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Local bifurcations on the plane with reversing point group symmetry
Pergamon 0960-0779(93)EOO22-4
Local Bifurcations
on the Plane with Reversing Point Group Symmetry .I. S. W. LAMB and H. W. CAPEL
Institute
for Theoretical
Physics, University
of Amsterdam, Netherlands
Valckenicrstraat
65, 1018 XE Amsterdam.
The
Abstract-Local bifurcations of fixed points, being symmetric with respect to a reversing point group consisting of mirrors, two- and fourfold rotations, are studied for dynamical systems in W These bifurcations are of interest in connection to symmetry breaking phenomena, in particular with respect to the occurrence of dissipative features in (weakly) reversible systems.
1. INTRODUCTION
In the theory of dynamical systems, currently the interest for (reversing) symmetry properties is steadily growing. Some two decades of studies on the consequences of symmetries for the global and local phenomena in dynamical systems have passed [l-3]. However. a systematic treatment of reversing symmetries in this context has started only recently [4-71, after some years in which the interest for reversing symmetries was mainly focussed on involutory ones [S-12]. The emphasis on involutory reversing symmetries is related to the presence of generalized time-reversal symmetries. Generalized time-reversal symmetries do not only exist in Hamiltonian systems in the absence of an external magnetic field so that the Hamiltonian is an even function of momenta, but also more generally in systems of coupled first-order differential equations if there is an involution which changes the direction of the motion. The discrete time version of this, i.e. a map, has generalized time-reversal invariance if there exists an involution which changes the map into its inverse. In this way any reversible map can be considered to be the product of two involutions. Interesting examples of generalized time-reversal symmetries are the externally injected laser system studied by Politi et al. [13], and the Nose-Hoover dynamics which has been successfully used in molecular dynamics calculations [15, 161. Both systems show the presence of periodic and/or strange attractors and repellors and are therefore essentially different from Hamiltonian systems. A general method to obtain reversible mappings that are not measure preserving has been discussed in [12] together with a variety of examples. Mom recently, it has become clear that apart from generalized time-reversal invariance there can be other interesting reversing symmetries which are generated by transformations of (even) order larger than two. Nice examples are the reversing symmetries in relation to the treatment of stochastic webs [6] and the fourfold reversing symmetry in Arnold’s cat map [17]. In the study of the influence of symmetries on dynamical systems, local bifurcations have received a lot of attention [l-3, 18, 191. In this paper we intend to give a contribution to a systematic inclusion of reversing symmetries in local bifurcation theory. We will discuss 271
‘72
J. S. W. LAMB
and H. U’. CAPEL
local bifurcations around fixed points of dynamical systems in R2 that are symmetric with respect to a reversing point group consisting of mirrors, two- and fourfold rotations. The special treatment of bifurcations in systems with a reversing symmetry can be of Interest for various reasons. First of all it has been known that conservative systems and reversible systems in R2 have many properties in common, such as the KAM-theorem. cf. Arnold and Sevryuk [lo], and the (conservative) values of the Feigenbaum constant S and scaling i’actor N in the period doubling of symmetric fixed points [12]. The dissipative features in non-measure-preserving reversible systems can arise by asymmetric fixed points moving in from infinity, but more interestingly also after bifurcations at symmetric fixed points [5. 12-141. Furthermore, it will turn out that certain bifurcation processes that mny occur accidentally for special choices of parameters in the absence of (reversing) symmetries become generic bifurcation processes in the class of systems having particular (reversing) symmetries. The bifurcations that we will consider in this paper are codimension one bifurcations, i.c. bifurcations occuring generically in one-parameter families of dynamical systems possessing a fixed point with the desired (reversing) symmetry properties. WC: will start analysing continuous time flows. Thereafter WC will discus3 discrete time maps,
2. CONTINUOUS TIME FLOWS
Let us consider a planar vector field F, defining a dynamical system by means of the ordinary differential equation (ODE) Ax = F(x), dt
x E R2.
Purthermore, Let U be a diffeomorphism of the plane. Then, by definition. symmetry of (1) if
(1, L’ is a
and a reversing symmetry of (1) if - $[u(x)]
= F 0 U(x).
(3)
All symmetries and reversing symmetries of (1) form a group under composition. This group is called the reversing symmetry group of (I) 141.The composition of’ a symmetry and a reversing symmetry js a reversing symmetry, whereas the composition of two symmetries or two reversing symmetries produces a symmetry. A dynamical system possessing a reversing symmetry is called weakly rever.~ihke in general and reversible in the case where it possessesan involutory reversing symmetry [ 101 (i.c. a reversing symmetry that is its own inverse). Such reversible systems can be regarded as having a generalized time-reversal invariance. e.g. the reversing involution may be a nonlinear transformation. For a rccenr review on reversible dynamical systems. we refer the reader to [12]. Let us focus on the neighbourhond of a fixed (or stationary) point of (1). i.e. a point x,; satisfying F(xO) = 0. Considering a reversing symmetry group of a dynamical system. the
Local bifurcations on the phne
isotropy subgroup TX,, of the fixed point x0 is the subgroup of the reversing symmetry group, such that for all U E XX0we have U(x,) = x0 [3]. The fixed points belonging to the fixed set of at least one reversing symmetry were called symmetric fixed points in [12]. The eigenvalues of the linearized vector field around such points are either zero or come in pairs {A, -A}. All other fixed points have been called asymmetric. Starting from an asymmetric fixed point with eigenvalues (hi} one obtains by application of any reversing symmetry U another asymmetric fixed point with eigenvalues { -&}. So in systems with reversing symmetries every attractor has to be accompanied by a repcllor. Because of the occurrence of various symmetries and reversing symmetries in the dynamical systems to be discussed in this paper we will USCthe work symmetric in a more general sense, indicating explicitly the (reversing) symmetries involved. Around a fixed point the local dynamics can be studied via the Taylor expansion of F around x(,. The object of this approach is to truncate the Taylor expansion at a certain order and obtain in this way a simple and ‘good’ description of the local dynamics, For vector fields of arbitrary dimension, it is far from obvious that a truncated Taylor expansion will give a ‘good’ picture of the local dynamics (see e.g. 1201).This depends on the stability of the truncated expansion with respect to higher order perturbations. However, for planar flows, which will be treated in the present paper, it has been shown that all Taylor expansions can be stabilized by a finite number of terms [Zl]. In connection with the local analysis we are going to perform, it would be convenient if the action of XX,,were linear. In this paper, we assume that XX,,is locally C-conjugate to a linear point group. (This is automatically the case if XC1is a compact Lie group,) In that case there is a smooth tranformation to a coordinate frame in which the action of X,,, on the Nth order Taylor expansion of F around x0, FCN), is linear. In that coordinate frame, we have the relation (J 0 F’“‘(x)
= F(N) 0 U(x),
in case U is a symmetry, and
in case U is a reversing symmetry. Considering a Taylor expansion FCN)possessing already a linear &, things can be made even more simple by transforming the expansion into normal form, i.e. considering a coordinate frame in which most of the Taylor coefficients are zero [22]. WC can remove the Taylor coefficients using polynominal coordinate transformations with linear part being the identity and nonlinear terms of degree 2, 3, . . . . However, in doing so, we would like the (linear) symmetry propertics of FCN)to be preserved. This can bc achieved by using solely symmetry-preserving transformations, i.e. transformations that commute with the (revetsing) symmetries [23]. In this way we are led to a so-called symnrefric rtormal fwrn. In a previous work [S], it was shown that such a symmetric normal form (in case of a vector field and a reversing symmetry of finite order) also can be derived directly from a general normal form, i.e. a normal form of a vector field possessing no (reversing) symmetries. imposing the linear reversing symmetry on this general normal form and satisfying all contraints resulting from it. The extension of this to the case of a finite reversing symmetry group is obvious, since the proof in ]5] is based on the existence of a unitary representation of a cyclic group, and any finite group has a unitary representation. 117studying local bifurcations, we consider local Taylor expansions of the vector fields al-ound fixed points of which the local dynamics is unstable with respect to weak pcrturhations. These perturbations serve as a paramctcrization for local bifurcations. Following terminology from singularity theory, such perturbations are usually called
27.!
J. S. W. 1.AM13 and H. W. CAPEL
unfoldings. Using this approach, we will study the local bifurcations that can occur in a dynamical system with a fixed point possessing an isotropy subgroup consisting of symmetries and reversing symmetries, considering only unfolding terms that are consistent with the isotropy subgroup. Considering local bifurcations of a fixed point x0 with isotropy subgroup XX,,, we are particularly interested in the isotropy subgroup of the fixed points that bifurcate from x,,. Of course also the symmetry properties of periodic orbits, limit cycles, and other objects are of interest. Periodic orbits and limit cycles may be studied on the basis of an appropriately chosen Poincare map, as being fixed points of such a map and its iterates. In this paper we will mainly focus on the symmetry properties of bifurcating fixed points. In the bifurcation processes new fixed points are born and they often have an isotropy group that is different from C,“. In the case that the isotropy subgroups of the fixed points that bifurcate from x0 are true subgroups of x0, we speak of symmetry-breaking bifurcations. Symmetry-breaking bifurcations with a change of stability properties are related to the phenomenon of spontaneous symmetry breaking. However, also the opposite phenomenon of symmetry increasing bifurcations may occur [24,25]. In this paper, we will restrict ourselves to local point groups possessing only mirrors (M), two- and fourfold rotations (R, and f$J. Other point groups require separate calculations and will not be treated here. Without loss of generality we will take the fixed point under consideration to be the origin. Moreover, we want at least one of the generators to be a reversing symmetry. This implies that we will deal with seven different point groups. These groups and their generators are given in Table 1, specifying also which generators are reversing symmetries and which generators are not. For the description of these groups, we borrowed the Shubnikov-Belov notation from magnetic point groups (see also [6]). For each of these groups we will discuss bifurcations of the fixed point that are likely to be observed in one-parameter families of dynamical systems. so-called generic codimension-1 bifurcations. In the forthcoming discussion of local bifurcations we will observe the occurrence of several kinds of saddle connections of which the stability is caused by the (reversing) symmetries. In order to clarify the discussion on these connections, we give some essential definitions. A saddle point x0 in R2 is characterized by the existence of a stable manifold W:) and an unstable manifold Wio that are defined as IV& = {x E lR2/lim~,,x = x0}, t--ra
(61
W& = {x E !??‘Ilim
(7)
in which @, stands for the time t evolution operator. Both W& and Wzo are smooth curves Table 1. Reversing point groups and their generators. M denotes a mirror and R, is a rotation over an angle N Group iti,
Reversing generator M
2' 2'm 4'
M M 47 47 KY'2
4’m
R+
3m’
Nonreversing -
& R,, M M
generator
Local bifurcations
on the plane
215
that pass through the saddle. Let x0 and x1 be saddles, then a point y (that does not coincide with x0 and/or x,) satisfying is called a heteroclinic point if x0 + x1, and a homoclinic point if xl1= x1. In the case of planar flows, the existence of a homoclinic (heteroclinic) point implies the existence of a smooth curve of homoclinic (heteroclinic) points that coincides with branches of the stable and unstable manifolds of the concerned saddle(s). Such a curve is called a homoclinic (heteroclinic) saddle connection.” After this introduction, we will start the local bifurcation analysis. We will discuss the reversing symmetry groups in the order of appearance in Table 1, i.e. we start with the groups that possessa reversing mirror. 2.1.
Point group symmetry m’
111the case of a single reversing mirror, that we will take to be the mirror in the x-axis 44,. the linear part of the vector field F1 should satisfy M, 0 Fl = -F, 0 M,,
(3
from which it follows that
corresponding to a hyperbolic or elliptic fixed point if /3r > 0 or 13~< 0, respectively. However, these points are structurally stable with respect to small perturbations that do not break the reversing symmetry and hence do not allow for local bifurcations in the neighbourhood of the origin. In these cases the local behaviour of the reversible flow as displayed by the symmetric normal form is of a conservative (non-dissipative) nature. This means that in the investigation of bifurcations around the origin we can restrict ourselves to the nilpotent case
(11) At a bifurcation, the linear part could also be zero, but this is exceptional and will therefore not be considered here. In the above cases we make use of the symmetric normal forms that were derived in [Sit (12)
and
‘hNotc that in the curves. A discussion .YlXc normal form form with respect to
case of discrete time maps, homoclinic and heteroclinic points usually do not come in smooth of this will he given in Section 3. (12) is identical to equation (67) of [S], whereas (13) is obtained from the reversible normal the reversing symmetry MY (equation (68) of [5]) by interchanging x and v,
J. S. W. LAMB and H. W. L‘APEL
27h
Considering only unfotdings that do not break the symmetry property, we find in the case of (12) a so-called saddle-node bifurcation. This bifurcation occurs in the unfolding
(14) where jl is the bifurcation parameter. (A Iinear term proportional to x in 3 can be transformed away by a shift x H x + const.) ITI case ,ug20< 0 there is a saddle-centre pair (saddle-node) that disappears as soon as jog,, > 0 (see Fig. I j. Both fixed points. resulting from the saddle-node bifurcation, are situated on the line 1’ = 0 and are hence symmetric with respect to M,. Note that fixed points that arc symmetric with respect to a reversing symmetry can be neither repelling nor attracting. Hence we find on the line !: = 0 a true centre that remains stable under all perturbations that preserve the reversing symmetry. whereas in non-reversible two-dimensional tlows a ccntre is structually unstable and will be transformed to an attracting or repelling focus under a small perturbation.
Notice furthermore the homoclinic orbit that surrounds the centre point. The stability of tht: homoclinic orbit arises because of the reversing mirror symmetry. One of the branches of the unstable manifold of the saddle that is situated on the fixed set of the reversing mirror has to intersect the fixed set of the reversing mirror in a point on the other side of the ccntre point for sufficiently small values of the bifurcation parameter. Hence this branch has to coincide with a branch of the stable manifold of the same hyperbolic point, as can be easily verified from Fig. I. The bifurcation diagram is given in Fig. 2. Note that this type of saddle-node bifurcation is similar to the one observed in two-dimensional Hamiltonian vector fields. Unfolding the other normal form, we typically find a pitchfork bifurcation (named after the form of the bifurcation diagram, see Fig. 4):
If $,,3 becomes negative. two fixed points bifurcate off the reversing mirror )’ = 0.
Fig. I. Local phase portrait of
m’ saddle-node hifurcatiorl
s (m') (74 C 3
C: (m’)
Fig. 2. Rlfurcation diagram of the m’ saddle-node bifurcation (Fig. 1). S denotes a saddle and CI a centre. The isotropy subgroup of the fixed points is indicated between brackets.
Local bifurcations on the
plane
z-77
In case fo3> 0, we have a saddle that becomes a centre, sending off two saddles (see Fig. 3(c)). The two saddles are connected via heteroclinic saddle connections. Note that although such intersections are not stable in general [21], they are stable in this cast because of the reversing symmetry. In fact, consider one of the saddles. Near the bifurcation point, one branch of the stable and one branch of the unstable manifold will intersect the fixed set of the mirror (i.e. the x-axis). Therefore, because of the reversing symmetry one finds that these branches must flow into the other saddle and hence must be hetcroclinic saddle connections. These heteroclinic connections form precisely the boundary of the set ol’ closed tlow lines around the center at the origin. Finally, we should remark that Fig. 3(c) has been drawn for the case of point-group symmetry 2172’(that is to be discussed in the next subsection), which is the special cast with f,, = 0 of (15). In the general case with f,] + 0, the flow lines will be not symmetric with respect to the mirror in the y-axis. However, the patterns of‘ local flow lines in both cases are topologically equivalent. In case f,,>< 0, a centre becomes a saddle, sending off two fixed points that arc foci, Because of the rcvcrsing symmetry, one of these foci must be attracting while the other is repelling. Moreover, one homoclinic saddle connection appears (see Fig. 3(a)). Although in ;I general two-dimensional tlow, such a connection would be unstable, here it is stable because of the reversing tnirror symmetry. This is caused by the fact that generically for sufficiently small [c the stable and unstable manifold of the saddle intersect the fixed set of the reversing symmetry (the line y = 0 in Fig. 3(a)) in a point, different from the saddle point itself. Because of the reversing symmetry we then find that this manifold must make a homoclinic connection. From Fig. 3(a) it is clear that this homoclinic connection scrvcs as the boundary of the basin of attraction of the attracting focus. In [13] this type of pitcht’ork bifurcation was used to illustrate the birth and occurrence of attractor-repeller pairs in reversible dynamical systems, cf. [14] for a similar treatment in the case of maps. The results obtained for the hetcroclinic and homoclinic saddle connections above follow immediately from the following proposition. (The proof is simple and will not be given here.) ProposilEon: Consider a reversing symmetry with fixed set $, then I Jf the stuhle or unstable manifold of a saddle, not belonging to the fixed set .Y. immects the fixed set E%then the intersection point is a heteroclinic point. * If the stable or unstable manifold of a saddle, belonging to the fixed set !+, intersects the fked set 9 then the intersection point is a homockc point if this point is not the .snddle itself. ‘The second part of this proposition of a more general theorem of Devaney [8], stating moreover that the homoclinic orbit is the limit of a one-parameter family of a symmetric closed orbits whose periods tend to ~0.
2.2.
Point group symmetry 2m’
In the case of a reversing point group generated by the reversing mirror M, and a twofold rotation around the origin, we are led to consider the following unfolding
J. S. W L.4MB
Fig. 3. Local pitchfork
bifurcations
and H. W. C’APEL
in the case of m’ ((a) and {c)) and 2~’
((b) and (c)).
Local bifurcations on the plaiic
The typical bifurcation here js again a pitchfork bifurcation. However, in this case we have two orthogonal reversing mirrors. The new fixed points are sent off one mirror but remain on the other one. Hence, if fO;< 0 we indeed have a centre bifurcating in one saddle and two centres instead of a saddle and two foci (see Fig. 3(b)). Note also the two homoclinic saddle connections Ihal occur in this pitchfork bifurcation. Their occurrence and stability follo\h tram our proposition, since the stable and unstable manifolds of the newborn saddle must intersect the fixed set of MY for sufficiently small values of the bifurcation parameter. This also gives rise to the fact that the intersections of the (un)stablc manifolds with the fixecl set of MI coincide with the saddle. In cast f,,?> 0 the pitchfork bifurcation is equivalent to the corresponding bifurcation in the case of pll’ (see Fig. 3(c)). The extra symmetry does not play an important role in this bifurcation. Ihe only difference being that in the case of m’ Fig. 3(c) would in general not be symmetric with respect to the mirror in the y-axis. Note that pitchfork bifurcations in which a centre is changed into a saddle and two centres, or a saddle is changed into a centre and two saddles, arc not generic in the case of two-dimensional Hamiltonian vector fields (without additional (reversing) symmetries). However, in the case of reversible ( /?z’) Hamiltonian vector fields these pitchfork bifurcations are generic [9]. In Fig. 4 WC present bifurcation diagrams of the pitchfork bifurcations we discussed in the cases m’ and 2~‘. In this figure the symmetry breaking can be followed easily, because of the isotropy subgroups that have been indicated. In relation with the text and Fig. 3. note that in the cast of Fig. 4(a) the fixed points with isotropy subgroup m’ refer to points on the fixed set of MA (the x-axis). whereas in the case of Fig. 4(b) the fixed points with isotropy subgroup m’ refer to points on the fixed set of Al, (the y-axis). 2.3.
Pointgrolq symmetry 4m’
In case the reversing mirror jW, is extended with a fourfold rotation. WC Uind the bifurcation picture to be changed drastically. The ODE having the origin as a fixed paint and possessing such a reversing symmetry group is given by i = f(X, y),
(17) 1 j = -f(y, X), J’), This implies that the origin is either a centre or where f(x. - JJ)= -f(x, y) = -f(-x. the linear part of the vector field is zero. (The nilpotent linear parts which we considered previously cannot occur here.) Hence, a local bifurcation will be possible only if the linear
(a?
c: [m’]
A (-! s (Ill’)
s I; I S [iri’j -E
R t-1
(’ (111’) (bj
c’ (am’)
s (h’) I: (777’)
c: (iII! 1 s i-1
s (1117 s (Zm’)
(I (h’) s (??,“I
Fig. 4. Bifurcation diagrams of pitchfork bifurcations in the case of m’ and 2~‘: (a) pitchfork in’ (Fig. 3(a) and 3(c)). and (h) pitchfork 2rn’ (Fig. 3(b) and 3(c)). S denoles a saddle. C a centre. A an attractor. and R a rep~llor. The isotropy subgroup of each fixed point is indicated between brackets.
2x1)
J. S. W. LAMB
and H. W. CAPEL
part of the vector field is zero. The first nonvanishing Taylor terms of (17) appear at third order. If we truncate there, we find around the bifurcation point i = py + f&y
+ f,,,3y3,
2 - f?,$, i. p = -/Lx - fo.1. where !L is a small bifurcation parameter. In cast p # 0, the origin is a centre. The type of bifurcation, occurring at p = 0, depends on f2, and fij3. We distinguish two main regimes. They are indicated in Fig. 5. WC will not discuss the marginal cases ]fijll = If_7,1or fo3= 0 here, because they are exceptional. In regime 1 of Fig. 5, WCfind a kind of multifork bifurcation. If /L goes through zero, the centre changes orientation of rotation and sends off four saddles and four centrcs. The new fixed points come in quadruples because of the fourfold rotation symmetry. The centres and the saddles are situated on two pairs of reversing mirrors (see Fig. 6(a)). One pair of reversing mirrors consists of IVY and MY and the other pair consists of M,,, and M,-._,_ i.c. the mirrors in the line x = )’ and x = -y, respectively. Note that the saddles are always on a different pair of perpendicular mirrors from the centres. This leaves us with two possibilities. One of which has been indicated in Fig. 6(a). Each of these possibilities occurs in two of the four regions labelled 1 in Fig. 5. However, since these two possibilities can be transformed into each other by a rotation over n/4 around the origin, these bifurcations are being regarded as equivalent. Note the eight heteroclinic connections that occur in Fig. 6(a). Their occurrence and stability can be easily shown from our proposition, since the branches of the stable and unstable manifolds of the four saddles all intersect fixed sets of reversing mirrors. In regime 11 of Fig. 5, we find a bifurcation of transcritical type. While pitchfork or multifork bifurcations are either supercritical (fixed points are born) or subcritical (fixed points are annihilated), the name transcritical bifurcation refers to the case that there is no net production or loss of fixed points at the bifurcation point. At the bifurcation point, four saddles, sitting on a pair of perpendicular reversing mirrors, are being absorbed at the origin whereafter they are sent off on the other pair of reversing mirrors. The local dynamics before and after the bifurcation are related through a rotation over n/4 and a change of orientation, as shown by the phase portraits in Fig. 6(b). Before and after the bifurcation we find locally four heteroclinic connections since, for every saddle, one branch of the stable and one branch of the unstable manifold intersect the fixed set of a reversing mirror. The other branches of the stable and unstable manifold do not have local intersections with the reversing mirrors around the bifurcation point. The bifurcation diagrams of the bifurcations in the regimes I and II of Fig. 5 are shown in Fig. 7. In connection with the text and Fig. 6 the fixed points with isotropy subgroup m’ refer to points on the fixed sets of M, and My for the saddles (or centres) and the fixed sets of M.LLJand MxyTyfor the centres (or saddles).
Fig.5. Bifurcationregimesof
(18) in the f,,-f,,
plane.
Local bifurcations on the plane
Fig. 6. Local bifurcations in the case of 4m’: phase portraits of (a) multifork (b) transcritical (regime II of Fig. 5) type bifurcations.
(regime I of Fig. 5) and
4 x s (m’) (4
c (4m’)
c (4m’) ---E
C (4m’)
4 x c (773)
4 x s (m’)
(b) 4 x s (m’) x
C (4m’)
Fig. 7. Local bifurcations in the case of 4~‘: bifurcation diagrams of (a) multifork (Fig. 6(a)) and (b) transcritical (Fig. 6.(b)).
2.4.
Point group symmetry 2’ and 2’m
If a fixed point has an isotropy subgroup that contains a reversing symmetry that is a twofold rotation R,, then it is easily verified that this fixed point can neither be a saddle, nor a centre. In fact, the linear part of the vector field around the origin will be zero.
2x2
J. S. W. LAMB
andH.
W. CAPEL
Hence, in the case where the origin is a fixed point, with an isotropy group that contains no nontrivial (reversing) symmetries other than the reversing point symmetry K,, the vector field F must satisfy F o R, = F. Therefore, the first nonzero Taylor terms in the expansion of the vector field appear at second order. Moreover, all second-order terms are allowed. Such a form can be unfolded by adding constant terms to the vector field. This destroys the fixed point at the origin. However, before considering the case of 2’, we will first deal with the slightly Icss complicated case of 2’m, where we take the mirror in the x-axis to be a symmetry. In this case WCfind some restrictions to the second-order terms leading to the unfolding (10) in which p is a small bifurcation parameter. From this unfolding we find two types of bifurcations: double-saddle nodes and transcritical bifurcations. In a double saddle-node bifurcation, two saddles and an attractor-repellor pair collide al the origin and then disappear. Note that because of the reversing twofold rotation a single saddle-node cannot occur at the origin. In the transcritical bifurcations we find that two saddles or two centres collide at the origin and pass through as, respectively, two saddles or an attractor-repellor pair. Let us first consider the double saddle-nodes. They occur if fiofoz> 0. The first type is found in case f2,,g,, < 0: two saddles on the x-axis and two centres on the y-axis are absorbed at the origin (see Fig. S(a)). In case f,,g,, > (I we have an attractor-repellor pair on the x-axis and a saddle pair on the y-axis that collide at the origin and then disappear (Fig. S(b)). In case fZDfo2 < 0 we have a transcritical bifurcation in which a pair of fixed points on one mirror makes a transition to the other one. In case f2,,gll i 0, we find two saddles on the reversing mirror that collide at the origin and proceed on the ordinary mirror (see Fig. 10(a)). In cast fZ,,fi12 > 0, two centres on the reversing mirror collide at the origin and become an attractor-repellor pair on the ordinary mirror (see Fig. 10(b)). Note that the latter change of character of the fixed points during the transcritical bifurcation is due to the fact that the centres move from a reversing mirror on which they are stable to an ordinary mirror on which centres cannot occur. Therefore, the centres have to change into an attractorrcpellor pair. (Saddles are not possible because of the conservation of Poincare index, see e.g. [%I.) In the case that there is only 2’ symmetry, i.e. there is only a twofold reversing rotation but no ordinary mirror, the situation becomes more complicated since there are no conditions on the second order terms of the Taylor expansion. By an appropriate choice of coordinates the unfolding can always be chosen as i = /1 + f&
+ f,,xy + f,,,y?
1 j = g2,x2 i- EhlXY + &12YZ7 where the constant term in jl has been removed by a shift and 1~is a small bifurcation < 0, the above unfolding leads to a complete disappearance parameter. In case g:i - 2g02gZ0 of the fixed point: no new fixed points are born from the origin. However, in case g:i - 2g,,,,g,,> 0, we find similar bifurcations as in the case of 2/m, i.e. double saddle-nodes and transcritical bifurcations. However, in these bifurcations the pairs of centres appearing in the case of 2’m are replaced by attractor-repellor pairs. This leads to transcritical bifurcations in which either a pair of saddles or an attractor-repellor pair
Ix-cal hifurcatiuns on lhc plant
(3)
----
I
Fig. 8. Local phase portraits 01 double saddknodc lxfurcations
in
the caw uf 2’~.
A C-J (4
2 x s (-)
(‘L’) E
@I
(2’7r1)
C
R C-1
h (m)
:!xS(m)
2 x s (rd)
(Ym) 2 x CI (Id)
E
K (,,I)
Fig. 9. Bifurcation diagrams of double saddle-node bifurcation? in Lhe case of (a‘) 2’ (Fig. 12), and (b) 2’171 (Fig. 8).
passes through the origin and double saddle-node bifurcations in which two saddles and an attractor-repeller pair are born at the origin. At this point we will refrain from giving a detailed analysis of the bifurcation types in terms of the six second-order parameters since the expressions that are involved are rather long and not very instructive.
J. S. W. LAMB and H. W. CAPEL
281
(b)
Fig. 10. Local phase portraits of bifurcations of traoscritical type in the case of 2’m
s C-1 (4 s(6)
s (m’) @I S (m’)
sC-1RC-1 sI-) AC-1
A(-) RC-J
S(mlc(4
A(ml
s Cm)
R (4
c Cm’)
Fig. 11. Bifurcation diagrams of bifurcations of transcritical type in the case of (a) 2’ and (b) 2’~n (Fig. 10)
In Fig. 12 we present the phase portrait of a 2’ double saddle-node that is only a slightly perburbed version of the first type of double saddle node in the case of 2’m (see Fig. 8(a)). Note that this phase portrait is topologically equivalent to the second type of double saddle-node in Fig. 8(b). The second type of double saddle-node of Fig. 8(b) is persistent under the breaking of the mirror symmetry. Hence, we find that there is only one type of double saddle-node bifurcation in the cas of 2’, with both saddles having one branch of the stable manifold connected with the repellor and one branch of the unstable manifold connected with the attractor. The bifurcation diagrams of the double saddle-nodes and transcritical bifurcations in the case of 2’ and 2’m are present in Fig. 9 and Fig. 11.
285
Local bifurcations on the plant
Fig, 12, Local phase portrait of a 2’ double saddle-node that is only slightly perturbed from the first type of double saddle-node in the case of 2’~ (Fig. R(a)).
2.5.
Point group symmetry 4’ and 4’m
In the case of reversing symmetry group 4’ with the rotation over 7r/2 as reversing generator the ODE is given by
i = f(x, y),
(21)
i j, = q-y, x),
where moreover f(x, y) = -f(-x, -y). Hence, the first nonlinear terms in the Taylor expansion occur at third order and there are no additional restrictions on these terms. But let us first consider the case of 4’m, where we take the mirror in the x-axis to be a symmetry, giving the additional conditions f(x, Y) = f(x, -y)
= -f(y,
x).
(22)
In this case, taking into account the Taylor expansion up to third-order terms, the local bifurcations around the origin can be studied on the basis of the following unfolding (23) in which p is a small bifurcation parameter. The origin is a saddle for all p + 0. This unfolding gives rise to bifurcations of multifork and transcritical type which occur if the saddle switches stable and unstable directions at p = 0. In the transcritical bifurcation four centres on the reversing mirrors, i.e. the lines n = y and x = -y , pass through the saddle at the origin and change to two attractor-repellor pairs on the x- and y-axis. These bifurcations occur in case f30(f30t- flz) < 0. The phase. portrait of this bifurcation is given in Fig. 13 and its bifurcation diagram is depicted in Fig. 14(a). Multifork bifurcations are found in case f,,,(f,, -t flz) > 0. In the case of 4’m we must distinguish between two types of multifork bifurcations. In case Iflz/ > Ifjo four saddles and four centres branch off the saddle point at the origin. The saddles are placed on the ordinary mirrors, i.e. the x-axis and y-axis, and the centers on the reversing mirrors, i.e. the lines x = y and x = -y (see Fig. 15(a)). The other type of multifork bifurcation occurs in case IfI < (f3,,).We then find four saddles on the reversing mirrors and two attractor-repeller pairs on the ordinary mirrors that branch off the saddle at the origin (see Fig. 15(b)).
.I. S. W. LAMB
Fig. 13. Local phase portrait
and H. W. c’APF,L
of the bifurcation
2 x A (-)
(a)
(b)
Fig. 14. Bifurcation
of transcritical
type in the case of 4’m
2 x R(-)
s (4’)
s (4’)
2 x R (-)
2 x A (-)
2 x c (m’
2 x R (m)
S (4’m)
S (4’m)
2 x C (m’
2 x A (m)
diagrams of bifurcations
of transcritical
type in the case of (a) 4’ and (b) 4'm (Fig. 13).
Note that the first type of multifork bifurcation gives rise to eight heteroclinic saddle connections (see Fig. 15(a)). The saddle connections along the x-axis and y-axis are enforced by the fact that the mirrors in these lines are symmetries. The other connections follow from our proposition because the mirrors in the lines x = y and x = -y are reversing symmetries. If there is only 4’ point group symmetry we have the unfolding i = px + fj($ + fi*X2y -t f,*.Xy2 + fo3y3, (24)
1 p = -py + f& - f,,x2y + f*,.Xy2 - f30y3, in which ,Uis a small bifurcation parameter. As in the case of 4’m the origin is a saddle for all ,U# 0. The type of bifurcations observed here are similar to the ones we discussed in the case of 4’m. In case f30(f30+ f12) + fo3(fo3f f2J < 0 we have transcritical bifurcations and in case f3,,(f30+ f12) + fo3(fo3+ fzl) > 0 we find multifork bifurcations. However, because the reversing mirror symmetries are absent the four centres that were advertised in case of the transcritical and first type of multifork bifurcation in the case of 4’m (see Fig. 1.5(a)) are replaced by two attractor-repellor pairs in the case of 4’. In Fig. 16 we present the phase portrait after a multifork bifurcation with symmetry 4’, only slightly perturbed from the first type of multifork bifurcation in the case 4’m. Note that this phase portrait is topologically equivalent to the second type of multifork bifurcation (see Fig. 15(b)) that is persistent with respect to the breaking of the mirror symmetry.
Local bifurcations
on the plane
(b) -IFig. 15. Local phase portraits
Fig. 16. Local phase portrait
of a 4’ multifork
of bifurcations
of multifork
that is only slightly perturbed case of 4'm (Fig. 15(a)).
type in case of 4’m.
from the first type of multifork
in the
Hence, we find in the case of 4’ only one type of multifork bifurcation. Note that a similar observation was made in Section 2.4 in the case of 2’ with respect to the two types of double saddle node bifurcations in the case of 2’~. The bifurcation diagrams of the transcritical and multifork bifurcations in the case of 4’ and 4’m are presented in Fig. 14 and Fig. 17.
J. S. W. LAMB
and H. W. CAPEL
4 x s (-)
(a)
s (4’)
--e
s (4’)
2 x A (-) 2 x R (-)
4 x s (m’) (b)
S (4’m)
4 x s (m’) ----E
2 x A (m)
--fE Fig. 17. Bifurcation
S (4’m)
S (4’m)
S (4’m)
4 x C: (m’)
2 x R(m)
diagrams of bifurcations
of multifork
3. DISCRETE
type in case of (a) 4’ (Fig. 16) and (b) 4’nt (Fig. 15).
TIME MAPS
In this section we will discuss the local bifurcations for discrete time maps on the plane. Such maps can be obtained from a continuous time flow as a Poincare map of periodically driven two-dimensional flows or autonomous three-dimensional flows, or as a stroboscopic picture of a planar flow. Furthermore discrete maps occur in many problems of physical interest, see e.g. trace maps for determining the energy spectrum of a quasiperiodic chain [27,28] and stationary state maps of chains with (next-) nearest neighbour interactions [29-311. Consider the dynamical system, governed by an invertible map L on the plane. Then, in analogy with the definitions for continuous time flows, the diffeomorphism U of the plane is a symmetry of L if uo
Lo
u-l
= L,
(25)
I/ 0 L 0 u-1 = L-l.
(26)
and a reversing symmetry of L if
It is obvious that if the isotropy subgroup of a fixed point of L possesses a linear symmetry, any truncated expansion of this map around that fixed point will also possess this symmetry. But in the case of maps linear reversing symmetries are, in general, not preserved exactly after truncation. * However, the Nth order Taylor expansion LcN) of a map L possessing a reversing symmetry U, is locally symmetric with respect to U up to order N [32,5], i.e. L(N) 0 u 0 L(N) (g) u
(27)
where (E’ denotes equality up to order N. In fact, in studying local bifurcations with reversing point group symmetry, it is sufficient to consider normal forms possessing locally the desired (reversing) symmetries. In a previous work [5], the derivation of such normal forms has been treated. In the investigation of local bifurcations around the origin we can restrict ourselves to the case of a two-fold eigenvalue one, taking into account the groups given in Table 1. *A similar problem occurs with truncated normal forms of measure preserving maps, see e.g. [32-341.
Local bifurcations on the plauc
The local bifurcation anaiysis in the case of m’ can be studied on the basis of the following unfoldings (with b # 0 and p being the bifurcation parameter) (28)
2
satisfying L 0 lqb 0 L ‘2 @* with the linear reversing symmetry G’,:
xr = x - by. (29)
: y’ = -y,
and x’ = x + py + f()J, y’ = y + bx + g,y’
+ (bh3 + &)y3
+ bpy.
(30)
satisfying L o 2; o L ‘2 @L with the linear reversing symmetry
Both I@~and I@; are mirrors in the x-axis. Note that the normal form (30) is very similar to (15) in the case of flows, the main difference being that from the invariant 2gor + f,! the term go2yzhas been kept, whereas (15) contains f,,xy. In the case of 2m’ we find (30) with go2= 0. The bifurcations occurring in these unfoldings are qualitatively the same as we found in the case of flows, i.e. the bifurcation diagrams of Fin. 4 are also found in the case of planar maps. For some explicit results on a modkl map, see [14]. In the case of fixed point bifurcations occurring at a fixed point of a map L with linear part L1 = Id a direct analogy is seen, since the time-l map of the flow of a homogeneous poIynomiaI vector field F of degree k with unfolding PC of a Iower degree is approximately (around the origin and for small cl) given by Lck) B = id + pC + F,
(32)
with Id denoting the identity map. Moreover: if (F + PC) 0 U = - U 0 (F, + PC) with a linear reversing symmetry U, we also have approximately LF’ o U o LF’ ‘=’ U. Locally the fixed points of F + yC and Lr’ are quantitatively the same in the normal form analysis, with the same coordinates and stability properties. Hence, we conclude that all bifurcation diagrams we have given for the fixed point bifurcation of planar flows also hold in the case of planar maps. The observed analogy between the fixed point bifurcations in planar maps and flows can also be understood via a theorem of Takens [35], implying that any truncated expansion of a map L around a fixed point with linear part (331
for any b E R (including b = 0), is also the truncated expansion of the time-l map of some planar flow around a stationary point of a vector field with linear part (34)
201)
5. S. W. LAMB
and I-l. W. CAPEL
Moreover, the unfoldings in both cases give rise to similar developments of local fixed points, implying that the bifurcation diagrams of the local bifurcations around fixed points with linear part (33) are qualitatively the same as the ones we observed in the previous section. However, although the bifurcation diagrams of the fixed-point bifurcations of planar flows and maps are the same, the local dynamics is not the same. For instance, in two-dimensional (autonomous) flows there cannot be chaos, while in two-dimensional (autonomous) maps chaos is a generic feature. This implies that for the case of maps we have to reconsider our statements about the local saddle connections. Since the stable and unstable manifolds of saddle points are smooth manifolds our proposition in Section 2 concerning the saddle connections of the previous section is also valid in the case of discrete time maps. However, whereas in the case of flows stable and unstable manifolds cannot have transversal intersections, in the case of maps this is well possible and famous for being a source for chaotic behaviour. This implies in the case of maps that the homoclinic and heteroclinic points on the fixed sets of reversing symmetries indicate in general transversal intersections of stable and unstable manifolds in stead of smooth saddle connections. Note, however, that the smooth saddle connections that are supported by ordinary mirrors in the flow, will persist in the case of maps. Another difference is found if one has a close look at the centres. In the case of maps in general they appear as complex objects with KAM-tori and chaotic bands. As an example, in Fig. 18 we depicted (part of) the phase portrait of the map [12, 141
,I I
x’ = [C - y][l + (y’ - l)?], y’
=
X
1 + (y + 1 - c>* ’ at C = 2.93, a little after the occurrence of a m’ pitchfork bifurcation (at C = 2 v 2). In comparison to the corresponding pitchfork bifurcation in the case of continuous time flows (cf. Fig. 3(a)), it is observed that the boundary of the domain of attraction of the newborn attractor is not any more formed by a smooth homoclinic curve. In fact, we found the domain of attraction to extend to the right side of the saddle point. The domain of attraction seems to be bounded by a KAM curve that has survived the pitchfork bifurcation. The existence of KAM curves before the pitchfork bifurcation (in the nearly integrable case) is proved by the reversible KAM theorem [lo]. Period-doubling and more generally period-tuplings that occur in the case of qresonances (q = 2,3, . .) will not be considered in the present paper. 4. CONCLUDING
REMARKS
In this paper we have studied local bifurcations in (weakly) reversible two-dimensional dynamical systems around fixed points possessing an isotropy subgroup consisting of mirrors, two- and fourfold rotations. We dealt with continuous time flows and discrete time maps. Using only local normal form expansions, we found the (reversing) symmetries determining not only the ‘local’ bifurcations of the fixed points but also some global features such as saddle connections. In particular we found that the birth of saddle points in the presence of reversing mirrors in many cases gives rise to homoclinic and/or heteroclinic points. In discussing two-dimensional reversible dynamical systems, analogies and differences with area-preserving (Hamilton) systems have been stressed by many authors [.5, 10, 12,32,33]. The similarity between two-dimensional Hamiltonian and reversible
701
Local bifurcations on the plane
3
2.5
2
1.5
1
0.5
‘, I
0 0
0.5
I
1
I
1.5
I
2
I
2.5
I
3
3.5
Fig. 18. Local phase portrait of the map (35) with C - 2.93, just after a the cxcurrrnce of 21YH’ pitchfork bifurcation.
systems can be observed already on the level of the normal forms. In this paper we find that some of the local bifurcations that occur in the presence of a reversing mirror are well known to occur in two-dimensional Hamilton systems as well. In particular, the rn’ saddle-node is well known to be generic in Hamilton systems. Moreover, the pitchfork bifurcation with a centre (saddle) changing into a saddle (centre) and two centres (saddles) has been discussed by Rimmer [9] in the context of reversible (rn’) Hamilton systems. The bifurcations displayed in Fig. 3(b), (c) and Fig. 6 are also familiar in symmetric Hamiltonian vector fields, see [3S]. Furthermore, we have observed various examples of symmetry breaking bifurcations. In the case that the isotropy subgroups of the newborn fixed points do not possessa reversing symmetry, often attractor-repeller pairs are born, bringing into evidence the dissipative features of the dynamical system. In pursuing the investigations of the attractor-repellor pair in the case of mappings one may find a period-doubling cascade with dissipative values of the Feigenbaum constant 6 and the scaling factor a, as well as the occurrence of strange attractors 1121.
J. S. W. LAMB and H. W. CAPEl.
202
The considerations given above can also be applied to more dimensional systems. It turns out that the (reversing) symmetries give substantial restrictions to the possible bifurcations. In this way higher-dimensional bifurcation phenomena with (reversing) symmetries are easier to handle than without (reversing) symmetries, the latter case being much more difficult to analyse. Local bifurcations with reversing point group symmetry in threedimensional flows and maps will be treated in a future paper [36]. Ackno&+gement
-It is a great pleasure to acknowledge various useful discussions with John A. G. Roberts. REFERENCES
I. D. H. Sattinger. Group Theoretic methods in B@rcation Theory. Lecture Notes in Mathematics. vol. 762. Springer, Berlin (1979). 2. A. Vanderbauwhede. Local Bifurcation and Symmetry, Research Notes in Mathematics. vol. 75. Pitman. London (1982). 3. M. Golubitsky, I. Stewart and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory. VU!. 2. Applied MathematicaI Sciences?vol. 69. Springer, New York (198X). 4. J. S. W. Lamb, Reversing symmetries in dynamical systems, .K Phys. A: Math. Gen. 25, 925 (1992). 5. J. S. W. Lamb, J. A. G. Roberts and H. W. Capel, Conditions for local (reversing) symmetries in dynamical systems, Physicu A 197, 379 (1993). 6. J. S. W Lamb, Crystallographic symmetries of stochastic webs, J. Pky~. A: Math. Gen. 26, 2921 (1993). 7. J. S. W. Lamb and G. R. W. Quispel, Reversing k-symmetries in dynamical systems, Physica D to appear. 8. R. L. Devaney, Reversible diffeomorphisms and flows, Trans. Am. Math. Sue. 218. 89 (1976). 9. R. J. Rimmer, Symmetry and bifurcation of fixed points of area preserving maps, J. D# Eq. 29, 329 (1978): Generic bifurcations for involutory area preserving maps, ~Mem.Am. Ma& Sot. 41(272). 1 (1983). 10. M. B. Sevryuk, Reversible systems, Lecture Notes in Mathematics, vol. 1211. Springer, Berlin (1986). 11. A. Vanderbauwhede, Subharmonic branching in reversible systems, SlAMJ. Math Anal. 21(4), 954 (1990). 12. J. A. G. Roberts and G. R. W. Quispel, Chaos and time-reversal symmetry: order and chaos in reversible dynamical systems, Phys. Rep. 216(2> 3), 63 (192). 13. A. Politi, G. L. Oppo and R. Bad&, Coexistence of conservative and dissipative behaviour in reversible dynamical systems, Phys. Rev. A 33, 375 (1986). 14. T. Post, H. W. Capel, G. R. W. QuispeI and J. P. Van der Weele. Bifurcations in two-dimensional reversible maps. Physica A 164, 625 (19!%I). 1.5. S. Nose. A molecular dynamics method for simulations in the canonical ensemble, bfol. Phys. 52, 255 (1984): Constant temperature molecular dynamics method, Prog. Theor. Phys. Suppl. 103. 1 (1991). 16. W. G. Hoover, Canonical dynamics: equilibrium phase space distributions. Phys. Rev. A 31. 1695 (1985): Reversible mechanics and time’s arrow, Fhys. Rev. A 37, 252 (1988). 17. V. I. Arnold and A. Avez, Problimes Ergodiques de la Mkcanique Ciassique. Gauthier-Villars, Paris (1967). 18. M. Field, Equivtiant dynamics. in Multiparameter Bifurcation Theon?. ContemDorarv . , Mathematics. vol. 56. p. 69. Amer.-Math. Sot.., Rhode Island (i986). 19. F. Takens, Singularities of vector fields, Publ. Math. IHES 43, 48 (1974). 20. H. W. Broer and F. Takens, Formally symmetric normal forms and genericity, Dynamics Reported 2. 39 (1989). 21. F. Dumortier, in Structures in Dynamics, Studies in Mathematical Physics, vol. 2, p. 161. North-Holland. Amsterdam (1991). 22. V. I. Arnol’d, deometrical Methods in the Theory of Ordinary Differerttial Equations, Grundlehren der mathematischen Wissenschaften, vol. 250. Springer, New York (1983). 23. H. W. Broer, Formal normal form theorems for vector fields and some consequences for bifurcations in the volume preserving case, in Dynamical Systems and Turbulence. Lecture notes in Mathematics, vol. 898, p. 54. Springer, Berlin (1981). Structure preserving normal forms at equilibrium points, Preprint. 24. P. Chassat and M. Golubitsky. Symmetry-increasing bifurcation of chaotic attractors, Physica D 32, 423 (1988). 25. M. Krupa and M. Roberts, Symmetry breaking and symmetry locking in equivariant circle maps, Fhysica D 57. 417 (1992). 26. D. H. Sattinger, Topics in Stability and Bifurcation Theory, Lecture Notes in Mathematics, vol. 309. Springer. Berlin (1973). 27. M. Kohmoto, L. P. Kadanoff and C. Tang, Locahzation problem in one dimension: mapping and escape, Ph.ys. Rev. Left. 50, 1870 (1983). 28. J. A. G. Roberts and M. Baake, Trace maps as 3d reversible dynamical systems with an invariant. J. Stat. Phys. 74, 829 (1994).
29. M. H#gh Jensen and P. Bak, Mean field theory of the three-dimensional anisotropic Ising model as a four-dimensional mapping, F!rys. Rev. B 27. 6853 (1983). 30. T. Janssen and J. A. Tjon, Bifurcations of lattice structures, J. Phys. A: Math. Gen. 16. 673 (1983).
Local bifurcations on the plane
143
31. J. S. W. Lamb, The ferro-antiferro bifurcation transition at the paraphase boundary in a microscopic model with incommensurate phases, J. Phys. A: Math. Gen. 25, 355 (1992). 32. G. R. W. Quispel and H. W. Capel, Local reversibility in dynamical systems. Phys. Letr. A 142, 112 (1989). 33. J. A. G. Roberts and H. W. Capel, Area preserving mappings that are not reversible, Phys. Len. A 162, 243 (1992).
34. A. Bazzani. M. Giovannozzi, G. Servizi, E. Todesco and G. Turchetti, Resonant normal forms, interpolating HamiItonians and stability analysis of area preserving maps, Physics D 64. 66 (1993). 35. F. Takens, Forced oscillations and bifurcations, Comm. Math. Inst. Univ. of Urrechr 3. I (1974). 36. J. S. W. Lamb. H. W. Cape1 and J. A. G. Roberts, in preparation.
Local Bifurcations
on the Plane with Reversing Point Group Symmetry .I. S. W. LAMB and H. W. CAPEL
Institute
for Theoretical
Physics, University
of Amsterdam, Netherlands
Valckenicrstraat
65, 1018 XE Amsterdam.
The
Abstract-Local bifurcations of fixed points, being symmetric with respect to a reversing point group consisting of mirrors, two- and fourfold rotations, are studied for dynamical systems in W These bifurcations are of interest in connection to symmetry breaking phenomena, in particular with respect to the occurrence of dissipative features in (weakly) reversible systems.
1. INTRODUCTION
In the theory of dynamical systems, currently the interest for (reversing) symmetry properties is steadily growing. Some two decades of studies on the consequences of symmetries for the global and local phenomena in dynamical systems have passed [l-3]. However. a systematic treatment of reversing symmetries in this context has started only recently [4-71, after some years in which the interest for reversing symmetries was mainly focussed on involutory ones [S-12]. The emphasis on involutory reversing symmetries is related to the presence of generalized time-reversal symmetries. Generalized time-reversal symmetries do not only exist in Hamiltonian systems in the absence of an external magnetic field so that the Hamiltonian is an even function of momenta, but also more generally in systems of coupled first-order differential equations if there is an involution which changes the direction of the motion. The discrete time version of this, i.e. a map, has generalized time-reversal invariance if there exists an involution which changes the map into its inverse. In this way any reversible map can be considered to be the product of two involutions. Interesting examples of generalized time-reversal symmetries are the externally injected laser system studied by Politi et al. [13], and the Nose-Hoover dynamics which has been successfully used in molecular dynamics calculations [15, 161. Both systems show the presence of periodic and/or strange attractors and repellors and are therefore essentially different from Hamiltonian systems. A general method to obtain reversible mappings that are not measure preserving has been discussed in [12] together with a variety of examples. Mom recently, it has become clear that apart from generalized time-reversal invariance there can be other interesting reversing symmetries which are generated by transformations of (even) order larger than two. Nice examples are the reversing symmetries in relation to the treatment of stochastic webs [6] and the fourfold reversing symmetry in Arnold’s cat map [17]. In the study of the influence of symmetries on dynamical systems, local bifurcations have received a lot of attention [l-3, 18, 191. In this paper we intend to give a contribution to a systematic inclusion of reversing symmetries in local bifurcation theory. We will discuss 271
‘72
J. S. W. LAMB
and H. U’. CAPEL
local bifurcations around fixed points of dynamical systems in R2 that are symmetric with respect to a reversing point group consisting of mirrors, two- and fourfold rotations. The special treatment of bifurcations in systems with a reversing symmetry can be of Interest for various reasons. First of all it has been known that conservative systems and reversible systems in R2 have many properties in common, such as the KAM-theorem. cf. Arnold and Sevryuk [lo], and the (conservative) values of the Feigenbaum constant S and scaling i’actor N in the period doubling of symmetric fixed points [12]. The dissipative features in non-measure-preserving reversible systems can arise by asymmetric fixed points moving in from infinity, but more interestingly also after bifurcations at symmetric fixed points [5. 12-141. Furthermore, it will turn out that certain bifurcation processes that mny occur accidentally for special choices of parameters in the absence of (reversing) symmetries become generic bifurcation processes in the class of systems having particular (reversing) symmetries. The bifurcations that we will consider in this paper are codimension one bifurcations, i.c. bifurcations occuring generically in one-parameter families of dynamical systems possessing a fixed point with the desired (reversing) symmetry properties. WC: will start analysing continuous time flows. Thereafter WC will discus3 discrete time maps,
2. CONTINUOUS TIME FLOWS
Let us consider a planar vector field F, defining a dynamical system by means of the ordinary differential equation (ODE) Ax = F(x), dt
x E R2.
Purthermore, Let U be a diffeomorphism of the plane. Then, by definition. symmetry of (1) if
(1, L’ is a
and a reversing symmetry of (1) if - $[u(x)]
= F 0 U(x).
(3)
All symmetries and reversing symmetries of (1) form a group under composition. This group is called the reversing symmetry group of (I) 141.The composition of’ a symmetry and a reversing symmetry js a reversing symmetry, whereas the composition of two symmetries or two reversing symmetries produces a symmetry. A dynamical system possessing a reversing symmetry is called weakly rever.~ihke in general and reversible in the case where it possessesan involutory reversing symmetry [ 101 (i.c. a reversing symmetry that is its own inverse). Such reversible systems can be regarded as having a generalized time-reversal invariance. e.g. the reversing involution may be a nonlinear transformation. For a rccenr review on reversible dynamical systems. we refer the reader to [12]. Let us focus on the neighbourhond of a fixed (or stationary) point of (1). i.e. a point x,; satisfying F(xO) = 0. Considering a reversing symmetry group of a dynamical system. the
Local bifurcations on the phne
isotropy subgroup TX,, of the fixed point x0 is the subgroup of the reversing symmetry group, such that for all U E XX0we have U(x,) = x0 [3]. The fixed points belonging to the fixed set of at least one reversing symmetry were called symmetric fixed points in [12]. The eigenvalues of the linearized vector field around such points are either zero or come in pairs {A, -A}. All other fixed points have been called asymmetric. Starting from an asymmetric fixed point with eigenvalues (hi} one obtains by application of any reversing symmetry U another asymmetric fixed point with eigenvalues { -&}. So in systems with reversing symmetries every attractor has to be accompanied by a repcllor. Because of the occurrence of various symmetries and reversing symmetries in the dynamical systems to be discussed in this paper we will USCthe work symmetric in a more general sense, indicating explicitly the (reversing) symmetries involved. Around a fixed point the local dynamics can be studied via the Taylor expansion of F around x(,. The object of this approach is to truncate the Taylor expansion at a certain order and obtain in this way a simple and ‘good’ description of the local dynamics, For vector fields of arbitrary dimension, it is far from obvious that a truncated Taylor expansion will give a ‘good’ picture of the local dynamics (see e.g. 1201).This depends on the stability of the truncated expansion with respect to higher order perturbations. However, for planar flows, which will be treated in the present paper, it has been shown that all Taylor expansions can be stabilized by a finite number of terms [Zl]. In connection with the local analysis we are going to perform, it would be convenient if the action of XX,,were linear. In this paper, we assume that XX,,is locally C-conjugate to a linear point group. (This is automatically the case if XC1is a compact Lie group,) In that case there is a smooth tranformation to a coordinate frame in which the action of X,,, on the Nth order Taylor expansion of F around x0, FCN), is linear. In that coordinate frame, we have the relation (J 0 F’“‘(x)
= F(N) 0 U(x),
in case U is a symmetry, and
in case U is a reversing symmetry. Considering a Taylor expansion FCN)possessing already a linear &, things can be made even more simple by transforming the expansion into normal form, i.e. considering a coordinate frame in which most of the Taylor coefficients are zero [22]. WC can remove the Taylor coefficients using polynominal coordinate transformations with linear part being the identity and nonlinear terms of degree 2, 3, . . . . However, in doing so, we would like the (linear) symmetry propertics of FCN)to be preserved. This can bc achieved by using solely symmetry-preserving transformations, i.e. transformations that commute with the (revetsing) symmetries [23]. In this way we are led to a so-called symnrefric rtormal fwrn. In a previous work [S], it was shown that such a symmetric normal form (in case of a vector field and a reversing symmetry of finite order) also can be derived directly from a general normal form, i.e. a normal form of a vector field possessing no (reversing) symmetries. imposing the linear reversing symmetry on this general normal form and satisfying all contraints resulting from it. The extension of this to the case of a finite reversing symmetry group is obvious, since the proof in ]5] is based on the existence of a unitary representation of a cyclic group, and any finite group has a unitary representation. 117studying local bifurcations, we consider local Taylor expansions of the vector fields al-ound fixed points of which the local dynamics is unstable with respect to weak pcrturhations. These perturbations serve as a paramctcrization for local bifurcations. Following terminology from singularity theory, such perturbations are usually called
27.!
J. S. W. 1.AM13 and H. W. CAPEL
unfoldings. Using this approach, we will study the local bifurcations that can occur in a dynamical system with a fixed point possessing an isotropy subgroup consisting of symmetries and reversing symmetries, considering only unfolding terms that are consistent with the isotropy subgroup. Considering local bifurcations of a fixed point x0 with isotropy subgroup XX,,, we are particularly interested in the isotropy subgroup of the fixed points that bifurcate from x,,. Of course also the symmetry properties of periodic orbits, limit cycles, and other objects are of interest. Periodic orbits and limit cycles may be studied on the basis of an appropriately chosen Poincare map, as being fixed points of such a map and its iterates. In this paper we will mainly focus on the symmetry properties of bifurcating fixed points. In the bifurcation processes new fixed points are born and they often have an isotropy group that is different from C,“. In the case that the isotropy subgroups of the fixed points that bifurcate from x0 are true subgroups of x0, we speak of symmetry-breaking bifurcations. Symmetry-breaking bifurcations with a change of stability properties are related to the phenomenon of spontaneous symmetry breaking. However, also the opposite phenomenon of symmetry increasing bifurcations may occur [24,25]. In this paper, we will restrict ourselves to local point groups possessing only mirrors (M), two- and fourfold rotations (R, and f$J. Other point groups require separate calculations and will not be treated here. Without loss of generality we will take the fixed point under consideration to be the origin. Moreover, we want at least one of the generators to be a reversing symmetry. This implies that we will deal with seven different point groups. These groups and their generators are given in Table 1, specifying also which generators are reversing symmetries and which generators are not. For the description of these groups, we borrowed the Shubnikov-Belov notation from magnetic point groups (see also [6]). For each of these groups we will discuss bifurcations of the fixed point that are likely to be observed in one-parameter families of dynamical systems. so-called generic codimension-1 bifurcations. In the forthcoming discussion of local bifurcations we will observe the occurrence of several kinds of saddle connections of which the stability is caused by the (reversing) symmetries. In order to clarify the discussion on these connections, we give some essential definitions. A saddle point x0 in R2 is characterized by the existence of a stable manifold W:) and an unstable manifold Wio that are defined as IV& = {x E lR2/lim~,,x = x0}, t--ra
(61
W& = {x E !??‘Ilim
(7)
in which @, stands for the time t evolution operator. Both W& and Wzo are smooth curves Table 1. Reversing point groups and their generators. M denotes a mirror and R, is a rotation over an angle N Group iti,
Reversing generator M
2' 2'm 4'
M M 47 47 KY'2
4’m
R+
3m’
Nonreversing -
& R,, M M
generator
Local bifurcations
on the plane
215
that pass through the saddle. Let x0 and x1 be saddles, then a point y (that does not coincide with x0 and/or x,) satisfying is called a heteroclinic point if x0 + x1, and a homoclinic point if xl1= x1. In the case of planar flows, the existence of a homoclinic (heteroclinic) point implies the existence of a smooth curve of homoclinic (heteroclinic) points that coincides with branches of the stable and unstable manifolds of the concerned saddle(s). Such a curve is called a homoclinic (heteroclinic) saddle connection.” After this introduction, we will start the local bifurcation analysis. We will discuss the reversing symmetry groups in the order of appearance in Table 1, i.e. we start with the groups that possessa reversing mirror. 2.1.
Point group symmetry m’
111the case of a single reversing mirror, that we will take to be the mirror in the x-axis 44,. the linear part of the vector field F1 should satisfy M, 0 Fl = -F, 0 M,,
(3
from which it follows that
corresponding to a hyperbolic or elliptic fixed point if /3r > 0 or 13~< 0, respectively. However, these points are structurally stable with respect to small perturbations that do not break the reversing symmetry and hence do not allow for local bifurcations in the neighbourhood of the origin. In these cases the local behaviour of the reversible flow as displayed by the symmetric normal form is of a conservative (non-dissipative) nature. This means that in the investigation of bifurcations around the origin we can restrict ourselves to the nilpotent case
(11) At a bifurcation, the linear part could also be zero, but this is exceptional and will therefore not be considered here. In the above cases we make use of the symmetric normal forms that were derived in [Sit (12)
and
‘hNotc that in the curves. A discussion .YlXc normal form form with respect to
case of discrete time maps, homoclinic and heteroclinic points usually do not come in smooth of this will he given in Section 3. (12) is identical to equation (67) of [S], whereas (13) is obtained from the reversible normal the reversing symmetry MY (equation (68) of [5]) by interchanging x and v,
J. S. W. LAMB and H. W. L‘APEL
27h
Considering only unfotdings that do not break the symmetry property, we find in the case of (12) a so-called saddle-node bifurcation. This bifurcation occurs in the unfolding
(14) where jl is the bifurcation parameter. (A Iinear term proportional to x in 3 can be transformed away by a shift x H x + const.) ITI case ,ug20< 0 there is a saddle-centre pair (saddle-node) that disappears as soon as jog,, > 0 (see Fig. I j. Both fixed points. resulting from the saddle-node bifurcation, are situated on the line 1’ = 0 and are hence symmetric with respect to M,. Note that fixed points that arc symmetric with respect to a reversing symmetry can be neither repelling nor attracting. Hence we find on the line !: = 0 a true centre that remains stable under all perturbations that preserve the reversing symmetry. whereas in non-reversible two-dimensional tlows a ccntre is structually unstable and will be transformed to an attracting or repelling focus under a small perturbation.
Notice furthermore the homoclinic orbit that surrounds the centre point. The stability of tht: homoclinic orbit arises because of the reversing mirror symmetry. One of the branches of the unstable manifold of the saddle that is situated on the fixed set of the reversing mirror has to intersect the fixed set of the reversing mirror in a point on the other side of the ccntre point for sufficiently small values of the bifurcation parameter. Hence this branch has to coincide with a branch of the stable manifold of the same hyperbolic point, as can be easily verified from Fig. I. The bifurcation diagram is given in Fig. 2. Note that this type of saddle-node bifurcation is similar to the one observed in two-dimensional Hamiltonian vector fields. Unfolding the other normal form, we typically find a pitchfork bifurcation (named after the form of the bifurcation diagram, see Fig. 4):
If $,,3 becomes negative. two fixed points bifurcate off the reversing mirror )’ = 0.
Fig. I. Local phase portrait of
m’ saddle-node hifurcatiorl
s (m') (74 C 3
C: (m’)
Fig. 2. Rlfurcation diagram of the m’ saddle-node bifurcation (Fig. 1). S denotes a saddle and CI a centre. The isotropy subgroup of the fixed points is indicated between brackets.
Local bifurcations on the
plane
z-77
In case fo3> 0, we have a saddle that becomes a centre, sending off two saddles (see Fig. 3(c)). The two saddles are connected via heteroclinic saddle connections. Note that although such intersections are not stable in general [21], they are stable in this cast because of the reversing symmetry. In fact, consider one of the saddles. Near the bifurcation point, one branch of the stable and one branch of the unstable manifold will intersect the fixed set of the mirror (i.e. the x-axis). Therefore, because of the reversing symmetry one finds that these branches must flow into the other saddle and hence must be hetcroclinic saddle connections. These heteroclinic connections form precisely the boundary of the set ol’ closed tlow lines around the center at the origin. Finally, we should remark that Fig. 3(c) has been drawn for the case of point-group symmetry 2172’(that is to be discussed in the next subsection), which is the special cast with f,, = 0 of (15). In the general case with f,] + 0, the flow lines will be not symmetric with respect to the mirror in the y-axis. However, the patterns of‘ local flow lines in both cases are topologically equivalent. In case f,,>< 0, a centre becomes a saddle, sending off two fixed points that arc foci, Because of the rcvcrsing symmetry, one of these foci must be attracting while the other is repelling. Moreover, one homoclinic saddle connection appears (see Fig. 3(a)). Although in ;I general two-dimensional tlow, such a connection would be unstable, here it is stable because of the reversing tnirror symmetry. This is caused by the fact that generically for sufficiently small [c the stable and unstable manifold of the saddle intersect the fixed set of the reversing symmetry (the line y = 0 in Fig. 3(a)) in a point, different from the saddle point itself. Because of the reversing symmetry we then find that this manifold must make a homoclinic connection. From Fig. 3(a) it is clear that this homoclinic connection scrvcs as the boundary of the basin of attraction of the attracting focus. In [13] this type of pitcht’ork bifurcation was used to illustrate the birth and occurrence of attractor-repeller pairs in reversible dynamical systems, cf. [14] for a similar treatment in the case of maps. The results obtained for the hetcroclinic and homoclinic saddle connections above follow immediately from the following proposition. (The proof is simple and will not be given here.) ProposilEon: Consider a reversing symmetry with fixed set $, then I Jf the stuhle or unstable manifold of a saddle, not belonging to the fixed set .Y. immects the fixed set E%then the intersection point is a heteroclinic point. * If the stable or unstable manifold of a saddle, belonging to the fixed set !+, intersects the fked set 9 then the intersection point is a homockc point if this point is not the .snddle itself. ‘The second part of this proposition of a more general theorem of Devaney [8], stating moreover that the homoclinic orbit is the limit of a one-parameter family of a symmetric closed orbits whose periods tend to ~0.
2.2.
Point group symmetry 2m’
In the case of a reversing point group generated by the reversing mirror M, and a twofold rotation around the origin, we are led to consider the following unfolding
J. S. W L.4MB
Fig. 3. Local pitchfork
bifurcations
and H. W. C’APEL
in the case of m’ ((a) and {c)) and 2~’
((b) and (c)).
Local bifurcations on the plaiic
The typical bifurcation here js again a pitchfork bifurcation. However, in this case we have two orthogonal reversing mirrors. The new fixed points are sent off one mirror but remain on the other one. Hence, if fO;< 0 we indeed have a centre bifurcating in one saddle and two centres instead of a saddle and two foci (see Fig. 3(b)). Note also the two homoclinic saddle connections Ihal occur in this pitchfork bifurcation. Their occurrence and stability follo\h tram our proposition, since the stable and unstable manifolds of the newborn saddle must intersect the fixed set of MY for sufficiently small values of the bifurcation parameter. This also gives rise to the fact that the intersections of the (un)stablc manifolds with the fixecl set of MI coincide with the saddle. In cast f,,?> 0 the pitchfork bifurcation is equivalent to the corresponding bifurcation in the case of pll’ (see Fig. 3(c)). The extra symmetry does not play an important role in this bifurcation. Ihe only difference being that in the case of m’ Fig. 3(c) would in general not be symmetric with respect to the mirror in the y-axis. Note that pitchfork bifurcations in which a centre is changed into a saddle and two centres, or a saddle is changed into a centre and two saddles, arc not generic in the case of two-dimensional Hamiltonian vector fields (without additional (reversing) symmetries). However, in the case of reversible ( /?z’) Hamiltonian vector fields these pitchfork bifurcations are generic [9]. In Fig. 4 WC present bifurcation diagrams of the pitchfork bifurcations we discussed in the cases m’ and 2~‘. In this figure the symmetry breaking can be followed easily, because of the isotropy subgroups that have been indicated. In relation with the text and Fig. 3. note that in the cast of Fig. 4(a) the fixed points with isotropy subgroup m’ refer to points on the fixed set of MA (the x-axis). whereas in the case of Fig. 4(b) the fixed points with isotropy subgroup m’ refer to points on the fixed set of Al, (the y-axis). 2.3.
Pointgrolq symmetry 4m’
In case the reversing mirror jW, is extended with a fourfold rotation. WC Uind the bifurcation picture to be changed drastically. The ODE having the origin as a fixed paint and possessing such a reversing symmetry group is given by i = f(X, y),
(17) 1 j = -f(y, X), J’), This implies that the origin is either a centre or where f(x. - JJ)= -f(x, y) = -f(-x. the linear part of the vector field is zero. (The nilpotent linear parts which we considered previously cannot occur here.) Hence, a local bifurcation will be possible only if the linear
(a?
c: [m’]
A (-! s (Ill’)
s I; I S [iri’j -E
R t-1
(’ (111’) (bj
c’ (am’)
s (h’) I: (777’)
c: (iII! 1 s i-1
s (1117 s (Zm’)
(I (h’) s (??,“I
Fig. 4. Bifurcation diagrams of pitchfork bifurcations in the case of m’ and 2~‘: (a) pitchfork in’ (Fig. 3(a) and 3(c)). and (h) pitchfork 2rn’ (Fig. 3(b) and 3(c)). S denoles a saddle. C a centre. A an attractor. and R a rep~llor. The isotropy subgroup of each fixed point is indicated between brackets.
2x1)
J. S. W. LAMB
and H. W. CAPEL
part of the vector field is zero. The first nonvanishing Taylor terms of (17) appear at third order. If we truncate there, we find around the bifurcation point i = py + f&y
+ f,,,3y3,
2 - f?,$, i. p = -/Lx - fo.1. where !L is a small bifurcation parameter. In cast p # 0, the origin is a centre. The type of bifurcation, occurring at p = 0, depends on f2, and fij3. We distinguish two main regimes. They are indicated in Fig. 5. WC will not discuss the marginal cases ]fijll = If_7,1or fo3= 0 here, because they are exceptional. In regime 1 of Fig. 5, WCfind a kind of multifork bifurcation. If /L goes through zero, the centre changes orientation of rotation and sends off four saddles and four centrcs. The new fixed points come in quadruples because of the fourfold rotation symmetry. The centres and the saddles are situated on two pairs of reversing mirrors (see Fig. 6(a)). One pair of reversing mirrors consists of IVY and MY and the other pair consists of M,,, and M,-._,_ i.c. the mirrors in the line x = )’ and x = -y, respectively. Note that the saddles are always on a different pair of perpendicular mirrors from the centres. This leaves us with two possibilities. One of which has been indicated in Fig. 6(a). Each of these possibilities occurs in two of the four regions labelled 1 in Fig. 5. However, since these two possibilities can be transformed into each other by a rotation over n/4 around the origin, these bifurcations are being regarded as equivalent. Note the eight heteroclinic connections that occur in Fig. 6(a). Their occurrence and stability can be easily shown from our proposition, since the branches of the stable and unstable manifolds of the four saddles all intersect fixed sets of reversing mirrors. In regime 11 of Fig. 5, we find a bifurcation of transcritical type. While pitchfork or multifork bifurcations are either supercritical (fixed points are born) or subcritical (fixed points are annihilated), the name transcritical bifurcation refers to the case that there is no net production or loss of fixed points at the bifurcation point. At the bifurcation point, four saddles, sitting on a pair of perpendicular reversing mirrors, are being absorbed at the origin whereafter they are sent off on the other pair of reversing mirrors. The local dynamics before and after the bifurcation are related through a rotation over n/4 and a change of orientation, as shown by the phase portraits in Fig. 6(b). Before and after the bifurcation we find locally four heteroclinic connections since, for every saddle, one branch of the stable and one branch of the unstable manifold intersect the fixed set of a reversing mirror. The other branches of the stable and unstable manifold do not have local intersections with the reversing mirrors around the bifurcation point. The bifurcation diagrams of the bifurcations in the regimes I and II of Fig. 5 are shown in Fig. 7. In connection with the text and Fig. 6 the fixed points with isotropy subgroup m’ refer to points on the fixed sets of M, and My for the saddles (or centres) and the fixed sets of M.LLJand MxyTyfor the centres (or saddles).
Fig.5. Bifurcationregimesof
(18) in the f,,-f,,
plane.
Local bifurcations on the plane
Fig. 6. Local bifurcations in the case of 4m’: phase portraits of (a) multifork (b) transcritical (regime II of Fig. 5) type bifurcations.
(regime I of Fig. 5) and
4 x s (m’) (4
c (4m’)
c (4m’) ---E
C (4m’)
4 x c (773)
4 x s (m’)
(b) 4 x s (m’) x
C (4m’)
Fig. 7. Local bifurcations in the case of 4~‘: bifurcation diagrams of (a) multifork (Fig. 6(a)) and (b) transcritical (Fig. 6.(b)).
2.4.
Point group symmetry 2’ and 2’m
If a fixed point has an isotropy subgroup that contains a reversing symmetry that is a twofold rotation R,, then it is easily verified that this fixed point can neither be a saddle, nor a centre. In fact, the linear part of the vector field around the origin will be zero.
2x2
J. S. W. LAMB
andH.
W. CAPEL
Hence, in the case where the origin is a fixed point, with an isotropy group that contains no nontrivial (reversing) symmetries other than the reversing point symmetry K,, the vector field F must satisfy F o R, = F. Therefore, the first nonzero Taylor terms in the expansion of the vector field appear at second order. Moreover, all second-order terms are allowed. Such a form can be unfolded by adding constant terms to the vector field. This destroys the fixed point at the origin. However, before considering the case of 2’, we will first deal with the slightly Icss complicated case of 2’m, where we take the mirror in the x-axis to be a symmetry. In this case WCfind some restrictions to the second-order terms leading to the unfolding (10) in which p is a small bifurcation parameter. From this unfolding we find two types of bifurcations: double-saddle nodes and transcritical bifurcations. In a double saddle-node bifurcation, two saddles and an attractor-repellor pair collide al the origin and then disappear. Note that because of the reversing twofold rotation a single saddle-node cannot occur at the origin. In the transcritical bifurcations we find that two saddles or two centres collide at the origin and pass through as, respectively, two saddles or an attractor-repellor pair. Let us first consider the double saddle-nodes. They occur if fiofoz> 0. The first type is found in case f2,,g,, < 0: two saddles on the x-axis and two centres on the y-axis are absorbed at the origin (see Fig. S(a)). In case f,,g,, > (I we have an attractor-repellor pair on the x-axis and a saddle pair on the y-axis that collide at the origin and then disappear (Fig. S(b)). In case fZDfo2 < 0 we have a transcritical bifurcation in which a pair of fixed points on one mirror makes a transition to the other one. In case f2,,gll i 0, we find two saddles on the reversing mirror that collide at the origin and proceed on the ordinary mirror (see Fig. 10(a)). In cast fZ,,fi12 > 0, two centres on the reversing mirror collide at the origin and become an attractor-repellor pair on the ordinary mirror (see Fig. 10(b)). Note that the latter change of character of the fixed points during the transcritical bifurcation is due to the fact that the centres move from a reversing mirror on which they are stable to an ordinary mirror on which centres cannot occur. Therefore, the centres have to change into an attractorrcpellor pair. (Saddles are not possible because of the conservation of Poincare index, see e.g. [%I.) In the case that there is only 2’ symmetry, i.e. there is only a twofold reversing rotation but no ordinary mirror, the situation becomes more complicated since there are no conditions on the second order terms of the Taylor expansion. By an appropriate choice of coordinates the unfolding can always be chosen as i = /1 + f&
+ f,,xy + f,,,y?
1 j = g2,x2 i- EhlXY + &12YZ7 where the constant term in jl has been removed by a shift and 1~is a small bifurcation < 0, the above unfolding leads to a complete disappearance parameter. In case g:i - 2g02gZ0 of the fixed point: no new fixed points are born from the origin. However, in case g:i - 2g,,,,g,,> 0, we find similar bifurcations as in the case of 2/m, i.e. double saddle-nodes and transcritical bifurcations. However, in these bifurcations the pairs of centres appearing in the case of 2’m are replaced by attractor-repellor pairs. This leads to transcritical bifurcations in which either a pair of saddles or an attractor-repellor pair
Ix-cal hifurcatiuns on lhc plant
(3)
----
I
Fig. 8. Local phase portraits 01 double saddknodc lxfurcations
in
the caw uf 2’~.
A C-J (4
2 x s (-)
(‘L’) E
@I
(2’7r1)
C
R C-1
h (m)
:!xS(m)
2 x s (rd)
(Ym) 2 x CI (Id)
E
K (,,I)
Fig. 9. Bifurcation diagrams of double saddle-node bifurcation? in Lhe case of (a‘) 2’ (Fig. 12), and (b) 2’171 (Fig. 8).
passes through the origin and double saddle-node bifurcations in which two saddles and an attractor-repeller pair are born at the origin. At this point we will refrain from giving a detailed analysis of the bifurcation types in terms of the six second-order parameters since the expressions that are involved are rather long and not very instructive.
J. S. W. LAMB and H. W. CAPEL
281
(b)
Fig. 10. Local phase portraits of bifurcations of traoscritical type in the case of 2’m
s C-1 (4 s(6)
s (m’) @I S (m’)
sC-1RC-1 sI-) AC-1
A(-) RC-J
S(mlc(4
A(ml
s Cm)
R (4
c Cm’)
Fig. 11. Bifurcation diagrams of bifurcations of transcritical type in the case of (a) 2’ and (b) 2’~n (Fig. 10)
In Fig. 12 we present the phase portrait of a 2’ double saddle-node that is only a slightly perburbed version of the first type of double saddle node in the case of 2’m (see Fig. 8(a)). Note that this phase portrait is topologically equivalent to the second type of double saddle-node in Fig. 8(b). The second type of double saddle-node of Fig. 8(b) is persistent under the breaking of the mirror symmetry. Hence, we find that there is only one type of double saddle-node bifurcation in the cas of 2’, with both saddles having one branch of the stable manifold connected with the repellor and one branch of the unstable manifold connected with the attractor. The bifurcation diagrams of the double saddle-nodes and transcritical bifurcations in the case of 2’ and 2’m are present in Fig. 9 and Fig. 11.
285
Local bifurcations on the plant
Fig, 12, Local phase portrait of a 2’ double saddle-node that is only slightly perturbed from the first type of double saddle-node in the case of 2’~ (Fig. R(a)).
2.5.
Point group symmetry 4’ and 4’m
In the case of reversing symmetry group 4’ with the rotation over 7r/2 as reversing generator the ODE is given by
i = f(x, y),
(21)
i j, = q-y, x),
where moreover f(x, y) = -f(-x, -y). Hence, the first nonlinear terms in the Taylor expansion occur at third order and there are no additional restrictions on these terms. But let us first consider the case of 4’m, where we take the mirror in the x-axis to be a symmetry, giving the additional conditions f(x, Y) = f(x, -y)
= -f(y,
x).
(22)
In this case, taking into account the Taylor expansion up to third-order terms, the local bifurcations around the origin can be studied on the basis of the following unfolding (23) in which p is a small bifurcation parameter. The origin is a saddle for all p + 0. This unfolding gives rise to bifurcations of multifork and transcritical type which occur if the saddle switches stable and unstable directions at p = 0. In the transcritical bifurcation four centres on the reversing mirrors, i.e. the lines n = y and x = -y , pass through the saddle at the origin and change to two attractor-repellor pairs on the x- and y-axis. These bifurcations occur in case f30(f30t- flz) < 0. The phase. portrait of this bifurcation is given in Fig. 13 and its bifurcation diagram is depicted in Fig. 14(a). Multifork bifurcations are found in case f,,,(f,, -t flz) > 0. In the case of 4’m we must distinguish between two types of multifork bifurcations. In case Iflz/ > Ifjo four saddles and four centres branch off the saddle point at the origin. The saddles are placed on the ordinary mirrors, i.e. the x-axis and y-axis, and the centers on the reversing mirrors, i.e. the lines x = y and x = -y (see Fig. 15(a)). The other type of multifork bifurcation occurs in case IfI < (f3,,).We then find four saddles on the reversing mirrors and two attractor-repeller pairs on the ordinary mirrors that branch off the saddle at the origin (see Fig. 15(b)).
.I. S. W. LAMB
Fig. 13. Local phase portrait
and H. W. c’APF,L
of the bifurcation
2 x A (-)
(a)
(b)
Fig. 14. Bifurcation
of transcritical
type in the case of 4’m
2 x R(-)
s (4’)
s (4’)
2 x R (-)
2 x A (-)
2 x c (m’
2 x R (m)
S (4’m)
S (4’m)
2 x C (m’
2 x A (m)
diagrams of bifurcations
of transcritical
type in the case of (a) 4’ and (b) 4'm (Fig. 13).
Note that the first type of multifork bifurcation gives rise to eight heteroclinic saddle connections (see Fig. 15(a)). The saddle connections along the x-axis and y-axis are enforced by the fact that the mirrors in these lines are symmetries. The other connections follow from our proposition because the mirrors in the lines x = y and x = -y are reversing symmetries. If there is only 4’ point group symmetry we have the unfolding i = px + fj($ + fi*X2y -t f,*.Xy2 + fo3y3, (24)
1 p = -py + f& - f,,x2y + f*,.Xy2 - f30y3, in which ,Uis a small bifurcation parameter. As in the case of 4’m the origin is a saddle for all ,U# 0. The type of bifurcations observed here are similar to the ones we discussed in the case of 4’m. In case f30(f30+ f12) + fo3(fo3f f2J < 0 we have transcritical bifurcations and in case f3,,(f30+ f12) + fo3(fo3+ fzl) > 0 we find multifork bifurcations. However, because the reversing mirror symmetries are absent the four centres that were advertised in case of the transcritical and first type of multifork bifurcation in the case of 4’m (see Fig. 1.5(a)) are replaced by two attractor-repellor pairs in the case of 4’. In Fig. 16 we present the phase portrait after a multifork bifurcation with symmetry 4’, only slightly perturbed from the first type of multifork bifurcation in the case 4’m. Note that this phase portrait is topologically equivalent to the second type of multifork bifurcation (see Fig. 15(b)) that is persistent with respect to the breaking of the mirror symmetry.
Local bifurcations
on the plane
(b) -IFig. 15. Local phase portraits
Fig. 16. Local phase portrait
of a 4’ multifork
of bifurcations
of multifork
that is only slightly perturbed case of 4'm (Fig. 15(a)).
type in case of 4’m.
from the first type of multifork
in the
Hence, we find in the case of 4’ only one type of multifork bifurcation. Note that a similar observation was made in Section 2.4 in the case of 2’ with respect to the two types of double saddle node bifurcations in the case of 2’~. The bifurcation diagrams of the transcritical and multifork bifurcations in the case of 4’ and 4’m are presented in Fig. 14 and Fig. 17.
J. S. W. LAMB
and H. W. CAPEL
4 x s (-)
(a)
s (4’)
--e
s (4’)
2 x A (-) 2 x R (-)
4 x s (m’) (b)
S (4’m)
4 x s (m’) ----E
2 x A (m)
--fE Fig. 17. Bifurcation
S (4’m)
S (4’m)
S (4’m)
4 x C: (m’)
2 x R(m)
diagrams of bifurcations
of multifork
3. DISCRETE
type in case of (a) 4’ (Fig. 16) and (b) 4’nt (Fig. 15).
TIME MAPS
In this section we will discuss the local bifurcations for discrete time maps on the plane. Such maps can be obtained from a continuous time flow as a Poincare map of periodically driven two-dimensional flows or autonomous three-dimensional flows, or as a stroboscopic picture of a planar flow. Furthermore discrete maps occur in many problems of physical interest, see e.g. trace maps for determining the energy spectrum of a quasiperiodic chain [27,28] and stationary state maps of chains with (next-) nearest neighbour interactions [29-311. Consider the dynamical system, governed by an invertible map L on the plane. Then, in analogy with the definitions for continuous time flows, the diffeomorphism U of the plane is a symmetry of L if uo
Lo
u-l
= L,
(25)
I/ 0 L 0 u-1 = L-l.
(26)
and a reversing symmetry of L if
It is obvious that if the isotropy subgroup of a fixed point of L possesses a linear symmetry, any truncated expansion of this map around that fixed point will also possess this symmetry. But in the case of maps linear reversing symmetries are, in general, not preserved exactly after truncation. * However, the Nth order Taylor expansion LcN) of a map L possessing a reversing symmetry U, is locally symmetric with respect to U up to order N [32,5], i.e. L(N) 0 u 0 L(N) (g) u
(27)
where (E’ denotes equality up to order N. In fact, in studying local bifurcations with reversing point group symmetry, it is sufficient to consider normal forms possessing locally the desired (reversing) symmetries. In a previous work [5], the derivation of such normal forms has been treated. In the investigation of local bifurcations around the origin we can restrict ourselves to the case of a two-fold eigenvalue one, taking into account the groups given in Table 1. *A similar problem occurs with truncated normal forms of measure preserving maps, see e.g. [32-341.
Local bifurcations on the plauc
The local bifurcation anaiysis in the case of m’ can be studied on the basis of the following unfoldings (with b # 0 and p being the bifurcation parameter) (28)
2
satisfying L 0 lqb 0 L ‘2 @* with the linear reversing symmetry G’,:
xr = x - by. (29)
: y’ = -y,
and x’ = x + py + f()J, y’ = y + bx + g,y’
+ (bh3 + &)y3
+ bpy.
(30)
satisfying L o 2; o L ‘2 @L with the linear reversing symmetry
Both I@~and I@; are mirrors in the x-axis. Note that the normal form (30) is very similar to (15) in the case of flows, the main difference being that from the invariant 2gor + f,! the term go2yzhas been kept, whereas (15) contains f,,xy. In the case of 2m’ we find (30) with go2= 0. The bifurcations occurring in these unfoldings are qualitatively the same as we found in the case of flows, i.e. the bifurcation diagrams of Fin. 4 are also found in the case of planar maps. For some explicit results on a modkl map, see [14]. In the case of fixed point bifurcations occurring at a fixed point of a map L with linear part L1 = Id a direct analogy is seen, since the time-l map of the flow of a homogeneous poIynomiaI vector field F of degree k with unfolding PC of a Iower degree is approximately (around the origin and for small cl) given by Lck) B = id + pC + F,
(32)
with Id denoting the identity map. Moreover: if (F + PC) 0 U = - U 0 (F, + PC) with a linear reversing symmetry U, we also have approximately LF’ o U o LF’ ‘=’ U. Locally the fixed points of F + yC and Lr’ are quantitatively the same in the normal form analysis, with the same coordinates and stability properties. Hence, we conclude that all bifurcation diagrams we have given for the fixed point bifurcation of planar flows also hold in the case of planar maps. The observed analogy between the fixed point bifurcations in planar maps and flows can also be understood via a theorem of Takens [35], implying that any truncated expansion of a map L around a fixed point with linear part (331
for any b E R (including b = 0), is also the truncated expansion of the time-l map of some planar flow around a stationary point of a vector field with linear part (34)
201)
5. S. W. LAMB
and I-l. W. CAPEL
Moreover, the unfoldings in both cases give rise to similar developments of local fixed points, implying that the bifurcation diagrams of the local bifurcations around fixed points with linear part (33) are qualitatively the same as the ones we observed in the previous section. However, although the bifurcation diagrams of the fixed-point bifurcations of planar flows and maps are the same, the local dynamics is not the same. For instance, in two-dimensional (autonomous) flows there cannot be chaos, while in two-dimensional (autonomous) maps chaos is a generic feature. This implies that for the case of maps we have to reconsider our statements about the local saddle connections. Since the stable and unstable manifolds of saddle points are smooth manifolds our proposition in Section 2 concerning the saddle connections of the previous section is also valid in the case of discrete time maps. However, whereas in the case of flows stable and unstable manifolds cannot have transversal intersections, in the case of maps this is well possible and famous for being a source for chaotic behaviour. This implies in the case of maps that the homoclinic and heteroclinic points on the fixed sets of reversing symmetries indicate in general transversal intersections of stable and unstable manifolds in stead of smooth saddle connections. Note, however, that the smooth saddle connections that are supported by ordinary mirrors in the flow, will persist in the case of maps. Another difference is found if one has a close look at the centres. In the case of maps in general they appear as complex objects with KAM-tori and chaotic bands. As an example, in Fig. 18 we depicted (part of) the phase portrait of the map [12, 141
,I I
x’ = [C - y][l + (y’ - l)?], y’
=
X
1 + (y + 1 - c>* ’ at C = 2.93, a little after the occurrence of a m’ pitchfork bifurcation (at C = 2 v 2). In comparison to the corresponding pitchfork bifurcation in the case of continuous time flows (cf. Fig. 3(a)), it is observed that the boundary of the domain of attraction of the newborn attractor is not any more formed by a smooth homoclinic curve. In fact, we found the domain of attraction to extend to the right side of the saddle point. The domain of attraction seems to be bounded by a KAM curve that has survived the pitchfork bifurcation. The existence of KAM curves before the pitchfork bifurcation (in the nearly integrable case) is proved by the reversible KAM theorem [lo]. Period-doubling and more generally period-tuplings that occur in the case of qresonances (q = 2,3, . .) will not be considered in the present paper. 4. CONCLUDING
REMARKS
In this paper we have studied local bifurcations in (weakly) reversible two-dimensional dynamical systems around fixed points possessing an isotropy subgroup consisting of mirrors, two- and fourfold rotations. We dealt with continuous time flows and discrete time maps. Using only local normal form expansions, we found the (reversing) symmetries determining not only the ‘local’ bifurcations of the fixed points but also some global features such as saddle connections. In particular we found that the birth of saddle points in the presence of reversing mirrors in many cases gives rise to homoclinic and/or heteroclinic points. In discussing two-dimensional reversible dynamical systems, analogies and differences with area-preserving (Hamilton) systems have been stressed by many authors [.5, 10, 12,32,33]. The similarity between two-dimensional Hamiltonian and reversible
701
Local bifurcations on the plane
3
2.5
2
1.5
1
0.5
‘, I
0 0
0.5
I
1
I
1.5
I
2
I
2.5
I
3
3.5
Fig. 18. Local phase portrait of the map (35) with C - 2.93, just after a the cxcurrrnce of 21YH’ pitchfork bifurcation.
systems can be observed already on the level of the normal forms. In this paper we find that some of the local bifurcations that occur in the presence of a reversing mirror are well known to occur in two-dimensional Hamilton systems as well. In particular, the rn’ saddle-node is well known to be generic in Hamilton systems. Moreover, the pitchfork bifurcation with a centre (saddle) changing into a saddle (centre) and two centres (saddles) has been discussed by Rimmer [9] in the context of reversible (rn’) Hamilton systems. The bifurcations displayed in Fig. 3(b), (c) and Fig. 6 are also familiar in symmetric Hamiltonian vector fields, see [3S]. Furthermore, we have observed various examples of symmetry breaking bifurcations. In the case that the isotropy subgroups of the newborn fixed points do not possessa reversing symmetry, often attractor-repeller pairs are born, bringing into evidence the dissipative features of the dynamical system. In pursuing the investigations of the attractor-repellor pair in the case of mappings one may find a period-doubling cascade with dissipative values of the Feigenbaum constant 6 and the scaling factor a, as well as the occurrence of strange attractors 1121.
J. S. W. LAMB and H. W. CAPEl.
202
The considerations given above can also be applied to more dimensional systems. It turns out that the (reversing) symmetries give substantial restrictions to the possible bifurcations. In this way higher-dimensional bifurcation phenomena with (reversing) symmetries are easier to handle than without (reversing) symmetries, the latter case being much more difficult to analyse. Local bifurcations with reversing point group symmetry in threedimensional flows and maps will be treated in a future paper [36]. Ackno&+gement
-It is a great pleasure to acknowledge various useful discussions with John A. G. Roberts. REFERENCES
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