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International Series of
Numerical Mathematics, Vol' 70
O 1984 Birkhiiuser Verlag Basel
1.
NUMERICAL STUDIES OF TORUS BIFURCATIONS
Iangford
The norrnal form equations for the interactlons of
a l{opf bifurcation and a hysteresls bifurcation
of stationary states can glve rise to an axisym-
metrlc attracting invariant torus. Nonaxisynnetrlc
perturbatlons are found to produce phase locklng'
period doubllng, btstability, and a farnily of strange
attractors.
Introduction
The rnajor unresolved problern of bifurcation theory today rnay well
be that of understandlng the transition to turbulence. This problem has
been the subject of several recent books and conferences, see [l-4]. The
so-called "main sequencet' [5r6] of bifurcations leading to turbulence beglns
with one or more bifurcations of statlonary states, followed by a Hopf
bifurcation to a periodic orbit, then a Nalnark-Sacker torus bifurcation
(and possibly a perlod doubling cascade), and flnall.y tlre appearance of a
strange attractor representiog turbulent flow. such sequences of
transitlons have been observed in experiments havlng sinple geometries, for
example Rayletgh-B6nard convection [7], Taylor vortices [8] ' and oscillating
chernlcal reactions [9]. Mathenatically, the first two or three bifurcations
in thls sequence are fairly well understood, and even explain some of the
experimenral data, but from the torus bifurcatlon <;rtward very 1lttle is
unders tood.
Recent studies of certain vector fields close to a singularlty
erith eigenvalues {0r*iurr-irrl} have revealed the existence ln thls context of
the following bifurcatl-on sequence: stationary bifurcation, Hopf
bifurcation, bifurcation to an invariant torus; see [10-20]' It is
convenient to think of these slngularities as resultlng fron the coalescence
of two "primary" bifurcations: a l{opf bifurcation (corresponding Eo i:ilc
eigenvalues *ior-iro) and a blfurcation of stationary states (corresponding
to the elgenvalue 0). Then it is not surprising to find stationary and l{opf
bifurcations on unfolding the slngularlty, however the appearance of an
286
invariant torus was unexpected. Its existence is guaranteed in a srna11 open
region of the unfolding-parameter space when certain inequalitles lnvolvlng
the low order nonlinear terms are satisfied. The prinary bifurcatlon of
stationary states entering lnto this coalescence rnay be any from a growlng
list of cases: saddlenode [12], transcritical [10], pitchfork [11 '13],
hysteresls or cusp [17], and lsola. Which case is to be expected in a given
problern depends on the number of parameters in the problem and on the
presence or absence of symnetries.
This paper is a continuation of the work in the previous
paragraph, in that it presents studles of bifurcations that occur after the
first appearance of an invariant torus. The approach is numerical, in
contrasE to the analytical results referenced above. The bifurcations in
question involve global dynamics, so complex that analytical techniques are
inadequate to give a fu11y detalled descrlptlon. One quite recent
analytical result, establlshed for the transcritlcal-Hopf case, is that in a
neighborhood of the si.ngular point the relative measure of parameter values
corresponding to quasiperiodic flow on the torus approaches unity as the
neighborhood shrinks to zero [18]. Another is the observatlon that, if as
the torus grows it approaches a heteroclinic saddle connection, then a
theorem of Silnikov rnay inply the presence of Smale's horseshoes and hence
chaotic dynamics [12]. Between these two extremes, very little is known
analytical1y. The present numerical studies help fill this gap and may
point the way for future theoretical work. Numerically we find strange
attractors wlth large basins of attraction, in contrast to the rrleakytt chaos
of the Silnikov mechanism.
2. The Model Equatlons
The numerical. results presented here are for the case of
lnteractl.ons of hysteresis and Hopf blfurcations' a case which is very rlch
in structure and is sti11 under investigation, but whlch promises to have
important applicatlons. We will assume that the spectrum conEains the
sirople eigenvalues {0, +io, -io} with the remainder in the negative half-
p1ane, and that a two-step prelirninary transformation of the system has been
perforrned, consisting of first a reduction to the 3-dirnensional center
287
nanifold, and then a transforrnation of this systen to its Poincar6-Birkhoff
nornal fonn. The resul-ting equations can be wrltten:
x'=(z -8)x-trly+h.o.t.
(1) y' = trrx + (z - B)y + h.o.t.
z' = \. * sz * ^"3 + b(t2 + y2) + h.o.t.
Here tr is the bifurcatlon parameter' and c, B are unfolding
parameters, all near 0. The frequency o comes from the original pure
irnaginary eigenvalue io. The a and b terms are "resonanttt, and "h.o.t.tl
stands for higher order terms, whlch do not affect the classification of
stationary and periodic solutions, but do influence the fu1l dynamics and
especially the flow on and near the torus. In principle, the higher order
terms can be transformed to be axisymnetric about the Z-axls up to
arbitrarily hlgh order, but the procedure does not converge in general; the
neighborhood of validity shrinks to zero as the order is increased.
Furthermore, axisymnetry is a highly nongeneric condltion. Therefore it
seems essential that nonaxisymmetry be retained if the conclusions are to
have any practical applicatlons.
The model equatlons used for numerical investigation were chosen
for computational economy while retaining the essential features of (1).
The variables and parameters were assumed rescaled to make the leading terms
in (f) of order one, then a and b were asslgned values -(i/3) and -1
respectively corresponding to one of the most interesting cases in the
general classification. The reuralning resonant cublc term in the z'
equation was retained along with a sinple nonaxisymrnetric fourth degree
monomlal to incorporate inportant nonaxl-symmetric effects. Hlgher order
terms ln the x' and y' equations have been dropped for the results presented
here, thelr presence does not seem to given bifurcation phenornena
qualitatively different from what has been found for (2).
x'=(z
Y' = ttx
- B)x - tty
+(z-B)y
= tr * qz - "3/3 - (t2 +y2)(f + pr) + ezx3
(2)
288
Here p and e are ttsmalltt, p determines the location of the
Nalmark-Sacker torus bifurcation, and e controls the nonaxisymnetry.
Analogous model equations have been derived from norrnal forms for the
transcritical*Hopf and pitchfork-Ilopf cases, and are described in [19,20]
and [11,17] respectively.
3. Nurnerical Results
Slnce it is the qualitatlve behaviour of soluEions that 1s of
interest, the results are presented here in graphical forno. The
computations were perforrned on an IBM personal computer, working in compiled
BASIC in double precision with an 8087 nunerical coprocessor, and the
graphics were produced on a Hewlett-Packard 7470A Plotter. The accuracy of
the computations is considered to be sufficient to show the location and
qualltatlve features of attractlng sets (o-lirnit sets), however the flow
lnside a strange attractor is characterlzed by divergence of trajectories
and sensitive dependence on intial conditlons, so thaE t.he numerical
solutions do not accurately predlct the location of a polnt on a trajectory
inslde a strange attractor over long time lntervals. Similarly the inltial-
value methods used here (Runge-Kutta and predictor-corrector) are lnadequate
for computing unstable orbits of saddle type.
Figures I to 13 show a series of computed solutions of system (2)
with parameter values c = 1, B = 0.7, tr = 0.6, to = 3.5, and p = 0.25 all
held fixed, whlle the axisynrnetry-breaking parameter € was slow1y increased.
Each flgure shows a slngle trajectory, plotted as a solid llne for X)0 and a
broken 1lne for X(0, after initial translents have died away. For each
flgure there ls a large basln of attracEion within which dtfferent cholces
of initlal point all lead to the same attractor. Prevlous studles of (1)
and analogous systems have traced the succession of bifurcations leadlng up
to an invarlant torus If7r19], so we begin here with the axisymmetric
lnvariant torus in Figure l. In Figure 2 wlth e = 0.025, one sees that the
trajectory has become more concentrated in one band around the torus and ls
less concentrated ln an adjacent band. Thls may be lnterpreted as a weak
resonant inte4Bction between the tvro osclllatlons on the torus. As e
increases, there appears to be a saddlenode bifurcallon of periodic orbits
289
Fig 3. Peri.od 4. Eps=& 04 Fig.4. Period & Eps=tl06
XEpe=0.07
Figl. Torus. Epe=0.0 Fig2. Tonus. Epe=0.025
Fi.g. A Pentod 16. Epe=0" 0675 Fig6. Period 32.
290
withLn the torus, giving rise to a stable lirnit cycle and an (unobserved)
unstable cyc1e. Figure 3 shows the period 4 1lmit cycle observed for e =
0.04, together with an initial transient portion of a typical solution
trajectory. we say that the system is now phase-1ocked, because the period
4 cycle is preserved if we vary the "forci-ng" frequency uJ over a small
interval; from the point of view of dynarnical systems theory, the system ln
Figure 3 is structurally stable. Increasing e to 0.06 glves the new
attractor in Figure 4, a cycle of period 8. The period 4 cycle evidently
still exists (between the two loops of the 8-cycle) but is now unstable, and
a period-doubling blfurcation has occurred between e = 0.04 and
e = 0.06. In the process, the smooth toroidal nanifold of Flgures I and 2
has been destroyed, because now a sma1l section containing the perlod 4 and
8 cycles is folded on itself by 1lJ0o every four revolutions, rather like a
Mobius band. Further increases of e produce a period doubling cascade, wlth
bifurcation values of e spaced more and more closely, see Figure 5 for
period 16 with e = 0.0675, and Figure 6 for period 32 with e = 0.07. We
have not tried to compute the Feigenbaum constant for this cascade, however
see [21]. Beyond this period doubling cascade, the solutions become even
nore complicated. For e = 0.09 there appears to be a "chaotic band" of
period 4, see Figure 7. Solutions from initial points in a large basln
approach this band quickly, but within the band the motion is aperiodic and
effectively unpredlctable over long tine intervals. Figure 8 shows the same
chaotie band in cross-section, cut by the x=0 plane. Figure 8 extends over
a much longer tine interval than Flgure 7; the trajectory (after translents)
has intersected the X=0 plane 1000 tirnes (500 Poincar6 maps), always falling
in one of eight "islands" (four for the Polncar6 rnap) which resemble
segments of a curve. The trajectory moves anong the islands ln the sequence
indlcated by the nunbers I to 4 in Figure 8. Note that ln this sequence the
island undergoes An S-fo1ding, then the "S" is flattened vertically onto
itself and stretched horizontally, and finally lt is roapped onto the
original island l. In this process the central portion of the island
reverses orientation, i.e. the outside becomes the inside, and the two ends
are folded inward. If the attractor is the limit of thls sequence as t+6
then it can not be siurply the short curve segment which it appears to be at
29t
\ l!:--_!-
t'--. d \-.1
Ftg 8. 500 Poincor6 l-lope.
Fig.9. Fo1ded Tonus. & 1Fig 10. 500 Poincor6 Mope.
\
I
I
I
Fig.7. Choottc Bond O0g
Ftg. 11. Turbulsnce. Eps=0.25 Figt2. 700 Poincon6 Mops.
292
first glance, but rather an lnfinitely folded curve of inflnite length, in
fact a fractal object [22]. A blown up portion of an island in Figure g
looks like a product of an interval with a cantor set, see [21] for an
analogous case.
Further increasing e causes the chaotic bands to widen and appear
to merge, eventually recreatlng the torus as in Figures 9 and 10, but now we
have not a smooth nanifold as in Figures I and 2, but a fractal object: a
thlck torus or bagel [23]. As e increases further, the torus vlsibly
thickens, and the flow becomes more and more ttturbulentt'or chaotic, see
Figures 11 and 12. Sti11 larger e causes the attractor to contact its basin
boundary, but it is not clear whether Newhouse sinks [24] are created in the
process. The attractor bursts through lts basin boundary, and due to the
nonaxisymmetry this occurs at first in a very srnal1 region which a
trajectory rnay fail to find for a long time. The result is 'transient
chaos'r, see Figure 13, with e = 0.28. A typical trajectory now resembles
the chaotic trajectory of Figure 1l for a long but finite time, but finally
escapes from the chaotic region and is draum to another attractor, in this
case a stable node on the negative Z-axis. This stable node coexists \,rith
the torus and chaotic attractors above for smaller values of e, and the
boundary separatlng their baslns of attraction can be extremely conrplicated,
probably another fractal set. Another exanple of coexistence of attractors
with cornplicated (fractal?) basin boundaries is shown in Figure 14, where e
= 0.07 and o = 5. Two limit cycles of periods 5 and 6, represented by the
symbols X and o respectively, coexi-st within the rrghost' of the former
invariant torus. very sma11 changes in initial points affect which cycle a
trajectory eventually approaches, and in fact preliminary work indicates
that the basin boundarles may be as cornplicated as the Julia sets of
iterated mappings of the complex p1ane.
4. Conclusions
Numerical computations have shown evidence of phase locking,
period doubling, coexistence of attractors, strange attractors varying from
a chaotic band to a thick torus, and transient chaos, all resulting from
nonaxi.symmetric perturbations of an invariant torus. This provides new
293
information on the unfoldings of the hysteresis-Hopf singularity. In
addition these rnodel equations may help in understanding the general problern
of bifurcaEions from invariant tori in 3D flows. Considerable recent effort
has gone into studies of bifurcations of rnaps of an interval [25], and of a
plane [26] I much of the motivation for that work comes from Poincar6 maps of
f1ows, but flows themselves have been considered too expensive for direct
study. The sirnple model equatlon (2) and those in [17,19] remove that
obstacle, opening the way to detalled computer-assisted direct studies of
the bifureations from tori to strange attractors.
Figures 7 to 12 show only a few sarnples of the variety of phase
portraits of chaotic or strange attractors for this system. It seems 1ike1,y
that these attraetors are topologically inequivalent and thus not
structurally stable in the classical sense. Yer ln a practical sense they
form a continuum, a smal1 perturbation of one of these strange attractors
yields a new strange attractor with sirnilar qualltative behavior. This
suggests that it may be necessary to devise a new more g1oba1 definition of
structural stability to deal with such strange attractors. The strange
atEractors studied here withstood perturbations far greater those that
destroyed the lnvariant tori which gave rlse to them. Thus the loca1
existence of quasiperiodic tori proven in [18] may be very loca1 lndeed,
whlle these strange attractors, whose existence has not yet been proven
rigorously, may in fact persist more globally.
ox o
Fi.g 14. Coextetence.
294
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lrl.F. Langford
Department of l'lathernaEics and Statistlcs
University of Guelph
Guelph, Ontario
Canada NlG 2W1
