Phenomenological Model of Chaotic Mode Competition in Surface

Haverford College
Haverford Scholarship
Faculty Publications
Physics
1985
Phenomenological Model of Chaotic Mode
Competition in Surface Waves
S. Ciliberto
Jerry P. Gollub
Haverford College, [email protected]
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Repository Citation
Ciliberto, S., and Jerry P. Gollub. "Phenomenological model of chaotic mode competition in surface waves." Il Nuovo Cimento D 6.4
(1985): 309-316.
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IL NUOVO CIBIENTO
VoL. 6 D, N. 4
Ottobre 1985
Phenomenological Model of Chaotic Mode Competition
in Surface Waves.
S. CILIBERT0
Istituto .~azionale di Ottiea - J~argo E. •ermi
J.P.
6, 50125 .Firenze, Italia
GOLLUB
Hayer/oral College - Hayer]oral, PA 19041
Department o] Physics, University o] Pennsylvania - Philadelphia, PA 19104
(rieevuto il 4 Giugno 1985)
Summary. - - W e present a four-variable model (two coupled parametrically forced oscillators) that describes many of the phenomena seen
in an experiment on surface waves in which the competition between
spatial patterns produces chaotic behaviour. The model reproduces
the route to chaos, the dimension of the attractor, the Kolmogorov
entropy and (approximately) the phase diagram.
PACS. 47.25. - Turbulent flows, convection and heat transfer.
PACS. 47.35. - Hydrodynamic waves.
1. - I n t r o d u c t i o n .
I n a r e c e n t p a p e r (1) we described a n e x p e r i m e n t on forced surface waves
in which chaotic behaviour is clearly p r o d u c e d b y t h e competition b e t w e e n
two spatial modes. We also suggested a phenomenologicul m o d e l t h a t explains
m a n y of t h e e x p e r i m e n t a l results. I n this r e p o r t we describe t h e p h e n o m enological model in greater detail. The e x p e r i m e n t itself is t h o r o u g h l y t r e a t e d
in a separate p u b l i c a t i o n (2).
(x)
(2)
S. CILIBERTO and J . P . GOLL~B: Phys. Bey. T~ett., 52, 922 (1984).
S. CILIB]~RTOand J . P . GOLLUB: J . _Fluid Meek, 158, 381 (1985).
309
~10
S. C I L I B E R T O a n d
g. P. GOLLUB
We recall only t h e results of t h e e x p e r i m e n t to allow a direct c o m p a r i s o n
with those of t h e model. The s y s t e m of i n t e r e s t is a cylindrical fluid l a y e r in
a container t h a t is subjected to a small v e r t i c a l oscillation. I t is well k n o w n (s)
t h a t , if the driving a m p l i t u d e exceeds a critical value Ao(lO), which is a f u n c t i o n
of frequency, t h e free surface develops a p a t t e r n of s t a n d i n g waves. T h e surface
d e f o r m a t i o n S(r, O, t) can t h e n be w r i t t e n as a snperposition of n o r m a l modes:
(1)
S(r, 0, t) ---- ~
a,,,~(t)Jdk~.~r) cos ~0,
lp~'tt
where J t are Bessel functions of order ~ a n d t h e allowed wave n u m b e r s kt,~
are d e t e r m i n e d b y t h e b o u n d a r y condition t h a t t h e d e r i v a t i v e J'z(kt,,~R)= O,
where R is t h e radius of the cylinder. The modes m a y be labeled b y t h e indices
(giving the n u m b e r of angular m a x i m a ) a n d m (related to t h e n u m b e r of nodal
circles). The mode a m p l i t u d e at.~(t) develops an instability w h e n t h e corresponding eigenfrequency (given b y t h e dispersion law for capillary g r a v i t y
waves) is a p p r o x i m a t e l y in resonance with half the driving f r e q u e n c y 1o a n d A
exceeds A~(1o). This p a r a m e t r i c i n s t a b i l i t y leads to standing waves in which
t h e mode a m p l i t u d e oscillates at Io/2. To t a k e into a c c o u n t t h e possibility of
a f u r t h e r slow modulation of the m o d e amplitudes, which, in fact, occurs due
to mode competition, we write each a m p l i t u d e in t e r m s of fast oscillations at
1o/2 and slow envelopes Cz(t) a n d Bz(t):
(2)
at(t) = Cz(t) cos (~]ot) + B~(t) sin (~t/ot) 9
We omit the second subscript because, in practice, only a single value of m is
significant for a given v a l u e of 1.
The behavior of t h e s y s t e m as a f u n c t i o n of A a n d ]o is shown in fig. 1.
Below the parabolic stability boundaries, the surface is essentially fiat. A b o v e
the stability boundaries, the fluid surface oscillates at half t h e driving f r e q u e n c y
in a single stable mode. The shaded areas are regions of m o d e competition,
in which the surface c a n be described as a superposition of t h e (4, 3) a n d (7, 2)
modes with amplitudes having a slowly v a r y i n g envelope in addition to t h e
fast oscillation a t ]o/2.
Our e x p e r i m e n t a l a p p a r a t u s , described in (1,~), allows us to s t u d y a fixed
linear c o m b i n a t i o n of t h e slow coefficients Or(t) and Bt(t), which we d e n o t e
b y a~(t). The d y n a m i c s of the slow oscillation was explored b y v a r y i n g A and ]o
s e p a r a t e l y inside of t h e i n t e r a c t i o n region. I n fig. 2 t i m e series a n d corresponding power spectra are shown for t h r e e different driving amplitudes b u t
fixed driving f r e q u e n c y of 16.05 Hz.
(3)
T . B . BENJAMIN and F. URS~LT.: PRO0. R. 8or London, Set. A, 225, 505 (1954).
P H ~ I ~ O M ~ N O L O G I C A ~ MOD~ZL
20o
.
OF CHAOTIC MOD~ C O M P E T I T I O N I N 8URFAC F. W A R S
.
.
.
.
.
.
.
125
I
15.6
~
~
~
?iZr\
251
.
I
I
I
15.8
I
I
I
I
I
I
15.9
16.0
fo(HZ)
.-.
~perioeLic
/ \
15.7
all
I
I
16.1
I
16'2
16.3
Fig. 1. - Phase diagram as a function of driving amplitude A and frequency J0. The
crosses are experimentally determined points on the stability boundaries. Stable
patterns occur in the regions labeled (4, 3) a n d (7, 2). Slow periodic and chaotic oscillations involving competition between these modes occur in the shaded regions.
Io
~
6
o.)
'
-I0
1~/ c)
iI
0
..............
I
SL~)
4L
~
A
J
f*/2
i
..
1.
I
r.
/
=149
o
-10
,Oo
0
~'~~'~ '"'~ '"
300
t(s)
600
0
50
100
150
f(lO-3Hz)
="~ 1
200
'
250
Fig. 2. - The transEt[ou from periodic to chaotic oscillation. Time series a n d corresponding power spectra of the slow oscillation are shown for ]o = 16.05 Hz and three
different driving amplitudes. Broad-band noise is associated with the appearance of
a subharmonic j*[2 o~ the d o m i n a n t oscillation.
~1~
S. CILIB:ERTO
and
Z. P. GOLL~3B
As t h e driving a m p l i t u d e is increased, ~ chaotic s t a t e w i t h ~ b r o a d p o w e r
s p e c t r u m is obtained. We characterized t h e chaotic beh~viour q u a n t i t a t i v e l y (2)
b y c o m p u t i n g f r o m the e x p e r i m e n t a l d~t~ ~he correlation dimension v of t h e
a t t r a c t o r (4) a n d u lower b o t m d K~ for t h e K o i m o g o r o v e n t r o p y K (5). W h e n
the oscillation is periodic (A ~ 121 ~m), we find v = 1.0 + 0.04 a n d K~ =
(0.0•
s -~. On t h e o t h e r h a n d , w h e n t h e slow oscillation is chaotic
(A ~- 190 ~m)~ v -~ 2.22-4-0.04 a n d K~ ~ ( 0 . 1 ~ 0 . 0 1 ) s -~. These m e a s u r e m e n t s
clearly d e m o n s t r a t e t h a t t h e a t t r a c t o r has a low (and fractional) dimension
a n d t h a t t h e r e is a t l e a s t one positive L y a p u n o v e x p o n e n t . T h e dimension
m e a s u r e m e n t s also show t h a t a f o u r - d i m e n s i o n a l p h a s e space is required to
represent the data.
2. - Formulation
of the model.
W e h a v e c o n s t r u c t e d a r e l a t i v e l y simple p h e n o m e n o l o g i c a l m o d e l t h a t
has a r e a s o n a b l e h y d r o d y n a m i c basis a n d a c c o u n t s for m o s t of our observations~
including t h e basic s t r u c t u r e of t h e p h a s e d i a g r a m . W e begin w i t h t h e f a c t
t h a t , in a linearized inviscid a p p r o x i m a t i o n , e a c h m o d e a m p l i t u d e az(t) follows
a M a t h i e u e q u a t i o n (8):
(3)
az(t) ~- (eo~ - - ~o~A cos COot) a~(t) : O,
w h e r e oJ~ is t h e eigenfrequency, ~ is a g a i n coefficient a n d ~oo ~-2~]o. W e
t a k e t h e p o i n t of v i e w t h a t one can a p p r o x i m a t e l y a c c o u n t for t h e effects of
d a m p i n g (due to all sources, including b u l k v i s c o s i t y a n d wall effects) b y
i n t r o d u c i n g a first-order t e r m ~ .
F u r t h e r m o r e , it is n e c e s s a r y to a d d a nonlinear t e r m to l i m i t t h e g r o w t h of t h e m o d e to finite a m p l i t u d e in t h e s t e a d y
state. T h e lowest-order n o n l i n e a r t e r m is cubic in t h e m o d e a m p l i t u d e , t h u s
we h a v e t h e following e q u a t i o n for t h e t i m e v a r i a t i o n of a n y m o d e :
(r
az ~- ~a~ ~- (co~ - - ~p~A cos eOot)a ~--~ ~za~.
We find t h a t this e q u a t i o n is sufficient to fit t h e ( a p p r o x i m a t e l y parabolic)
s t a b i l i t y curves in fig. 1 a n d to describe q u a n t i t a t i v e l y t h e v a r i a t i o n of t h e
s t e a d y - s t a t e m o d e a m p l i t u d e w i t h A a b o v e threshold.
N e x t we consider t h e p h e n o m e n o n of m o d e c o m p e t i t i o n . W e k n o w t h a t
o n l y two m o d e s a r e involved a n d t h a t t h e y i n t e r a c t . T h e r e f o r e , we consider
a m o d e l consisting of two coupled M a t h i e u oscillators. T h e r e are v a r i o u s w a y s
to i n t r o d u c e coupling p h e n o m e n o l o g i c a l l y . One m a y allow t h e coefficient of
(4) P. G~ASS~ROv.~ and I. P~OCACClA: Physica (Utrecht) 1), 9, 189 (1983).
(s) P. GRASSBERG:EIr ad/d I. PROCACCIA: ~hys. l~ev. A, 28, 2591 (1983).
PHENOMENOLOGICAL
MODEL
OF CHAOTIC
MODE
COMPETITION
IN SURFACE
WAVES
313
the driving t e r m for one mode to d e p e n d on t h e a m p l i t u d e of t h e o t h e r mode.
Alternatively, one m a y ~llow each d a m p i n g coefficient to depend on t h e ~mplitude of the o t h e r mode. Fin~lly, one m ~ y introduce nonlinear t e r m s containing b o t h mode amplitudes. We have tried all t h r e e ~pproaches, b a t we
present results only for the first one, simply because it provides a b e t t e r fit
to the experiments. Therefore, to describe t h e i n t e r a c t i o n of the Z -~ 7 a n d
I = 4 modes, we set
V4=~,+fl4,a~
(5)
and
V,=~,+fl,,a~,
where the coupling coefficient fl,4 is negative, while/54, is positive. The origin
of t h e sign difference is the observed phase difference b e t w e e n the two modes
during the oscillation. I n fact, a,o leads a 4o b y a b o u t 90 ~ (see fig. 4 of ref. (x)).
This implies t h a t the sevenfold m o d e p u m p s t h e fourfold mode, while the
fourfold m o d e damps t h e sevenfold mode. To solve t h e s y s t e m of t h e two coupled
Mathieu equations, we express t h e m o d e ~mplitudes b y eq. (2). We s u b s t i t u t e
eqs. (2) a n d (5) into (4), keeping o n l y t e r m s oscillating at ~]o a n d neglecting
those at 3~]o. We ~lso neglect J~z ~nd 0, because t h e t i m e scales of t h e fast
oscillation a n d mode c o m p e t i t i o n are v e r y different, so t h a t /~z<< 2~Oo0~ a n d
0, << 2eOoJ~,. Finally, we obtain t h e following four-dimensional s y s t e m for the
slow variables C4, B4, C, a n d B,:
O, .
. 89. c4.
I~ 4 = - - 89
(6)
O, =
.
rA4
VoA + ~4(C4
o , + B~) - - fl~,B27]B4,
-r [A 4 -4- yJ~ -r ~~ C~ -+- B**) + fl[, C~] C4 ,
89 c,.
[A,.
~o. A + L(c, + B,)
o
,
~
o
,
f174B
4]B, ,
B, = - 89 B, + [~, + , f A + ~,(C~ + B~) + ~40~,]C,,
where we have i n t r o d u c e d n o r m a l i z e d coefficients ~s follows:
(7)
rio, --_ f14,A /2oo '
r
=
ar
3. - Integration and properties of the model.
I n this s y s t e m most of t h e coefficients cnn be measured. The d a m p i n g
coefficients 7~ and gain coefficients ~o are a d j u s t e d to fit t h e parabolic stability
curves of fig. 1, and the nonlinear coefficients ~z are chosen to r e p r o d u c e the
m e a s u r e d saturation amplitudes, all in a region where o n l y a single m o d e is
314
s. CILIBEI~TOand J. P. GO~LUB
present. We numerically i n t e g r a t e the s y s t e m using t h e f o u r t h - o r d e r l~ungeK u t t a m e t h o d and find t h a t r e g e n e r a t i v e oscillations (both periodic and
chaotic) are, in fact, p r o d u c e d n e a r t h e intersection of the stability boundaries
for t h e two modes. We adjust t h e two mode-coupling coefficients to obtain
a n oscillatory d o m a i n similar in size to t h a t f o u n d in t h e e x p e r i m e n t s (fig. 1).
The space d i a g r a m p r o d u c e d b y this set of model equations is shown in fig. 3.
T h e v a l u e of t h e p a r a m e t e r s used in t h e m o d e l ~re
~* =
77 ~
0.40
s -~ ,
r ~ 0.10 s -1
~p~ - ~ 5 1 . 3
cm -~ s -1 ,
~fl~ - ~ 5 2 . 6
po = + (7.0 era-' s-~)A,
o~ : 49.59 r a d / s ,
~ ,o, _-
cm -a s -~ ,
-
(9.0 cm-'
~
----- 1 . 0 0
s -x ,
s-~)A,
m~ --~ 50.92 r a d / s .
(The mode amplitudes are t a k e n to be dimensionless with a scale set b y t h e
a r b i t r a r y chioee of ~o.)
200
-,oo \
9
,
,
,
9
/
9
~, '
,9
\
5O
fo(HZ)
Fig. 3. - Phase diagram obtained from the phenomenologieal model by numerical
integration. This figure should be compared with fig. 1.
The parabolic stability curves fit t h e e x p e r i m e n t a l d a t a to within a b o u t
10 % for A <150 t~m. However, t h e shapes of t h e periodic a n d chaotic regims
are different f r o m those f o u n d e x p e r i m e n t a l l y .
I n order to compare the b e h a v i o u r n e a r t h e onset of chaos with t h a t observed e x p e r i m e n t a l l y , we p r e s e n t {fig. 4) t i m e series of the slow c o m p o n e n t B
and corresponding power spectra f o r t h r e e different values of A~ b u t fixed
]0-~ 16.11 Hz. This figure m a y be c o m p a r e d with t h e e x p e r i m e n t a l d a t a in
fig. 2. The basic period of oscillation is different b y a f a c t o r of two, a n unex-
P1TEIXrOMRlq0LOGICAD MODEL OF CHAOTIC MODE COMPETITION IN SURFACE WAVES
3]5
plained d i s c r e p a n c y . However~ in b o t h cases we find t h e following f e a t u r e s :
a single s u b h a r m o n i c b i f u r c a t i o n of t h e slowly v a r y i n g m o d e a m p l i t u d e for
c o m p a r a b l e A a n d ]o; a n i n c r e a s e in t h e b a c k g r o u n d noise level a t or n e a r this
b i f u r c a t i o n ; a n d a loss of all s h a r p s p e c t r a l s t r u c t u r e a t higher A w i t h o u t
f u r t h e r bifurcations. Thus t h e onset of chaos s e e m s to b e quite similar in t h e
d a t a a n d in t h e model.
6
f*
b)
.
6
~)
f*/2
If*
|
O"
6
f
z,
2
150
t(s)
300
0
300
600
f (10-3Hz)
Fig. 4. - Time series and Fourier spectra obtained from numerical integration of the
model at throe driving amplitudes. This figure illustrates the transition from periodic
to chaotic oscillation and should be compared to the experimental data of fig. 2. Both
experiment and model show a single subharmonio bifurcation with associated broadband noise onset.
We also m e a s u r e d t h e c o r r e l a t i o n d i m e n s i o n v for t h e c h a o t i c s t a t e s of t h e
model. We find v = 2 . 4 1 ~ 0 . 0 4 for A - ~ 175 ~m, a n d t h e s a m e r e s u l t a t
A ~---155 v m . T h u s t h e s t r a n g e a t t r a c t o r p r o d u c e d b y t h e m o d e l h a s a b o u t
t h e s a m e dimension as t h a t f o u n d in t h e e x p e r i m e n t .
T h e L y a p u n o v e x p o n e n t s of t h e m o d e l e q u a t i o n s were also c o m p u t e d w i t h
t h e m e t h o d p r o p o s e d in (6). T h e K o l m o g o r o v e n t r o p y K (the s u m of positive
I ~ y a p u n o v e x p o n e n t s ) is 0.33 s -1 a t A = 175 ~zm. T h e r a t i o K/] of t h e K o l m o g o r o v e n t r o p y to t h e f r e q u e n c y ] of t h e slow oscillation is a p p r o x i m a t e l y 1.1
for b o t h t h e m o d e l a n d t h e e x p e r i m e n t .
(s) G.
21 -
BENETTIN,
I1 N t w v o
L. GALGANI and J . M . STRELCYN: _~ecca~v~ea,15, 9 (1980).
(7ime~fo D .
316
s. CILIB]gRTO and J. P. GOr~L~rB
4. - C o n c l u s i o n .
We have p r e s e n t e d a simple phenomenological model for chaotic-mode
competition in an e x p e r i m e n t a l s t u d y of p a r a m e t r i c a l l y forced surface waves.
The model describes m a n y of t h e e x p e r i m e n t a l results fairly well, including
t h e qualitative s t r u c t u r e of t h e p a r a m e t e r space (fig. 3), the r o u t e to chaos
(fig. 4), the dimension of the resulting strange a t t r a c t o r and t h e K o l m o g o r o v
e n t r o p y . (However, t h e model is n o t in complete a g r e e m e n t with t h e data.
The period of oscillation is off b y a f a c t o r of two, for example, a n d t h e shapes
of the stability boundaries are different.)
One might e x p e c t t h a t a correct model for this t y p e of p r o b l e m would be
derivable f r o m a H a m i l t o n i a n w h e n the damping t e r m s are e l i m i n a t e d as in (7).
T h e model described in this p a p e r does not satify this condition because t h e
coupling t e r m s m u s t have opposite signs to fit t h e data. As a n alternative,
we constructed a four-variable m o d e l s t a r t i n g with the H a m i l t o n i a n formulation proposed in (~), a n d adding damping. However, we could n o t o b t a i n a
good fit to the e x p e r i m e n t a l d a t a using this approach. (It is possible, however,
t h a t we were simply unable to find t h e correct p a r a m e t e r s empirically).
I n t h e f u t u r e , it would be desirable to investigate the relationship b e t w e e n
low-dimensional models and t h e a c t u a l h y d r o d y n a m i c equations, i n s t e a d of
using a phenomenological a p p r o a c h as we huve done. The phenomenologieal
model does, however, provide a good s u m m a r y of t h e e x p e r i m e n t a l results.
This work was s u p p o r t e d b y N a t i o n a l Science F o u n d a t i o n G r a n t CME8310933. S. C~L~ERTO acknowledges t h e s u p p o r t of a n Angelo della Riccia
fellowship during his t e n u r e at H a v e r f o r d College. J . P . Gollub appreciates
t h e ussistunce of a Guggenheim Fellowship. We acknowledge helpful discussions with J. G~CKE~m~I_v~, T. LxJ~v,~s~r, J. M~]~s, I. P~ocAccrA a n d
H. SWXNNEY, and technical assistance for t h e calculations of t h e L y a p u n o v
e x p o n e n t s of the model b y A. POLITI.
(~) J. MILa~s: J..Etu~d Mevh, 146,285 (1984)..
Q RIASSUNT0
8i presenta un modello a quattro variabili (due oscfllatori accoppiati forzati parametrieamente) ehe deserivo molti dei fenomeni visti in un esperimento sulle onde di superficie nel qualo la competizione fra lo configurazioni spaziali produce un comportamento
eaotico. I1 modello riprodueo la via al eaos, la dimcnsione doll'attrattoro, l'entropia
di Kolmogorov e (approssimativamonto) il diagramma di fase.
Pe3mMe He nony~eao.