Ames W.F., Roger C. Nonlinear equations in the applied sciences. Volume 185
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260
I
R,
,;
..........................
"".i
....
.'I.
1,
:
'1.
:
I'
.
'&I.,
,,$
.......................
Greg King
and
Ian
Stewart
r
=
radial
direction
z
=
axial direction
2
8
=
azimuthal
angle
Figure
1:
The Taylor-Couette apparatus. Here Ri,Ro denote the
angular velocities
of
the inner and outer cylinders respectively, and
q,
TO
denote their radii.
Symmetric
Chaos
261
A
B
C
Figure
2:
(a) Taylor vortices. (b) Modulated wavy vortices (near
ends) and wavy turbulence (middle). (c) Turbulent Taylor vortices
(b). [Pictures (a) and (c) courtesy of Harry Swinney and Randy
Ta€Sl
5.
Wavy turbulence.
The wavy interface becomes ever more com-
plicated and turbulence sets in.
6.
Turbulent Taylor vortices.
The stratified pattern
of
Taylor vor-
tices reasserts itself, but now the overall flow is turbulent and
the interface between vortices is ‘flat’ only ‘on the average’, in
the following sense: if the velocity of the flow is averaged over
a
period of time then along the ‘boundary’ between turbulent
vortices, the average
axial
velocity is zero. The comparison be-
tween Taylor vortices and Turbulent Taylor vortices is shown
in Figure
2.
262
Greg
King
and
Ian
Stewart
3
Symmetry in Dynamics
Throughout most of this paper we concentrate on discrete dynamical
systems:
Here
f
:
R"
x
R'
+
R"
is
a
C"
mapping, and
X
E
R'
is an
r-dimensional bifurcation parameter. Usually either
r
=
0
(no bi-
furcation parameter)
or
T
=
1.
Continuous systems
Xt+l
=
f(Zt,X).
(1)
have been widely studied, and the results of this section apply equally
to
them. Discrete systems arise
as
Poincard sections of continuous
ones, and continuous systems can be obtained by suspending discrete
ones. Continuous systems play
an
important role in sections
7
and
8.
More generally we may replace
R"
by
a
C"
manifold, for example
a
circle
or
a
torus.
A
subset
A
of
R"
(or
more generally of the ambient manifold) is
said to be an
attmctor
for
(1)
if
(b)
A
has
a
dense orbit.
(c) There exists
a
neighbourhood
U
of
A
such that for every
u
E
U
we have
~(u)
C
A,
where
~(u)
is the w-limit set of
u.
A
similar definition holds for the continuous case.
For
the defini-
tion of w-limit set see Guckenheimer and Holmes
[22].
Suppose that
a
compact Lie group
G
acts linearly on
R".
Let
y.x
denote the image of
z
E
Rn
under the action of
y
E
G.
We say
that
f
is
G-equiwariant
if
(4
f(A)
=
A.
for all x
E
R",y
E
G.
(More generally the actions of
G
on source
and target may differ, but we encounter this case only briefly in
section
6).
We require the following terminology:
(a) The
G-orbit
of x
E
R"
is
G.x
=
(7.2
17
E
G}.
Symmetric
Chaos
263
The concept
of
a
G-orbit is important, because if
x(t)
is any solu-
tion of the dynamical system
1
or
2
then
so
is
7.x(t).
That is,
all
dynamical phenomena come
as
group orbits.
In particular if
A
is an
attractor for the dynamics then
so
is
7.A
for all
E
G. We say that
7.A
is
conjugate to
A.
(b) The
isotropy subgroup
of
x
E
R"
is
This measures the degree of symmetry of
a
point
x.
We also require
a generalisation.
If
x
E
Rn
we let
O(x)
be the orbit
of
x,
and define
the
(orbital) symmetry group
of
x
to be
--
As
=
(7
E
G
I
TO(~)
=
O(Z)}.
If
A
is
an
attractor having
a
dense orbit
ft(x),
then the
symmetry
group
of
A,
written
AA,
is defined to equal
A,.
(c)
If
C
is
a
subgroup of G (which henceforth we write
as
C
5
G)
then the
jixed-point space
of
C
is
Fix(C)
=
{x
E
Rn
16.2
=
x
for all
Q
E
C}.
Fixed-point spaces are useful because of the following result.
It is
trivial
to
prove but fundamental and remarkably powerful:
Lemma
3.1
Iff is G-equivariant
and
C
<
G then
f(Fix(C))
C
Fix(C).
Proof
Let
x
=
6.2
for all
Q
E
C.
Then
a.f(x)
=
f(a.x)
=
f(x).
Corollary
3.2
For every subgroup
C
5
G,
the fixed-point space Fix@)
is
invariant
under the dynamics off in equations
(1,
2).
264
Greg
King
and
Ian
Stewart
Thus equivariant dynamical systems are naturally equipped with
a
set of invariant linear subspaces for the dynamics. Often their study
can be simplified (at least in some respects) by restricting them to
such subspaces. For example, finding solutions that break symmetry
to
a
subgroup
C
5
G is equivalent to finding solutions that lie inside
Fix(
C)
.
If
g
:
R"
-+
R
is such that
f(7.z)
=
f(z)
for all
z
E
R",7
E
G,
we say that
g
is
G-invuriunt.
Invariance and equivariance impose
strong restrictions on the form of mappings and functions, and these
restrictions are basic
to
the theory of dynamics with symmetry. The
main general result for
C"
maps is due to the combined efforts of
Hilbert, Schwarz, and Pdnaru:
Theorem
3.3
(a) There exist finitely many G-invariants
p1,.
. .
,pr
such that every
C"
G-invariant is of the form
g(p1,.
. .
,pr)
for
a
C"
map
g
:
R'
-,
R.
(b) There exist finitely many polynomial G-equivariant maps
hl,
.
. .
,
h,
:
R"
+
R"
such that every G-equivariant is of the form
glhl
+
. . .
t
gah,
where the
gj
axe
Coo
G-invariants.
We call
p1,.
.
.
,pr
a
Hilbert
basis
for the invariants and
hl,
. . .
,ha
a
system
of
generators
for the (module
of)
equivariants (over the
invariants)
.
Warning
Computing the functions
pi
and mappings
hj
is not always easy, and
is normally done on
a
case by ease basis. There is no simple direct
general method.
Example
Let G
=
Dn
acting on
C.
The group is generated by
C
and
K,
where
C.Z
=
eaniInz and
K.Z
=
2.
A
Hilbert basis for the invariants is given
by the functions
aZ,
Re(?'),
Symmetric
Chaos
265
and
a
system of generators for the equivariants is
See Golubitsky
et
al.
[19]
p.52.
4
Numerical Examples
In
order
to
set the scene and exhibit some
of
the new phenomena
that arise in the chaotic dynamics of symmetric systems, we discuss
a
variety
of
numerical experiments.
4.1
Cubic
Logistic
Map
The simplest nontrivial group action is
22
acting on the line by
z
w
-2.
The
logistic
map
z
H
Lz(1
-
z)
has been widely studied,
but is not equivariant for this Z2-action. However,
a
minor variant,
the
cubic logistic
map,
is. It takes the form
z
w
kz(
1
-
z2).
Because
this is odd in
2,
equivariance is immediate. See
also
Rogers and
Whitley [30] and Chossat and Golubitsky
[9].
Figure
3a
shows
a
numerical bifurcation diagram for the cubic
logistic map. It is obtained by fixing the value of
k,
iterating the
map for long enough to remove transients, and then plotting the
next
100
iterates of
2.
After that,
k
is increased slightly and the
process is repeated.
We
see
an initial bifurcation from
a
Z2-symmetric fixed point
z
=
0
to an asymmetric fixed point (unlike the initial bifurcation
in the logistic map, which is
to
a
period-2 point). Then there is
a
period-doubling cascade (of asymmetric points), leading to chaos,
with the usual periodic windows interspersed. A period-3 window is
clearly visible.
After that, however, there is
a
new phenomenon:
a
sudden ‘ex-
plosion’
of
the attractor, from an asymmetric interval contained in
the positive half-line into
a
symmetric interval containing both neg-
ative and positive values of
2.
This is an example of
a
phenomenon
that Grebogi
et
al.
[21]
call
a
crisis.
What has happened is that
two disjoint strange attractors have collided and ‘fused’ into
a
single
266
Greg
King
and
Ian
Stewart
(c)
Figure
3:
Bifurcation diagrams for cubic logistic map. (a) Start-
ing from positive
x.
(b) Starting from negative
x.
(c)
Showing
all
attractors.
Symmetric
Chaos
267
attractor. Recall that
an
attractor, by definition, is
indecomposable,
that is, it cannot be split into two components, each of which is itself
an attractor. This is
a
consequence
of
the ‘dense orbit’ condition (b)
of section
3.
It is important to realise that the symmetric attractor
that exists after the crisis is not just
a
union of two separate asym-
metric attractors: it is indecomposable and has its own dense orbit.
To prove this we must use symbolic dynamics: see section
6
below;
but it is easy to convince oneself numerically,
as
follows. Figure 3(a)
is obtained by starting each iteration (for‘each
k)
at
a
positive value
of
x.
If
instead we start at
a
negative value, we obtain Figure 3(b).
This is Figure 3(a) upside down. Indeed this is
a
consequence of the
equivariance: if for some value of
k
we know that
a
susbset
A
C
R
is
an attractor, then
so
is its conjugate
-A.
If
we now combine the two
bifurcation diagrams into one, Figure 3(c), we see that the sequence
prior to the crisis actually involves pairs
of
symmetrically related
attractors, and that these do indeed merge at the crisis point.
The curious feature of such an event is that it
incmases
the sym-
metry of the attractor. Normally, bifurcations tend to
break
sym-
metry. It is true that reversing the bifurcation parameter trivially
turns
a
symmetry-breaking bifurcation into
a
symmetry-increasing
one, but generally there is
a
‘natural’ direction for the bifurcation
parameter, in which the degree of nonlinearity increases, often lead-
ing from order into chaos; and for non-chaotic states the bifurcations
in that direction normally lead
to
less and less symmetry. However,
when strange attractors
are
present, symmetry-increasing crises are
common. They have been reported several times in the literature: see
Kitano
et
al.
[24],
Sakaguchi and Tomita
[31],
Szab6 and T61
[33],
Chossat and Golubitsky [9],[10]. We postpone detailed discussion
until section
5.2,
and instead seek further examples.
4.2
Dihedral Symmetry
A
rich source of examples is afforded by mappings
of
the plane equiv-
ariant under the natural action of the dihedral group
D,.
The equiv-
ariants for this group action are stated in section
3
above. Chossat
and Golubitsky
[lo]
have studied the family of D,-equivariant map-
268
Greg
King
and
Ian
Stewart
pings
Field and Golubitsky
[
161 have produced high-resolution computer
pictures of the attractors of
3
in which pixels
are
colour-coded ac-
cording to how many times the point lands on them. This gives an
impression of the invariant measure on the attractor. One
of
Field’s
pictures was used
as
the cover for the
SERC
Nonlinear Systems Ini-
tiative brochure [32], and
a
second such picture drawn by Andrew
Cliffe appears inside.
We here fix
n
=
3.
To set the scene we first give some samples of
numerically plotted attractors in Figure
4.
These (and most other
attractors in this paper) were plotted using
Kuos,
a
dynamical sys-
tems package for
SUN
workstations written by Swan Kim and John
Guckenheimer
.
Symmetry-increasing crises readily occur in these maps. An ex-
ample is shown in Figure
5,
where three copies of
a
Z2-symmetric
attractor collide to create
a
D3-
symmetric attractor.
As
observed by Grebogi
et
ul.
{21], when
a
crisis occurs, there is
a
short period after the collision during which time the state
x
flips
intermittently between the remnants of the original three attractors.
This intermittency occurs here and
also
in other examples below.
Another example includes some regular dynamics
as
well, Fig-
ure 6(a-f). Here (a) shows not two invariant circles,
as
might appear
from
a
static view, but
a
single Z2-symmetric quasiperiodic orbit.
Successive points hop alternately between the two loops. In
(b)
this
breaks down (in what appears to be the usual manner for invariant
circles, but symmetrized) to form
a
curious attractor that appears
to consist of thin curves but actually has extra fine structure not
shown here. The conjugate attractors to this, shown in (d), merge
in (e) to from
a
single fully symmetric attractor. By stage (f) this
has acquired considerable extra structure.
To some extent the structure visible in these pictures is controlled
by the fact that the mapping
(3)
is in general not
a
diffeomorphism.
There is
a
tendency for the
gross
structure
to
be closely related to the
critical values
of the mapping, that is, the points
x
at which its Jaco-
bian is singular. For example we show in Figure
7
the set of critical
f(2,
A)
=
(a21
+
pv
+
A)z
+
7P-1.
(3)
Symmetric
Chaos
269
Figure
4:
Attractors
for
the mapping
(3)
with the following pa-
rameter values:
(a)
a
=
-1.1,/3
=
.17,7
=
-.79,X
=
1.89.
(b)
.944911,X
=
1.
(d)
a
=
1,p
=
O,7
=
.944911,X
=
1.
(Close up
of
centre
of
Figure 6(c).)
(Y
=
1.8,/3
=
O,7
=
1.34164,X
=
-1.89.
(c)
(Y
=
-1,p
=
O,7
=