Direction-reversing traveling waves

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PhysicsLetters A 159 (1991) 17-20
North-Holland
PHYSICS LETTERS A
Direction-reversing traveling waves
A.S. L a n d s b e r g a n d E. K n o b l o c h
Department of Physics, Universityof California at Berkeley, Berkeley, CA 94720, USA
Received 29 March 1991; revised manuscript received 22 July 1991; accepted for publication 31 July 199 I
Communicatedby D.D. Holm
A simple mechanism for generating traveling waves which reverse their direction of propagation in a periodic manner is presented. This mechanism is generic in systems possessing 0(2) symmetry, and corresponds to a codimension-one symmetrybreaking Hopf bifurcation from a circle of nontrivial steady states.
entire
I. Introduction
circle
of nontrivial
steady
states with
I Z l '~ O (].gI/2 ), and parameterized by the spatial phase
The symmetries of a physical system provide a key
to much of its dynamical behavior. As a consequence
equivariant bifurcation theory has met with remarkable success in a wide range of physical applications,
including fluid dynamics, laser physics, and chemical oscillations. (See, for example, refs. [1,2].)
In this note we consider physical systems depending on a parameter/~ that are translation and reflection invariant in one dimension, and subject to periodic boundary conditions. Such systems have O (2)
symmetry. When a trivial (i.e., O (2) -invariant ) state
loses stability a t / , = 0 to a steady state with nonzero
wavenumber k (i.e., the instability breaks 0 ( 2 )
symmetry), the steady state may be written in the
form
~tr/(X) = zeikx+ z e - i k x + O (Z 2 ) ,
( 1)
where z(/,) is the complex mode amplitude. The approach to this state is described by a dynamical
equation whose structure is restricted by the requirement that it commutes with the standard action of
0 ( 2 ) on C,
z~eiktz,
Z--,~.
(2)
Thus
dz
d--~= f ( I z l 2 ' ~ ) z "
(3)
It follows that the bifurcation a t / , = 0 gives rise to an
O = arg (z). Owing to the Z (2)-equivariance of the
problem, eq. (3) and others like it [3,4] possess a
real, reflecfion-invariant subspace. We may therefore classify solutions of such equations according to
their symmetry type. In the first category we have
those solutions which are reflection symmetric. In
the terminology of refs. [3,4], these include, for example, steady states (SS) and standing waves ( SW );
since the group action of SO (2) generates an entire
family of reflection-invariant subspaces these solutions are not isolated (cf. eq. ( 3 ) ) . The second class
of solutions consists of those which break the reflection symmetry. These correspond to various types of
traveling waves.
In this note we examine a particular instability associated with the circle of steady states, and describe
how it gives rise to a new type of traveling wave which
we call a reversing traveling wave (RW). This instability is a H o p f bifurcation from a circle of nontrivial steady states which breaks reflection symmetry. Steady state symmetry-breaking bifurcations
from a circle of SS are well understood, and give rise
to (nonreversing) traveling waves ( T W ) [5-7];
those from a circle of SW produce two-frequency
(quasiperiodic) waves called modulated traveling
waves ( M W ) . In the absence of translation invariante the same bifurcations would give rise to a pair
of asymmetric steady states, and a pair of asymmetric oscillations, respectively. The RW described
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17
Volume 159, number 1,2
PHYSICS LETTERSA
below are a natural generalization of these ideas.
30 September 1991
cation is described by the normal form [ 6 ]
dx =2x+ x3
(6a)
dO
-x.
dt
(6b)
dt
2. Derivation of the normal form equations
Consider an O (2)-equivariant dynamical system
on C n. Such a system is naturally written in terms of
coordinates {zi}, i = 1..... n, each of which transforms
under O (2) as in (2) for some ki. The presence of
the one-parameter translation symmetry SO (2) enables us to reduce the order of the system by one [ 8 ]
through the introduction of canonical coordinates (r,
O, w~, w2.... , w~_ ~), where O specifies the spatial
phase of the pattern. Owing to marginal stability of
all patterns with respect to translations, the O coordinate must decouple from the rest. The dynamical equations then take the general form
dr
Thus solutions with x = 0 represent reflection-symmetric states that do not drift (i.e., SS), but those
with x # 0 represent nonsymmetric drifting states
(TW).
We now examine the case where the steady state
loses stability via a H o p f bifurcation which breaks
the reflection symmetry, i.e., the polynomial q(s) has
a factor s2+o92. The center manifold is then threedimensional, and a center manifold reduction can be
performed to reduce eqs. (4) to the form
(4a)
(dx/dt~_(O
dy/dt]-\o9
dt -g(r, w),
(4b)
dO =f3( x , Y) mod 27r,
dO
=h(r, w),
dt
(4c)
where, owing to the remaining reflection symmetry,
dt =f(r, w) ,
dw
--
where w= (w~ ..... wn_ ~) are complex amplitudes and
r, O are real. The family of reflection-invariant subspaces is distinguished by {wi} real for all i = 1, ...,
n - 1 . The drift velocity of the pattern, dO]dt=-O,
must vanish in these subspaces. The Z(2)-equivariance of the equations manifests itself as equivariance under the operation (w, ..... w,_ 1, O) ~ (Wz, ...,
w._,, -0).
A real fixed point of the reduced system (4a,b)
corresponds to a circle of steady states in the original
problem. To investigate the stability of these solutions we examine the eigenvalues s of the fixed point.
The characteristic equation factors as follows:
sp(s)q(s) = 0 .
(7a,
y)]
(7b)
f, ( - x , - y ) = -f~ (x, y ) ,
f2( --X, --y) = --f2 (X, y ) ,
A( - x , - y ) = -f3(x, y ) .
(8)
Here, the coordinates (x, y) represent the components of the wave which break the reflection symmetry. Note that the motion in the reduced center
manifold (7a) is simply that of a standard H o p f bifurcation. However, since dO~dr represents the drift
velocity of the wave, the oscillatory motion produced by the H o p f bifurcation induces a periodic reversal in the direction of propagation of the wave.
(5)
The zero eigenvalue originates from the neutrally
stable direction associated with translations (eq.
(4c)). The second factor determines the stability of
the solution to perturbations within the reflectioninvariant subspace, while the third describes stability out of the subspace. In the case when q ( 0 ) = 0 ,
q' (0) # 0, i.e., a steady-state symmetry breaking bifurcation takes place, the dynamics near the bifur18
%o9)(~) + (f2~((x: y) ~ ,
3. An example: the triple zero bifurcation with 0(2)
symmetry
A particularly simple example of the above instability arises in the triple zero bifurcation with O (2)
symmetry. Assuming that the linearization of the bifurcation is nilpotent, it is described by the equations, truncated at third order,
Volume 159, number 1,2
PHYSICS LETTERSA
30 September 1991
dzl
dt --z2,
(9a)
dz2
dt =z3,
(9b)
dz3 =2Zl + 922 +~]z 3 + a [ z 1 [2Z 1 +b[z 2 [2z l
dt
+clz, 12z2+dlz112z3 +ez2~ +fz2~
+ gz2 z~l + h [z2 ]Zz2 +jzl ~2z3 + kZl z2~
+/Iz212z3 + mz2~ ,
(9c)
where z~, z2, Za are complex amplitudes. These equations are equivariant under the following action of
the group 0 ( 2 ) :
SO(2): (zl,z2, z3)~(eiOzl,ei~z2, ei~'z3),
Z(2):
(zl,z2, z 3 ) ~ ( ~ , ~ , ~ ) .
(10a)
(lOb)
For purely illustrative purposes we restrict to the
simple case (b, c ..... m) =0. The stability of the circle of steady states at [Zl [2= -2/a, z2=z3=0 is determined by
S(S3--qS 2 - 9s+ 22) ( s 2 - q s - 9) = 0 .
( 11 )
A symmetry-breaking Hopf bifurcation occurs for
r/=0, 9<0. A transformation to canonical coordinates followed by a center manifold reduction near
this instability yields equations of the form (7) with
X=½ Im(z2/zt),
1
y=-- - ~ I m ( z a / Z l ) ,
to=x~--g.
(13)
where
A = N/~-- 2
a'
B=
/ 4(993-22)
~/
9-~g2
q.
Fig. 1. A reversingtravelingwave from eqs. ( 13 ) and (14) with
()., o, t/) = ( -0.075, -0.0015, 0.0001), k= 3.14, Om=02 =0.
1. The wave reverses its direction of propagation with
a half-period ~ / x ~ - v and is strictly periodic, in contrast to the MW mentioned in section I. Note, however, that the steady state solution at q = 0 is already
unstable to perturbations within the invariant subspace, so that the resulting RW will not be stable.
Stable RW require the retention of additional terms
in (9c), in particular, the z 2 ~ term. A detailed discussion of the normal form (9) will be given
elsewhere.
In contrast to the reversing ("blinking") waves
observed in binary fluid convection, the RW are
present in an O (2)-symmetric system and do not require breaking of translation symmetry for their existence [ 9 ].
(12)
Sub/supercriticality of the bifurcation is governed
by the sign of 2. Near onset, [~/[ << 1, the streamfunction ( 1 ) then takes the form
7t=Acos[Bsin(x/~-vt+Ol)+kx+02] ,
X
(14)
Here, q represents the bifurcation parameter and 0~,
02 determine the temporal and spatial phase of the
wave pattern. The solution (13) is illustrated in fig.
4. Conclusion
The triple zero bifurcation has a number of applications in fluid dynamics. A particularly extensive
study of these equations by Arneodo et al. [ 10 ] and
their application to rotating thermohaline convection [ 1 1 ] has focused on the dynamics in the reflection-invariant subspace, i.e., on steady states, and
on standing waves and their transition to chaotic
standing waves via the Shil'nikov mechanism. Indeed in refs. [ 10,1 1 ] it is shown that such chaotic
standing waves occur arbitrarily close to the initial
bifurcation in the (2, v, q) parameter space. This
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PHYSICS LETTERS A
conclusion relies on a scaling that reduces eqs. (9)
to the case where all the coefficients except a vanish,
and thus holds in a certain cone in the parameter
space. In the more realistic case when periodic
boundary conditions are imposed these states may
be unstable to traveling wave disturbances. We have
shown that in the cone studied by Arneodo et al. the
symmetry-breaking H o p f bifurcation does not lead
to stable RW near such a bifurcation. However, in
other regions of the parameter space near (0, 0, 0)
the full equations (9) apply and stable RW are
possible.
One may speculate as to what other systems may
exhibit direction-reversing traveling waves. A necessary condition is that the dimensionality of the
space perpendicular to the reflection-invariant subspace must be at least three. This allows for the two
dimensions subsumed by the H o p f bifurcation and
the additional dimension associated with neutrally
stable translations. Although this condition is satisfied by the truncated modal equations governing
convection in a rotating layer [ 12 ], convection in a
horizontal or vertical magnetic field [ 13 ], and lasers
with a saturable absorber [ 14 ], these systems do not
exhibit the necessary instability.
20
30 September 1991
Acknowledgement
This work
DMS8814702.
was
supported
by
NSF
grant
'References
[ 1 ] M. Golubitsky, I. Stewart and D.G. Schaeffer, Singularities
and groups in bifurcation theory, Vol. II (Springer, Berlin,
1988).
[2] J.D. Crawford and E. Knobloch, Annu. Rev. Fluid Mech.
23 (1991) 341.
[3] E. Knobloch, Phys. Rev. A 34 (1986) 1538.
[4] G. Dangelmayr and E. Knobloch, Philos. Trans. R. Soc. A
332 (1987) 243.
[ 5 ] J. Greene and J.-S. Kim, Physica D 33 ( 1988 ) 99.
[ 6 ] E. Knobloch and D.R. Moore, Phys. Rev. A 42 (1990) 4693.
[7] M. Krupa, SIAM J. Math. Anal. 21 (1990) 1453.
[8] G.W. Bluman and S. Kumei, Symmetries and differential
equations (Springer, Berlin, 1989 ).
[9] G. Dangelmayr and E. Knobloch, Nonlinear evolution of
spatic~temporal structures in dissipative continuous systems,
eds. F.H. Busse and L. Kramer, NATO ASI Series B, Vol.
225 (Plenum, New York, 1990).
[10] A. Arneodo, P.H. Coullet, E.A. Spiegel and C. Tresser,
Physica D 14 (1985) 327.
[l 1 ] A. Arneodo, P.H. Coullet and E.A. Spiegel, Geophys.
Astrophys. Fluid Dyn. 31 (1985) I.
[ 12 ] E. Knobloch and M. Silber, Geophys. Astrophys. Fluid Dyn.
51 (1990) 195.
[ 13] E. Knobloch and D.R. Moore, Eur. J. Mech. B ( 1991 ), in
press.
[ 14 ] G. Dangelmayr and M. Neveling, J. Phys. A 22 (1989) 129 I.
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