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 0375-9601/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved. 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 19 Volume 159, number 1,2 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. 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