 0.
Inserting the expression for R~ 3'3) following from the eqs. (9) to (11) yields the eq. (21a, b) whose solutions are discussed in the text. We now assume that eqs. (B.5) have been solved up to order ezk- ~, and that the coefficients/~(t) and co(t) (l_< 2 k - 2) as well as the ~t) (1 < 2 k - 1) have been determined, with the exception of the complex amplitudes X (2k-1), y(2k-1). Then, the solvability condition (B.5), which represents two complex equations, together with the two real eqs. (20a, b), constitute a system of six real equations which are linear in the six unknowns R e X (2k-1), Im X (2k-l), Re Y (2k-D, I m Y (2k-1), ~(2k), 60(2k). Since the determinant is generically nonzero, the system has a unique solution. Thus, by induction we have shown that the bifurcation equations (B.1) may be solved successively to arbitrary order.
K.-H. Hgcket aL/ DegenerateHopfbifurcation
355
Appendix C
Stability analysis of limit-cycle solution By inserting the expansions (46) into (43), we obtain the perturbation equations
,Oo[t +
=o,
(ca)
el:
~o[I + in1] °~(n1) -~-g(1)~(n0),
(C.2)
ez:
oa0[l+inl]'~{, z)=-me<*)o ~, - L ~ ) ' ~ ° ) - L ~ ) ' ~ ° - ) : - L ~ ½ "$m)-.+2+ [°(2)-in°a{2~]'~ m.
(C.3)
Eq. (C.1)has nontrivial solutions for n _+ 1, yielding in lowest order the form of the perturbation given in eq. (48). Eq. (C.2) yields o 0) = 0, and eq. (49) arising as the solubility condition for (C.3) are finally transformed into the eigenvalue problem (50) where the matrix M has the structure
[ Mn
M12 I M13 MI, I
/M*
M * ', M * q _
M,,
M*|
/
(C.4)
)1433 M3a,/
M~'~ M3~ ,, M3'I
M3']]
with matrix elements given by Mu= }[2C/-(1-02)F
'] - 2 1 2 C ; , - 0 - 0 2 ) F " ] ,
Mz~= ½12c;-(1-02)r
,] +
½12c{,-(1-02)F
,,] = M ~ ,
M33= - ½12(C/- C ; ) - 3(1 - p 2 ) r ' ] + 2 [ 2 ( C { ' - C~')- 3(1 - p2)F"], M34 = - ½ [ ( 1 - p2)F' + 2C3'cos4ep - 2C~' singq0] + 2 [ ( 1 - p2)r"+ 2c~' cos4q0 + 2c3'p sin4q~ ] , M~4 = ½~/1 - O Z [ o F " - s i n 4 ~ C ; ] M~3 = - V/1 - Oz [ o F " +
- 2 ~/1 - O z [ O r ' +
sin4~C3"],
sin4,~C3'] - i~1 - O~ [ o F ' - sin4~C3"],
M31 = ~1 - O2 [ p F " - sin4q)C3'] + i l l -
p 2 [OF'+ sin4qoC3"],
where
F' = C { - Cg - C; cos 4q0,
(1
F " = C [ ' - C ~ ' - C3" cos 4%
(1 - p 2 ) r " = - 2 ~
-
p2)/-,,
=
2C{, (2) + 2C{'.
2/~2) +
(c.5)
For 0 < O2 < 1, the additional relations derived from (21a, b) hold,
pF'+C~'sin4cp=O, such that M14 = M31 = 0.
oF"-C~'sin4ep=O,
(C.6)
356
K.-H. H6ck et aL/ Degenerate Hopf bifurcation
Solving (50), we obtain
a~=O,
a2 = - 2 / t (2)
(c.7)
and o3/, = ReM3~
+__¢-
(tin M . ) 2 + IM~,I 2 .
(c.8)
For classes II-IV, the last equation directly reveals from the block-diagonal form of M, whereas for class I the relations (C.6) have to be used.
References [1] K.-H. HiSck, P. Jordan and H. Thomas, J. Phys. C (1987), in press. [2] See, e.g., .I.E. Marsden and M. McCracken, The Hopf bifurcation and its application, Applied Mathematical Science 19 (Springer, New York, 1976). [3] M. Golubitsky and I. Stewart, Arch. Rat. Mech. Anal. 87 (1985) 107. [4] S. Thiesen and H. Thomas, Z. Phys. B65 (1987) 397. [5] J.P. Serre, Linear Representation of Finite Groups (Springer, New York, 1977). [6] G.Ya. Lyubarskii, The Application of Group Theory in Physics (Pergamon, Oxford, London, New York, Paris, 1960). [7] C.J. Bradley and A.P. Cracknell, The Mathematical Theory of Symmetry in Solids (Clarendon Oxford, 1972). [8] G. Ioos and D.D. Joseph, Elementary Stability and Bifurcation Theory (Springer, New York, 1980). [9] J.W. Swift, Ph.D. Thesis, Univ. California, Santa Cruz (1984). [10] J.W. Swift, Four coupled oscillators: Hopf bifurcation with the symmetry of the square, submitted to: Dynamics and Stability of Systems (1986).
D E G E N E R A T E H O P F B I F U R C A T I O N IN T H E P R E S E N C E OF S Y M M E T R Y * K.-H. H O C K and D. H A C K E N B R A C H T Institut fftr Festkbrperphysik, Technische Hochschule Darmstadt, Hochschulstr. 2, D-6100 Darmstadt, Fed. Rep. Germany P. J O R D A N and H. T H O M A S lnstitut fftr Physik, Universitilt Basel, Klingelbergstr. 82, CH-4056 Basel, Switzerland Received 15 September 1986 Revised manuscript received 20 February 1987
We investigate the bifurcation of time-periodic states from a stationary state destabilized by the undamping of a set of modes associated with a degenerate pair of complex-conjugate frequencies. This problem is of particular interest for bifurcations in driven systems with symmetry whose order-parameter dimension n is even and n > 4. For this case of a degenerate Hopf bifurcation a star of symmetry-equivalentlimit cyclesbifurcates in analogy to the star of symmetry-related domains arising at a symmetry-breakingphase transition in equilibrium systems. We illustrate this fact by analyzinga concrete example with n = 4. Within the framework of an amplitude expansion, we explicitly construct the time-periodic states and discuss their stability. In particular, it is shown that fairly general conclusions for the bifurcation behaviour can be drawn on the sole basis of the knowledgeof the order-parameter symmetry.
1. Introduction In a recent paper [1] we have investigated the role of symmetry in autonomous, nonlinear dynamical systems in which soft-mode or hard-mode instabilities give rise to bifurcations of new states from a stationary reference state. It has been demonstrated that the application of group-theoretical methods to bifurcation problems provides important pieces of information on phase transitions in driven systems. For example, instabilities associated with the undamping of normal modes of a given stationary state may be classified by the type of representatio n associated with the order parameter of the system. As a consequence, symmetry criteria can be formulated for phase transitions to new stationary states or to permanently time-dependent structures. In particular, symmetry considerations lead to distinguish two types of hard-mode stabilities (coupling-induced or symmetry-induced hard-mode instabilities) which we briefly recall in section 2. Each of these instabilities gives rise to a degenerate Hopf bifurcation for even values of the order-parameter dimension n with n >__4. In that case, the well-known Hopf theorem [2] cannot be applied to assure the existence of stable limit cycles bifurcating from a stationary reference state. Very recently, Golubitsky and Stewart [3] succeeded in generalizing the Hopf theorem to the case of the coupling-induced, degenerate Hopf bifurcation. For the case of the symmetry-induced, degenerate H o p f bifurcation, on the other hand, the knowledge on the bifurcation behavior is still the subject of current research.
*Work supported by the Sonderforschungsbereich65 Frankfurt-Darmstadt and by the Swiss National Science Foundation. 0167-2789/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
K.-H. HSek et aL/ Degenerate Hopf bifurcation
339
In this paper, we investigate a representative example for this type of bifurcation based on a four-component order parameter. Unfortunately, such a case does not occur in systems described by spatially uniform tensorial quantities in three-dimensional space. Four-dimensional physically irreducible representations may occur in the case of wave instabilities of translationally invariant systems (order parameter components ~ exp (ikx), ~ exp ( - i k x ) [4], or as space-group representations of crystals or other spatially periodic structures. In the present paper, we consider specifically an order-parameter representation with matrix group Q A C2 (Q = quaternion group), which is realized by several space-group representations (see section 6). In section 2, we define the class of systems to be studied and construct the equation of motion for the order parameter on the basis of symmetry considerations. To investigate the bifurcation of time-periodic states, we derive the bifurcation equations by power series expansion and solve them explicitly. The results of the calculations are discussed in section 4 from the symmetry point of view. Section 5 is devoted to the stability analysis of the bifurcating solutions and section 6 concludes with a brief discussion of our results. 2. Construction of the equation of motion
We investigate an autonomous, non-linear dynamical system in which a hard-mode instability gives rise to bifurcations of time-periodic states from a given stationary state with symmetry group G. Within a Landan-type analysis, the bifurcation behaviour of the system is described in terms of an n-component order parameter (OP) ¢ which represents deviations from the stationary reference state q, = 0. The dynamics of the OP ~ is governed by an equation of motion of the form q~= B(O; #).
(1)
The real velocity field B(¢; #) is non-linear, analytic function on the OP-space ~2 satisfying B(0; #) = 0 for all values of the real parameter #, which can be externally controlled. Taylor expansion around ¢ = 0 yields 6 = L(/~) "q~ Y'~ o(k)(o; #).
(2)
k>l
Here, L(#)--VB]¢= o is the Jacobian whose eigenvalues X(#) determine the type of bifurcation. The higher-order terms O(k)(rk; ix), on the other hand, determine the position of the bifurcating solutions in OP-space $2, the variation of their amplitude with the control parameter/~, and their stability. Symmetry requires B(qD; #) to be equivariant with respect to the (real) representation F = ( F ( g ) J g ~ G) of the symmetry group G acting in OP-space/2, i.e. r(g)
•
= n(r(g)
• q,;
Vgea.
(3)
In particular, the linear operator I.(#) in (2) satisfies [L(/z), F ( g ) ] = 0
Vg~G.
(4)
From representation theory it then follows that the type of F uniquely determines the structure of L(#)
[51.
K.-H. Hgck et al./ Degenerate Hopf bifurcation
340
We are interested in the case that the stationary state ~ = 0 looses stability at the critical point/~c = 0 by' virtue of a degenerate pair of complex-conjugate eigenvalues of I.(/z) crossing the imaginary axis with nonvanishing imaginary part. As discussed in ref. 1, such a situation may occur in OP-dimensions n - 4, 6, 8 . . . . for three different types of OP-representation lr: Either F is irreducible over R but decomposes over C into two complex-conjugated irreducible representations 3', 3' *, i.e. F = V • 3' *, with (a) 3' equivalent to 3'* (type 2) or (b) F inequivalent to 3' * (type 3), o r / ' is reducible and decomposes into two equivalent representations (c) /" = 27 (type 1.2) with 3' irreducible over R and C, For cases (a) and (b), their instability is called a symmetry-induced hard-mode instability, while (c) is called a coupling-induced hard-mode instability. The latter has been analyzed recently in detail by Golubitsky and Stewart [3] and thus will not be discussed further in this paper. For types 2 and 3, the linear operation L(/~) has the following structure [1]: Type 2 L(/~) = a~°)(/x)l + a[1)(/~)l + a[2)(F)m + at3)(/~)n.
(5a)
Type 3 (5b)
L ( # ) = a[°)(#)l + a[1)(#)l, with at(°) real, and the matrices 1, I, m,
n
generate the quaternion algebra,
12= m 2 = n 2 = - 1 , I. m = - m . I = n
(6a) and cyclic.
(6b)
As pointed out in ref. 1, the structure of the matrices I.(/Q for type 2 and type 3 are closely related, i.e. (5a) can always be brought into a form identical to (5b) by a suitable choice of coordinates. As a representative example of a symmetry-induced hard-mode instability, we thus construct a specific model in OP-dimension n = 4 by considering an OP-representation lr of type 3 of the symmetry group G = Q/x C2 which is the semi-direct product of the quaternion group Q and the cyclic group of order 2. It is worth pointing out that this model comprises a whole class of systems with different symmetry groups G' which are determined by theorem 1 of ref. 1. The group G = Q/x C2 consists of 16 elements
G = ( +_e, +i, +j, + k , +l, +il, +jl, + k l }
(7a)
with multiplication rules defined by 1) i 2 = j 2 = k 2 = I 2 = _ e ;
(7b)
2) il= li,
k l = lk;
(7c)
and cyclic.
(7d)
j l = lj,
3) ij = - j i = k
It is convenient to choose a basis { e i } in OP-space ~2 such that the (real) OP-representation F = { F(g)lg
341
K.-H. HOck et al./Degenerate Hopf bifurcation
G } is generated by the three 4 × 4 matrices:
0
-1', '
1 J x
1
0',I
. . . . . . .
-4- . . . . . . . . I
0
i I-1
0
0
/°___1i.........
0
/
0
r
0
/
°/
',a
"=
-1/
1 0
°i
Ol -ll
0
o/ -1
/
.
(8)
/ /
Since F is of type 3, there exists exactly one nontrivial matrix commuting with all matrices of F [5]. Therefore the linear operator k(/~) has the form (5b) with I given in (8). Next we determine the form of the leading higher-order term O (k) in the Taylor expansion (2). The total number N ( k ) of basic equivariants 0 (k'j), j = 1. . . . . N(k), of given order k contributing to O (~) may be evaluated from a well-known formula in group theory [6] (see Appendix A). For the group G = Q A C2 under consideration, one obtains N(1) = 2,
N(2) = 0,
N(3) = 10,
N(4) = 0 . . . . .
However, their form has to be found by inspection, since no algorithm for a systematic construction is yet available. This task is greatly simplified by the observation that if O (k'j) is an equivariant of order k, the same is true for I • 0 (k'j). For the group G = Q A C2, we therefore have to determine five basic equivariants of order three. These can be derived as gradient terms from suitably chosen invariants of order four, i.e. 0 (3,j)(+) ___~gr+i(4,j)(+),
(9)
with 1 +14 "~ +42 "1=+34 + +4), 1 ( 4 ' 1 ) - 4( 1 +1+3 2 2 + +i+ 2 ), 1(4'3) = ~(
1 +1+2 2 2 ={-+2+2), •(4,2) : ~(
2 2 "q- +2+3 2 2 ), i(4,4) ~__.1 ~( +1+~
0o)
1(4'5) = +1+2+3+4"
Thus, up to third order the equation of motion takes the form 5
q~: l o t ) "+ :
E [a~2+-l)(~) 1 + a~2+)(~) I] • 0(3'+)(+) + o(1015) •
(11a)
j=l
For simplicity, we take the coefficient a[°)(/z) as control parameter:
a~0)(~) = ~, which satisfies the Hopf transversality condition d al(°)/d# ~ O. Further we assume a}t)(/~) = 0~0, a~2J-1)(#) = c2j-1,
a~ZJ)(l~) = c2j
to be independent of the bifurcation parameter/z.
(alb)
342
K.-H. Hi~cket al./ Degenerate Hopf bifurcation
3. Calculation of the bifurcating solutions 3.1. Destabilization of the stationary state The normal modes of the stationary state ~ = 0, ~o(k)(t) = A e -i~'k',
(12)
are determined by the eigenvalue problem [t-(t0 + io~11 "I = 0.
(13)
Since L(/~) is specified by (5b) and (llb), one finds the degenerate pair of eigenfrequencies tox,2 = + O~o+ i/*, ~3,4 = - °~o + i~t
(14a)
with eigenvectors
1 [x=~-
-i 1 00'
1 [2=~
-
0 0 1i' .
/3:/1",
f4=[2 ""
(14b)
Thus on approaching the critical point/~: = 0 from below, the normal modes become undamped, and for ~o ~ 0 there occurs a hard-mode instability of the stationary state q~= 0 against critical perturbations of the form - A sin(o~0t + a) ]
A cos( ot + ¢3¢b(t) = R
[
B c o s ( 6 % t + fl) l ,
(15)
- B sin (~Oot +/3) ] where A 2 + B 2 = 1, such that ISq,(t)l = R is the mode amplitude. For later convenience, we set a = 0 + %/3 = 0 - q0 and choose 0 = O. Furthermore, we define the parameter p = ( A 2 - B 2 ) / ( A 2 + Bz), such that A=~(1+0)/2,
B=((1"0)/2.
(16a)
Eq. (15) describes a circular motion 6 ~ ( t ) = R [ u l ( p , ~ ) cos o~0t + u2(p, ~) sin O~ot]
(16b)
with radius R in a plane through the origin of the OP-space 12 spanned by the unit vectors u l ( p , ~ ) = - A ( e 1 sin~ - e2cos ~) + B(e3 cos ~ + e 4 sin ~),
u2(P, cp) = - A ( e 1 cos ~0+ e 2 sin 9~) + B ( e 3 sing~ - ¢4 COS 99).
(16C)
K.-H. Hi~cket aL/ DegenerateHopf bifurcation
343
Its orientation is determined by the two parameters p and tp. Thus, at # = 0 the stationary state becomes unstable against circular motions in a two-parameter family of planes through the origin of the OPspace ~2. Based on these results, we expect the bifurcating solutions to emerge at # = 0 + as a circular motion of frequency too in a plane given by (16c). With increasing g, the radius R will increase from its value R = 0 at # = 0, the frequency to will shift away from too, and higher harmonics will appear. The set of parameters (P, q0, R, to) will be determined by the higher-order terms in the equation of motion (11). 3.2. Bifurcation of time-periodic states We assume that the bifurcating solution is a periodic function, `#p(t) = `#p(t + T) with period T = 2~r/to, represented by the Fourier series
`#p(t, #) = )-~`#. (#)
(17a)
e -inwt
n
with the reality condition
We adopt the method of power series expansion [8] to construct explicitly the bifurcating solutions. The invariance of the equations of motion (1) against time translations leads to an indeterminateness of the phases of the Fourier coefficients ¢,(g): The transformation (17b)
,#,,(It) ~ `#,,(g) exp ( - i m T )
generates a solution differing from (17a) by a time shift • = ,//to. It is a common technique in bifurcation theory not to use the control parameter # as the expansion parameter, but the amplitude of the solution bifurcating from the stationary state `# = 0. Thus, a small real parameter e is introduced as a measure of this amplitude. Correspondingly, one then expands the Fourier amplitudes `#n, the frequency to, and the bifurcation parameter # into power series in e. Because of symmetry, the solution `# has to change sign with e, whereas # and to have to be invariant under e ~ - e. Therefore, we write
`#. = E
,
(lSa)
k>O
#=
2~ #~2k)eZk,
(18b)
k>l to = 620 +
E tot2k)e2k" k>l
(18C)
The expansion parameter e is identified with the modulus R=[`#+I[ of the first Fourier amplitude. Therefore, the expansion coefficients `#~k) satisfy the equations
I`# 1)12 -4- I`#O 1 =
---
1,
(19a)
k
E `#(12k-2j+l)'(`#(2j+l)) * = 0 j~O
( k = 1,2 . . . . ).
344
K.-H. Htck et al. / Degenerate Hopf bifurcation
Further, since the phase 71 in (17b) may depend in an arbitrary way on e, each expansion coefficient q,]k) is indeterminate with respect to an additive change by iaq~(1a) (a real). We remove this freedom by requiring Im[*{2k+*)'(*(11))* ] = 0
(k= 1,2,...).
(19b)
By substituting the expansions (17a) and (18) into the equation of motion (11) and equating the coefficients of the powers e k to zero, we obtain the perturbation equations listed in Appendix B. There it is shown that this set of equations represents a hierarchy which may be solved successively to arbitrary order in e. In lowest order, the branching solution is of the same form as the instability mode, eq. (16b), with amplitude e ~p(t, e) = e[ul( O, cp) cos 0aot + u2(p, ~) sin ,a0t ] ,
(20)
where (u 1, U2) are given by (16c), and the parameters p and ~o are yet unspecified. From solvability condition of the perturbation equation in e 3 we find that the two parameters P and cp determining the orientation of the plane as well as the second-order contributions oa(2) and/,(2) of (18) are given by the two coupled equations A [i:~o(2) + Ix(2) + C1A 2 + C2 B2 + C 3 B 2 e 4i~°] = 0,
(21a)
I
B [i,~a(2) +/,(2) + C1B 2 + C2A 2 + C3A2e-4i~] _ 0,
(21b)
where the remaining phase is fixed by a = - f l = ~. The three complex constants, Cj = C / + iCj", are defined by C 1 = ¼[(3q + c 3 ) - i ( 3 c 2 + c4)], (21c)
C 2 = ½[(c s + c 7 ) - i ( c 6+ c8)], C3 = ¼ [ ( - c 5 + c7 + c9) - i ( - c 6 + c8 + c10)],
where the c k, k = 1,..., 10, are the coefficients of the third-order equivariants defined in (11). For a given set of coupling constants (Cj }, a number of different limit cycle solutions (20) is allowed by (21). Each of these solutions may be appointed to one of the four classes listed in table I according to its symmetry properties discussed in section 4. The limit cycles of classes II-IV, which bifurcate on planes whose orientation in OP-space f2 is independent of the choice of the coupling constants, exists for all values { Cj } for which/,(2) >_ 0 is satisfied. In those cases, the solutions ~(t) are confined to the respective planes to any order of the perturbation expansion, i.e. the out-of-plane components are always zero. For the solutions of class I, on the other hand, the values of the parameters (p, ~) explicitly depend on the values of the coupling constants { Cj ), i.e. p2 =
(C3'2 + C~"2) 2 sin24~° [c ; , ( c ; c;(c;
cos4q0 =
- c;( c;,-
(22a) 2'
+
(C3, 2 + C3,,2)
(22b)
The existence region of these solutions is limited by the conditions 0 < p 2 < 1 (16a) and - 1 < cos 4~ < 1,
K.-H. Hrck et at~Degenerate Hopf bifurcation
345
i.e. the variation o f the coupling constants { C/, C/'} is restricted by c ; 2 + c ; "~ < ( c ; - c ~ ) ~ + ( c ; ' - c ~ ' ) 2
(22c)
[C3' :g ½ ( C [ - C~)] 2 + [C3" T- ½(C{'- C[')]2 > ¼ [ ( C / - C~) 2 + (C{'- C(')2].
(22d)
and
Each class may be divided into two subclasses (a) and (b). Let a bifurcation plane of (a) be spanned by the vectors u~(#,, %), i = 1,2. Then a bifurcation plane of (b) is spanned by the vectors u~(#u, ~b), i = 1,2, simply related to those of (a) by ili(Pb, ~b) -~ U,(--Pa, ~0a-t- '/1"//2).
(23a)
The set of orthonormal vectors
u,(o, ~p), u,(-O, ~P+ ~r/2), i = 1,2
(23b)
represents a convenient basis of the OP-space for the discussion of symmetry and stability of the bifurcation solutions. Finally, it can be shown that the higher-order terms q~tk) in the perturbation expansion (18) do not alter the symmetry properties of the branching solutions (20).
4. Symmetry properties of the bifurcating solutions We now discuss what happens on passing through the instability from the symmetry point of view. It is a basic feature of hard-mode instabilities in driven systems that the resulting phase transition may not only be associated with a symmetry change in OP-space 12 but also with a spontaneous breaking of time-translation symmetry. As explained in detail in ref. 1, for the discussion of the symmetry properties of the bifurcating solutions it is advantageous to augment the group G acting in OP-space 12 by the group .Y" of time translations, 0r: t ~ t + z, yielding c . = c × 3-,
(24)
the high-symmetry group (H-group) of the system. We use the convenient Seitz notation (gl'r) [7] to represent the symmetry operations of augmented groups, but with multiplication rule simply given by (gx[¢x)(g21~'2) = (gxg2l~'l + ¢2).
(25)
The group action of G H in 12 is defined by [(gl¢)~] (t) = gtk(t- ¢) = r ( g ) . q~(t- ~-), with I'(g) given in (8).
(26)
K.-H. HSck et a l l Degenerate Hopf bifurcation
346
Clearly, the stationary state q~= 0 is invariant under G and the continuous group oq"of time-translations. The bifurcating solutions (20), on the other hand, are invariant only under a subgroup G L of GH, the low-symmetry group (L-group) of the system. We now discuss in detail the various L-groups, i.e. all possible solutions, which are predicted by symmetry provided that the bifurcating solutions are of the form of the critical mode (16b), i.e. q~(t) = R [ux(p, cp) cos to0t + u2(p, rp) sin to0t] .
(27)
It is evident from the transformation law (26) that these solutions are only invariant under the subgroup J-r c J - of discrete time-translations, t ~ t + nT, i.e. = ( o.
1. = o, + 1 . . . . },
(28)
where T = 2~r/to is the period of the bifurcating limit cycle. Thus on passing through the hard-mode instability, a discrete "time lattice" is generated, spanned by the "basis vector" T. In this sense, the L-groups of time-periodic systems have the structure of a "space group". Correspondingly, a "reciprocal lattice '" is generated by the "basis vector'" to. Thus, we may introduce the concept of the Brillouin zone of the space group, which turns out to be helpful, for example, in the discussion of bifurcations from given time-periodic states. In order to derive the L-groups, we proceed to investigate the transformation behaviour of the limit cycles (27) under the group action of G. To find all possible L-groups, we first determine those elements of G which leave invariant a bifurcation plane with arbitrary parameters (O, qo). These elements are given by
g,,i( p, ) = Er, j( g ),,j( p, ),
(29)
with i = 1, 2 and comprise a subgroup G 1 of G. In general, there are specific planes in 12, i.e. specific sets of (p, ~), which are left invariant by a symmetry group Gi c G of higher order than G v In order to find these planes, we have to consider all (maximal) subgroups G i for which eq. (29) holds. For a given group G i c G, the elements g ~ G - G~ (i.e. g ~ G, g ~ G~) rotate the plane (that is left invariant by G~) in 12. If we factorize G into left cosets with respect to the subgroup G~, i.e. q
G = Y'~ g~G,,
q = [GI/IGi{,
(30)
a=l
the coset representatives g, generate a set of distinct bifurcation planes spanned by the basis vectors
g..,(o,
= .i,.(o,
(31)
For any fixed a, the subspace $2, c 12, spanned by the vectors (31), is invariant under the subgroup G}") c G, conjugate to G i, i.e. G}") = g . G , g ; 1.
(32)
An dement g of G: "), which leaves the subspace ~2, invariant, may change the associated limit cycle solution by shifting its phase. However, partial time translations ~o = n T / m (n, m integer) in combination
K.-H. H~ck et aL~Degenerate Hopf bifurcation
347
Table I The parameters p and q0, determining the orientation of a bifurcation plane, as well as the second-order corrections to the control p a r a m e t e r , / t (2), and to the frequency, (o(2), of the time-periodic state for the various classes of solutions. Class
p
(p
I
(a)
0
0 < qo < 3~r//4
(b)
O-" -O
qo --* qo + ~r//2
II (a)
w(2)
- C { + ½(1 - p2)(C~ - C~ - C~ cos4~o)
G'-
-c~
c;'
½(1 - p 2 ) ( C : ' - G ' -
G ' cos4qo)
1 arbitrary
(b)
/L(2)
-
1
ni (a)
o o
-G + flG - G - G)
c ; ' - ' ~(G" - G" - G')
- C ; + ~1 ( G t- G + G )
G ' - ~~(G" - G ' + G')
1
(b)
•
t
I
,~/2
~/4
IV (a) 0
(b)
3~r/4
with the elements g ~ G~(") can be found such that the augmented elements (g],rp) leave the limit cycle invariant, i.e. [(glvp)O](t)=O(t)
(33)
•
Thus, in general, the L-group GLi of a time-periodic state exhibits the structure of a "non-symmorphic space group", i.e.
(34a)
GL,= E
g~G, °
with elements of the form (gl% + n T ) . For given Gi, the conjugate groups G }") give rise to conjugate L-groups
G ~ )=
E
(glzp)-q'r.
(34b)
g ~ G('~)
Thus the left coset representatives of G i in G generate a star of symmetry-equivalent limit cycles. If we apply the procedure described above to the group G --- Q A C2, we find the following results: (1) For arbitrary (p, (p), a bifurcation plane is invariant under the subgroup
G, = {_+e, ±t} = C.,
(35)
which is isomorphic to the cyclic group of order )C4I = 4. With q = IG]/IC4] = 4, we obtain for a given limit cycle a star of four symmetry-equivalent solutions. The stars generated from the solutions with (p, q0) and ( - p , cp + ,r/2) are distinct.
348
K.-H. HOck et al./ Degenerate Hopf bifurcation
Table II The coset representatives'enteringthe low-symmetrygroup GLi associatedwith G,. Group G~
Coset-representatives of o~:v in GLi
61
((e#), (- el{ T), (11{T), (- 11¼T)}
G2
{(el0), ( - el½ T), (i{¼ T), ( - i[¼ T), (ll¼ T), ( - 11¼T), (il[0), (-
G3
{(e~0), (-el{T),
(kl¼ T), (- kl¼T),
(q¼T), (-II3T), (kll½T),
ill½T)} (- kl#)}
{(el0)' (_el{T), ( j i l T ) ' ( - j I.3z T), ( lhl T), (-/[~T), 3 • 1 T), (_jll0)} ( fl[~
(2) There are three pairs of special planes described by parameters (p, ¢p), i.e. {(1, arbitrary), ( - 1, arbitrary)},
{ (0,0), (0, ~r/2)) and {(0, rr/4), (0, 3~'/4) }. Each of the corresponding planes is invariant under a subgroup G i c G (i = 2, 3, 4) which is isomorphic to Caw Since IC4hl = 8, we find in all three cases a star of two symmetry-equivalent limit cycle solutions. A star generated from a solution with parameters (p, ¢p) already comprises the solution with ( - p, q~+ ~r/2). F r o m table II all possible L-groups for G = Q A C2 can be constructed, since we give the coset representatives (g]~-) that enter G[~) in (34) for one member of the set G~(~) of conjugate groups, i-- 1,2,3,4. The groups Gi may serve to define the various classes of solutions. For G = Q A C2, the solutions can therefore be appointed to one of the four classes listed in table I. Let us finally discuss the reduction of the OP-representation/" if we restrict the elements of G to G;. For i = 1,/" decomposes into two (over R) irreducible E-representations of C4, i.e. £ = E • E,
(36a)
while we find in all other cases that F decomposes into two inequivalent representations of C4h, i.e. F= Eg~E
u.
(36b)
This block-diagonal decomposition is achieved in the (parameter-dependent) basis given in (23b).
5. Stability of the bifurcating solutions In this section we analyze the stability of the (small amplitude) limit cycles q~p(t, ~) = q~p(t + T ( e ) , ~) found in section 3, because only stable solutions have physical significance. For this purpose, we study the dynamics of small deviations from this state, ~ ( t , e) -----cbp(t , e) = ~ ( t , e),
(37)
K.-H. Hi~ck et al. / Degenerate Hopf bifurcation
349
and linearize the equation of motion (11) about ~p(t, e), which yields 8q~(t) = L(~p(t, e))" 8q~(t).
(38)
The (real) linear operator L = WB]#~is periodic in time with period T(e) = 2~r/o:(e) and o:(e) given in (21) *. We therefore apply the Floquet theory [8] to solve (38). With the ansatz 8~(t)=~(t)e
°',
~(t)=~(t+T),
(39)
eq. (38) is converted into an eigenvalue problem for the Floquet exponent o = Re o + i Im e,
(
d
_ol).~(t)
=0,
(40)
where I m a is restricted to the first "Brillouin zone", i.e. - lo: _< I m o < ½o:. Since we deal with an autonomous system, the "Goldstone" mode 8~(aM)(t) = ~p(t) = ~(~M)
(41)
is always a solution to (8) associated with the Floquet exponent o~ra = 0. The stability of a limit cycle Op(t) is then established if all other exponents determined by (40) have negative real part. Since L(t) and ~(t) are periodic in time, they may be represented by a Fourier series L(t) = )"~Lne - i ~ ' ,
~ ( t ) = E ~ , e -i~°'',
(42)
with o: = o:(e). Inserting the expansion (42) into (40) yields an algebraic equations for the Fourier components ~,,
E L,-~,-r - (a -
ino:)~, = 0.
(43)
r
The perturbation expansion (18) of the periodic solution ~p(t, e) induces a corresponding one for the linear operator L, i.e.
L.= E
ekL(nk),
(44)
k>_O
and from the results of section 3 we derive the general structure of L up to e2: L =
L~ ) + e2(L~ ) +
L(22)e -2i°:t + --2'-/(2) , 2 i ~ , }.
(45)
With L(o°)= Oo|, L~) real and L~½= (L~)) *, L~) and L(~) depend on the specific limit cycle under consideration. *Since ~k (t + T/2) = -q~p(t) and L is even in ~v,Lp actually has period T/2. It is more convenient, however, not to take this fact explicitly into acount, We only have to note that the Fourier expansion of L in (43) below contains only even terms (see eq. (45)),
350
K.-H. HSck et aL/ Degenerate Hopf bifurcation
Consequently, we also expand ~ and o into power series of e, ~,=
Z ~k~}k),
0 = Y'~ e%(k),
k>_O
(46)
k>_O
where the unperturbed Floquet exponent has to be taken as zero, o(°) = 0, since the eigenfrequency o~o of the stationary state is at a reciprocal lattice vector of the time lattice defined by the limit cycle. The calculation, whose details are given in Appendix C, then yields in the lowest order t(t)
= t t 0) e-i~0t.-~ 4(._01ei~0 ',
(47)
where the Fourier components ~o) are found to be linear combinations of the orthonormal vectors (ui } given in (23b): I~°) = bl(u , + iuz) + b3(u 3 + iu4),
(48)
~0~ = b2(u 1 - iu2) + b 4 ( u 3 - iu4). (For convenience, we have defined u3(p, ~) = u l ( - p , ~o+ ~r/2) and u4(o, ~) = u z ( - p , ep + ~r/2).) Furthermore, we find that the first nonvanishing contribution to the Floquet exponents is of the order e2. It is determined together with the coefficients b i from the solubility condition for the inhomogeneous second-order perturbation equation,
L~). ~ o ) + L~). ~ = (0(2)_ i~(2)) ~o), L~). t(_o) + L~), . ~0) = (0 (2) + i0~(2))t(ol.
(49)
These coupled equations can be cast into an eigenvalue problem M" + = a(z)qJ
(50a)
b2
(50b)
(hi/
with
~b=-
b3
b4 and the matrix IVl given in Appendix C. As expected from symmetry (see end of section 4), the matrix IVI becomes block-diagonal in the basis ui(o, q~) for classes II-IV. For class I, on the other hand, such a decomposition is not expected, but it turns out that the matrix elements M31, M32, M41, M42 vanish. For each class of solutions we solve the eigenvalue problem. The results are listed in table III. We now discuss the conclusions we can draw from the explicit expressions for the Floquet exponents oj: (1) For bt(2) > 0, all solutions are (at least marginally) stable against small perturbations lying in the plane of the limit cycle under consideration. These "in-plane"-deviations are the phase mode (Goldstone mode) and the amplitude mode (Hopf mode), associated with the real Floquet exponents o 1 = O~M = 0 and (I2 = O H =
--2~ (2) <
0.
K.-H. H6ck et al./ Degenerate Hopf bifurcation
351
Table III The non-trivial Floquet exponents and the border of stability for the stable solutions. Class
o3,4 (2)
u
- ( c [ - c~) _+[-(C('
-
C~') 2 +
C; 2
02(2)
0 (2) = 0 b
2c(
c~ 2 + G '2 = (c~ - c~) 2 + ( c ; ' - c d ' ) :
+ C;'21 ~/2
}[(c/- c~) - 3c~1
III
[ c ; - ½ ( c ~ - c~)] 2 + [ c ; ' - ½ ( c ; ' - c ~ ' ) ] 2
~+~+~ +_ ~ [ ( c ;
IV
l[(c;
- c~ + c ; ) + 8 c ; ' ( c 7
- c~) + 3c;1
+_~ [ ( c ;
- c~ - c;)
= ¼[(c; - c~) 2 + (Cf' - cf) 2]
- c ~ ' - c ~ ' ) ] ~/~ It;
- 8c~'(c;'
- Q'
- c;')U
2
+ l(c;
- c~)] 2 + [ c ; ' + ½ ( c ; ' - c~')] 2
1 t = c~t + ~1 = ~[(C1 - c-;) 2 + ( c 7
c~') 2 ]
aa2(2) o H -- 2/~(2). bBorder of stability. =
=
(2) The stability is therefore determined by the (real or complex) Floquet exponents 03 and o4 associated with out-of-plane deviations. In particular, we find for class I, that the real part of one of these exponents is always positive. Therefore these solutions are unstable. It turns out, that the solutions of classes II-IV are stable within a limited region in the parameter space
(q}. We now briefly describe the types of instabilities which occur when the coupling constants { C/, C/' } are changed in such a way that the boundary of the stability region is crossed. If a real Floquet exponent changes sign, then the original limit cycle becomes unstable against a pitchfork bifurcation of a pair (generally: a multiple) of new limit cycles with the same period. If a pair of complex conjugate Floquet exponents crosses the imaginary axis with nonvanishing imaginary parts, then the original limit cycle becomes unstable against the formation of a torus. It should be noted, however, that the validity of the perturbation theory is limited to Floquet exponents of order e2. Therefore, the two frequencies of the motion on the torus differ only by an infinitesimal amount. For the same reason, an instability against period-doubling requiring Ime = -T- ½~00 cannot occur within the region of validity of the perturbation expansion.
6. C o n c l u s i o n s
In this paper we have analysed an example of a degenerate Hopf bifurcation associated with a four-component order parameter which is supposed to transform according to a single, physically irreducible representation of the symmetry group of the system. We have shown that fairly general conclusions can be drawn from symmetry considerations. The transformation behaviour of the order parameter completely specifies the structure of the linear operator which determines the form of the destabilizing mode. The solutions bifurcating at the critical point form stars of symmetry-equivalent limit cycles, each star being characterized by a distinct low-symmetry group. The existence of these solutions for the particular group G = Q A C2 is proved by explicitly solving the
K.-H. Htck et aL / Degenerate Hopf bifurcation
352
bifurcation equations. The stability analysis yields the result that the solutions with the lowest low-symmetry group are unstable, while all others (with low-symmetry groups of higher order) are stable. It is worth pointing out that the orientation of the bifurcation planes in OP-space ~2 associated with the stable solutions are determined by symmetry alone, while for the unstable solutions the orientation depends on the coupfing constants of the leading higher-order terms in the equation of motion. Though we do not known of any experimental data, such a bifurcation behaviour is expected to occur for the particular case G = Q A Cz, for example, in crystals with orthorhombic, tetragonal or cubic symmetry at certain high-symmetry points of the Brillouin zone. A simple example is represented by the (symmorphic) tetragonal space group I 4 / m (C~h) [7], where the complete space-group representation resulting from the P-point, k = ¼ (1,1,1), of the Brillouin zone with small representation of type E has Q A Cz as matrix group. Other representative space groups where either a small or a complete space-group representation leads to the matrix group Q A C2 are listed below: I) Orthorhombic systems 1212121 Pcca Pbcm Pbca
(D9), D8 (Zh), ~D n~ \ 2hi, is (D2h), (D 2h 27~J,
Ibca
Pnna Pccn Pbcn Pnma Imma
(D 26h ) ' (D10~ ~ 2hj, /DX4"~ \ 2h J, (D16-~ ~ 2h), (D~).
P4/ncc
(DSh),
F4132
(04).
II) Tetragonal systems 14122 P42/ncm
(Dam),
(D~°).
III) Cubic systems Pa3
(T6),
Acknowledgement We are indebted to Dr. J.W. Swift for letting us use his Ph.D. thesis [9] and a paper [10] prior to publication.
Appendix A Number of equivariants
The number of equivariants of order k is given by 1 N ( k ) = ~-~ E X t r p ( g ) x ( g ) ,
(A.1)
g~G
where Xtrl~ denotes the character of the symmetrized kth power of the representation /" and x ( g ) =
K.-H. HOck et aL/ Degenerate Hopf bifurcation
353
Tr F(g). X[rl~(g) is computed by X [ F ] k ( g ) --
E xi l(g--~)"-'xi'~(+k) it,J2 ..... ik 1'ti1!2'2i2! ... k'kik!
(A.2)
with the constraint Z~=tl. ik = k. Thus, for the first few k we have
Xtrl3(g) = {x3(g) + ½x(g)x(g 2) + ½x(g3),
(A.3)
X[r]'(g) = ~ x 4 ( g ) + ¼xZ(g)x(g z) + ½x(g)x(g 3) + {x2(g z) + ¼X(g4) •
Appendix B
Perturbation expansion for limit-cycle bifurcation Substituting the expansions (17a) and (18) into the equations of motion (11) and equating the coefficients of successive powers e k to zero yields the following set of perturbation equations: El:
- ~0[I + in1] "~,~1) --
=0,
•
lit
n
k k e2k+l: -- OOO[|"4-in1] °l/brn(2k+l)-~E fig{2j) q-inO0(2j)] °Cb(rgk-2j+l)-{- E R(2k+l'2j+l)' j=l j=l
(B.1)
Here R~,k' 1) denotes the contribution of the lth order equivariant O (o(~,) to the order e k of the equation of motion for the n th Fourier coe~cient. We now show that eqs. (B.1) may be solved successively to arbitrary order. For n = +1, the operators occurring on the left-hand sides are singular in the directions of the eigenvectors fl, 12 and I3, 14, respectively. We therefore decompose the vector space 9 into invariant subspaces 12(+) spanned b y / t , I2, and 12(-) spanned b y / 3 , I4, I2 =/2 (+) • 12(-),
(B.2)
with corresponding projection operators P(-+ ), stich that I + in1 =
i[(n - 1)P {+) + (n + 1)P(-)].
(B.3)
354
K.-H. HOck et al./ Degenerate Hopf bifurcation
In order e~, one finds
0(?= 1__[ v~- X(1)1"1+
Y(l>h] E
(B.4)
,/,.(~)=0 (n4= +1), with complex amplitudes X ~1)= A exp ( - ia), Y(]) = B exp ( - ifl). In all higher orders, the Fredholm alternative yields the solvability condition
P(+)"
r(12k+1) = 0,
(B.5)
where ¥(2k+1) denotes the fight-hand sides of (B.1). If (B.5) is satisfied, the solution is obtained in the form
= 1
[
+ r
+ % ] + "i p ( _ ) . r(2k+l),
(B.6)
n
tO0
with complex amplitudes X (2k+l), y(2k+ l) It remains to show that eq. (B.5) together with eqs. (20a) and (20b) determine successively all coefficients x ( 2 k + l ) , y(2k+l), a n d ~t(2k), 03(2k).
In order e 3, the solvability condition (B.5) takes the form [, (2) + iw(2))[ X(X)fl + yO)fz ] + p(+), R~3,3)= 0.
Inserting the expression for R~ 3'3) following from the eqs. (9) to (11) yields the eq. (21a, b) whose solutions are discussed in the text. We now assume that eqs. (B.5) have been solved up to order ezk- ~, and that the coefficients/~(t) and co(t) (l_< 2 k - 2) as well as the ~t) (1 < 2 k - 1) have been determined, with the exception of the complex amplitudes X (2k-1), y(2k-1). Then, the solvability condition (B.5), which represents two complex equations, together with the two real eqs. (20a, b), constitute a system of six real equations which are linear in the six unknowns R e X (2k-1), Im X (2k-l), Re Y (2k-D, I m Y (2k-1), ~(2k), 60(2k). Since the determinant is generically nonzero, the system has a unique solution. Thus, by induction we have shown that the bifurcation equations (B.1) may be solved successively to arbitrary order.
K.-H. Hgcket aL/ DegenerateHopfbifurcation
355
Appendix C
Stability analysis of limit-cycle solution By inserting the expansions (46) into (43), we obtain the perturbation equations
,Oo[t +
=o,
(ca)
el:
~o[I + in1] °~(n1) -~-g(1)~(n0),
(C.2)
ez:
oa0[l+inl]'~{, z)=-me<*)o ~, - L ~ ) ' ~ ° ) - L ~ ) ' ~ ° - ) : - L ~ ½ "$m)-.+2+ [°(2)-in°a{2~]'~ m.
(C.3)
Eq. (C.1)has nontrivial solutions for n _+ 1, yielding in lowest order the form of the perturbation given in eq. (48). Eq. (C.2) yields o 0) = 0, and eq. (49) arising as the solubility condition for (C.3) are finally transformed into the eigenvalue problem (50) where the matrix M has the structure
[ Mn
M12 I M13 MI, I
/M*
M * ', M * q _
M,,
M*|
/
(C.4)
)1433 M3a,/
M~'~ M3~ ,, M3'I
M3']]
with matrix elements given by Mu= }[2C/-(1-02)F
'] - 2 1 2 C ; , - 0 - 0 2 ) F " ] ,
Mz~= ½12c;-(1-02)r
,] +
½12c{,-(1-02)F
,,] = M ~ ,
M33= - ½12(C/- C ; ) - 3(1 - p 2 ) r ' ] + 2 [ 2 ( C { ' - C~')- 3(1 - p2)F"], M34 = - ½ [ ( 1 - p2)F' + 2C3'cos4ep - 2C~' singq0] + 2 [ ( 1 - p2)r"+ 2c~' cos4q0 + 2c3'p sin4q~ ] , M~4 = ½~/1 - O Z [ o F " - s i n 4 ~ C ; ] M~3 = - V/1 - Oz [ o F " +
- 2 ~/1 - O z [ O r ' +
sin4~C3"],
sin4,~C3'] - i~1 - O~ [ o F ' - sin4~C3"],
M31 = ~1 - O2 [ p F " - sin4q)C3'] + i l l -
p 2 [OF'+ sin4qoC3"],
where
F' = C { - Cg - C; cos 4q0,
(1
F " = C [ ' - C ~ ' - C3" cos 4%
(1 - p 2 ) r " = - 2 ~
-
p2)/-,,
=
2C{, (2) + 2C{'.
2/~2) +
(c.5)
For 0 < O2 < 1, the additional relations derived from (21a, b) hold,
pF'+C~'sin4cp=O, such that M14 = M31 = 0.
oF"-C~'sin4ep=O,
(C.6)
356
K.-H. H6ck et aL/ Degenerate Hopf bifurcation
Solving (50), we obtain
a~=O,
a2 = - 2 / t (2)
(c.7)
and o3/, = ReM3~
+__¢-
(tin M . ) 2 + IM~,I 2 .
(c.8)
For classes II-IV, the last equation directly reveals from the block-diagonal form of M, whereas for class I the relations (C.6) have to be used.
References [1] K.-H. HiSck, P. Jordan and H. Thomas, J. Phys. C (1987), in press. [2] See, e.g., .I.E. Marsden and M. McCracken, The Hopf bifurcation and its application, Applied Mathematical Science 19 (Springer, New York, 1976). [3] M. Golubitsky and I. Stewart, Arch. Rat. Mech. Anal. 87 (1985) 107. [4] S. Thiesen and H. Thomas, Z. Phys. B65 (1987) 397. [5] J.P. Serre, Linear Representation of Finite Groups (Springer, New York, 1977). [6] G.Ya. Lyubarskii, The Application of Group Theory in Physics (Pergamon, Oxford, London, New York, Paris, 1960). [7] C.J. Bradley and A.P. Cracknell, The Mathematical Theory of Symmetry in Solids (Clarendon Oxford, 1972). [8] G. Ioos and D.D. Joseph, Elementary Stability and Bifurcation Theory (Springer, New York, 1980). [9] J.W. Swift, Ph.D. Thesis, Univ. California, Santa Cruz (1984). [10] J.W. Swift, Four coupled oscillators: Hopf bifurcation with the symmetry of the square, submitted to: Dynamics and Stability of Systems (1986).