glima|r
Uynumia
9 Springer-Verlag 1988
Self-generated aperiodic behaviour in a simple climate model
J M S a l a z a r 1 and C Nicolis 2
1 Facult6 des Sciences, Universit~ Libre de Bruxelles, (CP 231), B-1050 Bruxelles, Belgium 2 Institut d'A~ronomie Spatiale de Belgique, Avenue Circulaire 3, B-1180 Bruxelles, Belgium
Abstract. Climatic variability arising from the
coupling between ocean temperature and sea-ice extent is studied in a spatially distributed system.
A spatial degree of freedom is crudely introduced by the coupling, through energy transfer, of two
"box models" each of which describes a different space region. The evolution equations are cast into a normal form and some qualitative features of this general class of models are predicted. It is shown, both analytically and numerically, that in- ternally generated complexity in the form of aper- iodic behaviour can be a natural consequence of spatially distributed systems.
1 Introduction
The complexity of the climatic variability over the past million years is a well accepted fact. Its ori- gin remains, however, still largely open. Accord- ing to one view, the sharp 100-kyr maximum of the variance spectrum and the less pronounced maxima near 20 and 40 kyr arise essentially from coupling between the climatic system and exter- nal periodic forcings of astronomical origin (see e.g. Berger 1981 and references therein). A second possibility, suggested by the existence of a strong aperiodic background in the spectrum, is that in- ternally generated complexity plays a prominent role (Nicolis and Nicolis 1984). Clearly then, the development of simple mathematical models should be of great help in assessing the merits and drawbacks of these views, or even whether they are complementary.
In recent years several authors have analysed the effect of external periodic forcings on simple
Offprint requests to: C Nicolis
climate models giving rise to periodic oscillations (Le Treut and Ghil 1983; Saltzman et al. 1984a;
Nicolis 1987). The main motivation of these stud- ies was to see whether a "basic" internally gener- ated oscillation can give rise to a climatic signal reminiscent of the record. Ample evidence of complex dynamics in the form of aperiodic be- haviour has been produced. In particular, Nicolis (1987) has established the possibility of chaotic attractors arising from the coupling of the sea- ice-ocean temperature system with astronomical forcings of periodicities in the range 103-105 years.
Implicit in the above series of works is the idea that the 100-kyr peak of the paleoclimatic spectrum indicates the existence of basically peri- odic dynamics. Now, in many instances it may happen that in a system undergoing aperiodic be- haviour (which may be quasi-periodic as well as chaotic) the power spectrum may still display a distinguished maximum, along with certain addi- tional features such as a continuous background or a splitting of spectral lines. Our purpose in the present paper is to report on the first known un- forced climate model giving rise to internally gen- erated complexity in the form of aperiodic behav- iour.
The model is presented in Section 2. In Sec- tion 3 the equations of evolution are cast into a normal form which allows us to undertake the study of qualitative behaviour and to sort out some general features independent of the details of the particular model. In the bifurcation analy- sis reported in Section 4, the full list of transition phenomena predicted by the normal form is de- rived. Most prominent among them is the possi- bility of quasi-periodic behaviour, whose charac- teristics are analytically determined. In Section 5 the analytical results are supplemented with nu-
106 Salazar and Nicolis: Self-generated aperiodic behaviour in a simple climate model merical simulations. Additional results, applica-
ble outside the range of validity of the normal form, are also reported. A short discussion is pre- sented in Section 6.
2 The model
We focus our attention on climate models de- scribing the interaction between sea-ice extent and mean ocean temperature. Let 0 be the devia- tion of the sine of latitude of the sea-ice extent from a reference state and 0 the excess of mean ocean temperature. As suggested by Saltzman et al. (1982), the following set of equations describes the essence of these interactions:
- - = u2g- ulO- 302g dO
d i (1)
dr~ = - ~ 2 ~ + ~ 1 0 d f
in which all the parameters are positive. It has been shown that (1) give rise to a H o p f bifurca- tion, beyond which a stable limit cycle is gener- ated. In view of the uncertainties related to the values of parameters, its periodicity cannot be de- termined very sharply. A lower b o u n d seems to be in the 103-year range, while the upper b o u n d is 105 years, the characteristic time of Quaternary glaciations.
In a previous work (Nicolis 1984a) it has been suggested that the lumped oscillator in the form of (1) cannot provide a fully satisfactory view of the dynamics. As a first step towards the study of spatially distributed oscillators, the coupling be- tween two Saltzman oscillators (representing, for instance, the two hemispheres or even two regions along a longitude belt) has been considered. Ad- opting the rather reasonable assumption that the strength of coupling is proportional to the tem- perature inhomogeneities, one is then led to the following set of coupled equations
1984a):
d f = - {/)~1) O1 ']- ~{1) O1
dO1 __ ~21)/~1 -- ~p-~l) 01 - I/I(31) 0 2 O1 dT
+ D1(02-0,) d[
dO2 _ l/_/(22) 02 -- t/J{2) 02 -- I/'/(32) 0 2 (~2 d{
"[- 0 2 (01 -- 02)
(Nicolis
(2)
These equations constitute the starting point of the present study. We first switch to scaled varia- bles (Nicolis 1984b) defined by
qC
t
0~ (3)
The result is
O1 =bl O1-al /]1- /]2 01 + d 1 ( 0 2 - 0 0 0.1 = -/]1 + 01
02 = b2 02 -- a2772 - - / ] 2 02 + d2 (01 -- 02) 02 = --/]2 "4" 02
(4)
where the following combinations of initial pa- rameters have been introduced"
a i - ( ~ ) 2
b,- (5)
d i - 6i
and for simplicity it has been assumed that the two subsystems differ solely in the values of D, ~1 and hu2. Despite these simplifications, the com- plexity of the scaled system (4) remains considera- ble. This calls, therefore, for a qualitative study, to which we now turn our attention.
3 Reduction to normal form
In the absence of coupling and provided that bi > ai, it can easily be shown (Nicolis 1984b) that each individual oscillator in (4) has a single stea- dy-state solution/]i = Oi = 0, undergoing a H o p f bi- furcation at a critical value bit = 1. It is therefore natural to expect that some essential characteris- tics of the behaviour of the coupled oscillators could be viewed as resulting from the interaction of two H o p f bifurcations. This type of problem has been investigated extensively in modern liter- ature of dynamical systems (Guckenheimer and Holmes 1983). Because of the nonlinearities, glo- bal solutions cannot be determined. But under
certain conditions a local analysis, amenable to familiar perturbation methods, can be carried out.
One way to secure these conditions is to place on- eself in a range of parameter values in which small amplitude solutions occur for all four varia- bles. This means that (4) should undergo two coalescent H o p f bifurcations simultaneously. This degenerate situation implies in turn that the sys- tem's parameters should simultaneously satisfy two conditions (instead of bi~ = 1 o r b2c = 1 men- tioned above): for this reason one speaks of a codimension two bifurcation.
Let us determine these conditions for (4).
Choosing for simplicity d~ = d 2 = d , we can write them in the compact form:
b2r = [(4 (d + 1) + (a2 -- a 1))
+r al) 2 - 1 6 d 2 1 / 4 bl~ = [(4 (d + 1) + (a2 -- a 1))
-T-l//(a2 -- al) 2 -- 16d2]/4 (9) The oscillation frequencies tom and 092, for the interacting H o p f bifurcations at the critical point [bl~,b2~l, are determined by the critical eigenvalues 2~,2 = + ico~ and 2~,4 = + ic02 of the equation
e(2)=/~4 + 22Bc--}- Dc=O (10)
where Bc and Dc are obtained by substituting (9) into (10):
\0-1 r12
displaying the linearized operator
(6)
0
R ---- 0 b2 - d (7)
0 1
and the nonlinear contributions L ~ = - r / 2 0 i for i = 1 , 2 .
N o w we search in the parameter plane [bl,b2]
for a point [b~,b2r in which the stability operator R has two pairs of purely imaginary eigenvalues.
It is expected that interesting bifurcation phe- nomena such as transitions from periodic to qua- si-periodic motion can arise in the neighbourhood of [ble, b2c]. Direct algebra gives us the following characteristic polynomial for the matrix R:
P(2) = ) 4 + A~3 + B22 q_ C2 + D
A = 2 ( d + 1 ) - ( b i - b 2 )
B = - b ~ d + b2bl + 1 - d b 2 + a2al - 2b2 - 2bl + 4d
C = 2 d + a l + a 2 - ( a l b z + a 2 b l ) + d ( a l +a2) + 2bib2 - 2d(bl + b2) - (bl + b2) D = - b i d + b l b 2 - b l a 2 - a l b 2 - db2
+ a l d + da2 + ala2
(8) with
The conditions to have coalescence of H o p f bifur- cations are A = 0 and C = 0, which lead to
Br = a2 -Jr- al + 1 + 4 d - d(blc -'}- b2r - 2(b1~ + b2~) + blr
Dr = blcb2c - d(blr + b2c) - blca2 - b ~ a l + d(al + a2) + ala2
We are now in the position to transform (4) into a normal form. The basic philosophy is to re- strict oneself to the vicinity of [bit, bee] and try to eliminate, by means of an adequate transforma- tion, as many nonlinearities as possible. The pro- cedure may be d e c o m p o s e d into three steps (Ar- nold 1980; Wang and Nicolis 1987).
L i n e a r transformation
The aim of this transformation is to diagonalize the linear operator R. This is achieved by the change of variables
(ol) (zi /
T]I Z~
02 = Q z2
'h Iz~]
(11)
where the columns Qi of the matrix Q are the ei- genvectors of R corresponding to the critical ei- genvalues 27. Straightforward algebra gives
Q~ =
(a2 - [s -27111 + 271)/d (a2- [s-27111 +271)/(d(1 +27))
(1 -~27) for i = 1,4 (12) with s = b 2 c - d . Notice that the new variables zi are complex valued. The form of Q - 1 needed to carry out the diagonalization is given in Appendix A.
108 Salazar and Nicolis: Self-generated aperiodic behaviour in a simple climate model The transformed form of (4) reads
Z, = R ' Z + L'[Z]
with Z=[zl,z~,z2,z*] R'=Z[fi/j and Q -1L[QZ]-= k ( Z ) .
(13)
L'[Z] = -
Nonlinear transformation
We seek new variables y, related to Z through zi =y~ + h~(yfl, i,j = 1,... 4 (14) in terms of which as many nonlinearities as possi- ble are eliminated from (13). It can be shown (Ar- nold 1980) that this can be achieved if hi is a poly- nomial containing quadratic and cubic terms. Not all the nonlinearities can be eliminated however:
those that subsist are associated with resonance phenomena, whose details need not be developed here. Suffice it to mention that such terms are o f the form Yl lyll 2, Yl ly212, Y2 [yll 2 and y2 [Y2l 2.
Yl =#AlYl -A21ooYl lyll 2 - A l o u Y l ly212 p2=#Z)y2_Boo21Y21Y212_Bllmly~12y 2 (16) The explicit form of the coefficients Aokt and Bo.kt is given in Appendix C.
Taking advantage of the relative simplicity of (16) as compared to the original equations, (4), we shall now explore, using standard bifurcation analysis, the various modes of behaviour accessi- ble to the system.
4 Bifurcation analysis
Let us, for convenience, transform (16) to polar coordinates. We set
y~=r~ exp(i(&) for i = 1,2 (17a) and
A2100 = a30 + ifl30, A 1011 = ~12 + ifll2
B0021 = a03 + ifl03, BlUO = 0~21 + if121 (17b) With these substitutions we can write (16) as:
Unfoldin9
In practice, the computation of the normal form is carried out exactly at the bifurcation point. The vicinity of this point is subsequently explored by displacing slightly the parameters from this crit- ical situation
b i = b i c q - b i l , bil ,< 1 (15a)
and, consequently, the eigenvalues of the linear operator R from their critical values Z~,
2i ~----Xi -[-#/q~i , # "~ c 1 1 (15b)
The explicit expressions for Z~ are given in Ap- pendix B.
The effect of these deviations on the normal form, known as unfolding, has not yet been fully assessed for the kind of codimension two bifurca- tion considered in the present paper. Neverthe- less, if one is close enough to the bifurcation point and one is concerned with the dynamics in a small neighbourhood of the unstable steady state r/i=0~=0, the size o f which is typically lyel _#1/2, it is easy to show that the unfolding affects only the linear terms of the equations. After a tedious calculation one finally arrives at the normal form equations given by
/'1 ~ # l r l - - ~ 3 0 r l - - o~12rlr 2 3 2 1 ~-" 0"1 - - ] ~ 3 0 r 1 - - ] ~ 1 2 r 2 2 2
i'2 = (/11 - - # 2 ) r 2 - - a 0 3 r 3 - - a 2 1 r 2 r 2
- 2 2
0 2 = O"2 - - flo3r 2 -- f121r l
(18)
where /21 = IRA, I, # 2 = IRA 1, o-1 = Z ] + HAl and 02 = Z ; + llZ3 ~. A general classification of bifurca- tion diagrams for (18) may be found in Iooss (1981).
The analytic study of these equations predicts the following cascades of bifurcations (cf. Table 1). First, the trivial steady state (s.s.) (rbr2)= (0,0) becomes unstable. This can happen at # 1 > 0 , leading to a limit cycle bifurcation at # 1 = 0 , whose radius is rls = (#1/a3o) 1/2, the value r2, of I"2 being r2,=0; or at #1 >#2, leading to a limit cycle (T 1) bifurcating at #1=#2, whose radius is
r2s=((#1--#2)/a03) 1/2, the value rl, of rl being rl, = 0. At the next step, a secondary bifurcation can give place to two possibilities: either the coex- istence of Jr1,, 0] and [0,r2s]; or through the insta- bility [rl~,0] or [0,r2A to a state (rl,,r2,) in which both rl and r2 are nonzero. On inspecting the equations for the phases ~Pl and ~02 in (18), we conclude that this solution should be, typically, a quasi-periodic regime since in most cases the fre- quencies 01=~bl and c02=~b2 would be noncom-
Table 1. C a s c a d e o f b i f u r c a t i o n s
Stationary state (0~0) Primary b~furcations T ~
(r1~)2 = #~ (rz~)z #~-~t2
O~30 /~l)~
S u p e r c r i t i c a l i f a30 > 0 S u p e r c r i t i c a l i f a03 > 0 S u b c r i t i c a l i f a30 < 0 S u b c r i t i c a l i f a03 < 0
ro~ =a~-f13o #~ COz=Oz-flo3 #~ -/22
~30 ~03
Secondary bifurcations T 2
#1 = a 3 0 - ff2 - 0~30 - - r ]
(ra,)2 = a o 3 / - z l - - a 1 2 ( # 1 - - / - t 2 ) a300~03 -- ~120~21
T e r t i a r y b i f u r c a t i o n T 3
~30 (a03 - - a a 2 ) + a03 (a30 -- a21)
# 2 = # ~
a3o(ao3 -a,2)
w i t h c~c~o3 - c~l~t2~ > 0
# 2 = (a12__ a03) # 1 O~12
(r~) 2 _ a3o(u~ - ~ ) - a2~#~
~30a03 - - ~12a21
2 2
r ~ = a 2 - i l l , (r,~) - flo3 (r2,)
mensurate. The corresponding attractor would be a two-dimensional torus, (T2). Although this type of attractor cannot display the sensitivity to initial conditions typical of chaotic dynamics, it nev- ertheless can give rise to complex behaviour as long as resonance between co~ and c02 is avoided.
The appearance of a third frequency correspond- ing to a bifurcation T 2 ~ T 3 implies several condi- tions on the coefficients aa of the system (18) (see Table 1). However, in the range of parameters studied in the present work, the possibility of such a transition is excluded.
5 N u m e r i c a l s i m u l a t i o n s
Local behaviour near criticality
In this section we compare some representative numerical simulations carried out for the system (4) with the theoretical predictions given by the normal form (18).
Our system has five parameters: ai, b; and d with i = 1,2; bl and b2 being our bifurcation pa- rameters. On the other hand, the distance from the H o p f bifurcation is given by the perturbations bi~ = b~-b~c, which according to the normal form analysis must be very small. In what follows we proceed by varying a b a2 and d under the restric-
tion that the value of blc be comparable to that ob- tained for the case of a unique lumped oscillator [be ~ 1 (Nicolis 1984b)].
The parameters al and a2 were varied from 5.4 to 12.4, and the parameters b11, b2a from
- 1 x 10 -5 to 0.1. In terms of the original model with q~-I N 600 years (Saltzman et al. 1982) and in the vicinity of the H o p f bifurcation, this leads to a self oscillation ranging from a periodicity P - 1 2 0 0 to P ~ 2 0 0 0 years. For the range of pa- rameters studied, the analysis of Section 4 pre- dicts the existence of either periodic or quasi-peri- odic motion. In the case of quasi-periodic motion we observe that the difference in ~ot and ~2 grows when the difference in parameters aL and a~ in-
c r e ~ s e s .
Let us now consider a particular case with a a = 6 . 4 (Saltzman et al. 1982), az---10.4 and d = 0 . 4 , which exemplify a transition from a stationary state to a periodic motion and a sec- ondary transition to a quasi-periodic motion ( s . s ~ T ~ - + T 2 ) . In the case of our example, the ei- genvalues at the bifurcation point (b~,b2~)=(1.3165, 1.4834) are Aa =2.3880i, Z3=2.9884i and the normal form has the follow- ing expression:
kl = / - q r l - 6.369 r 3 - 0 . 1 4 8 3 r l r 2
/'2 = (ill - - , / 1 2 ) / ' 2 - - 0.4766rZr2 - 0.4923r~ (19)
110 Salazar and Nicolis: Self-generated aperiodic b e h a v i o u r in a simple climate model
For bal=0.1 and b21=0.05 corresponding to a small deviation from critically in #1=0.053 and
#2=0.021, respectively, the normal form predicts a supercritical bifurcation with r2,= 0.008, which starts to bifurcate at #a = 0. This bifurcation corre- sponds to a periodic solution (T a) of frequency, in dimensionless units, o91=2-4629 ( P ~ 1600 years).
From the branch ras, a secondary branch arises at /-tl = 0.023. This secondary branch corresponds to a T 2, with frequencies ogas = 2.454 ( P ~ 1600 years) and o92s = 3.0126 ( P ~ 1300 years). As the two latter frequencies can be regarded as noncommensu- rate, it is expected that quasi-periodic motion will arise.
When the perturbation is bH=0.01 and b2a = - 0 . 0 5 , the normal form presents a supercri- tical bifurcation with (r2s)2=0.1756. This branch starts to bifurcate at 0 on/-.s =l-~2 and leads to a limit cycle with frequency o92p = 2.4578. The possi- bility of a secondary bifurcation from this branch (r2,) is excluded, given that the numerical value for r2~ is negative.
A tertiary bifurcation leading to a T 3 could take place if certain conditions are satisfied (see also Table 1). This would correspond to the ap- pearance of a small third frequency in the Fourier power spectrum. This possibility was not ob- served in the range of parameters studied in the present work.
The above theoretical predictions are con- firmed by numerical simulations. Figure 1 a, b and c shows, respectively, the time evolution of r/a, a two-dimensional projection of the attractor in the 0a, 02 plane and the Fourier spectrum of 7/2 for the parameter values corresponding to a quasi-peri- odic solution; Fig. 2a, b, c is obtained for param- eter values corresponding to a periodic solution.
We see that in the vicinity of the critical point the qualitative agreement with the normal form anal- ysis is excellent. This is no longer true when one places oneself in a parameter region for which the system is far from the bifurcation point. For this reason we will next explore this region by numer- ical simulations.
Complex dynamics far from the bifurcation point The bifurcation analysis and the numerical simu- lations reported above show that if the values of al and a2 are close and, in addition, da and d2 are taken to be identical, the characteristic frequen- cies o9a and 092 of the coupled system are of the same order of magnitude. For reasonable values of the characteristic relaxation times of ice this
r] 1 05
0.0
-0.5
20 40 60 80
t , , , , I , , , , I , , , T I , , , , I , , , , I , , , ,
8z b
1
[
-1
l l t l l l l ' l l l l ' l l l l I R l l 1 ' J l l = l l
--2 0 2
01
P(Q)
t . :
tO-S~
25 7 5 Q 12.5
Fig. 1. a Time evolution of ~/1. b Two-dimensional projection of the quasi-periodic attractor in the 01, 02 plane, e Fourier spectrum of the r/2. Parameters used: a ] = 6 . 4 , a2=10.4, bl = b l c + 0 . 1 , b2=b2c+O.05, dl=d2=0.4
leads to overall periodicities of a few thousand years.
A major problem in climatology is to explain the origin of long-term variability in a system in
I ]
01 1]1
05 1
_ii i
If2 0.05
0.00
-0.05
P(Q) tO-Z
10-4
20 /,0 t 60 80
_ , , , , I , , , , I , , , , t , , , , I , , , , I , [ , b
l , ~ , l J , , , l J J ~ , l , , , J l , l l l ] , , , ,
-0.2 00 03
ql ' l l ' l ' ' ' ' l ' l i l l ' l ' ' l l l t = l t l '
C
0
, I II J J J I ~ L , I
2.5 5.0 7.5 Q 10.0 12.5
Fig. 2. a Time evolution of 01. b Two-dimensional projection of the periodic attractor in the r/l, r/2 plane, c Fourier spectrum of 02. Parameters used: as in the Fig. 1 except bl =bac+0.01 and b2=b2c-0.05
20 t,0 60 80
t
j Fig. 3. Time evolution of r/1. Parameters used: as in Fig. 1 ex- cept a2=3.06, bl =2.0, b2=2.2, d2=0.5
Berger 1981). We now show that, far from bifurca- tion, our coupled oscillator model can also show similar behaviour. In addition, our simulations will reveal other kinds of exotic behaviour o f po- tential interest.
In all cases considered, keeping the values of al and a2 comparable, we relax the assumption dl = d2. According to the third relation (5), dl and d2 should typically be different and it is this type of flexibility that we would like to explore pres- ently.
Figure 3 reports one interesting situation. Glo- , bally speaking the behaviour is quasi-periodic, as in Fig. 1. But in contrast to this latter case, we ob- serve a slowly varying envelope superimposed on a faster oscillation. A second type of new behav- iour is shown in Fig. 4a. The signal for 02 is very erratic, and the corresponding Fourier spectrum, Fig. 4b, has an intricate structure. This is a strong indication that the quasi-periodic attractor has evolved to a chaotic one. On the other hand, the signal for 01 and its Fourier spectrum (Fig. 4c and d) are more regular. This illustrates the possibility of asynchronous behaviour and varying phase lag in climatic changes between different regions o f the globe (see also Nicolis 1984a).
As the parameters o f the system are varied, one observes that aperiodic behaviour is fre- quently interrupted by periodic windows, arising through phase locking. Figure 5 illustrates this be- haviour. As in Fig. 4, we also observed asynch- roneity and phase lag: the time series for box 1 (Fig. 5a) are very simple and those for box 2 show considerable complexity (Fig. 5b), while remain- ing periodic. In addition, the frequency of the lat- ter box is about 8 times smaller. This difference is which the time scales of the individual processes
are much shorter (see Saltzman et al. 1984b). A popular idea is that this kind of variability is in- duced by astronomical forcings (see, for instance,
112 Salazar and Nicolis: Self-generated aperiodic behaviour in a simple climate model
iI . . . . I . . . . I . . . . I . . . . I . . . . I ' ' ' ' ~
02 PIP . b
10-4
_ 10 -6
/,', , , I I , I , I , , I i I , , , i I j II, i I , ; , i
131 20 L,O t 60 80 P d 2 5 5 0 7
i
20 40 t 60 80 2.5 5.0 7.5 f~ 10.0 12.5
Fig. 4. a Time evolution of 02. b Fourier spectrum of 02. c Time evolution of 01. d Fourier spectrum of 01. Parameters used: as in Fig. 3 except bl=b2=2.0, d1=0.2 and d2=0.6
also reflected in the two-dimensional projection of the attractor in the 0t, 02 plane (Fig. 5c).
6 Discussion
In this paper we have explored the role of aper- iodic behaviour in the interpretation of long-term climatic variability. We have shown both analyti- cally and numerically that aperiodicity can be generated by the coupling between spatial degrees of freedom. In most of the cases analysed it corre- sponded to a quasi-periodic attractor with two ir- rationally related frequencies. In addition to the irregular time dependence of the variables, a qua- si-periodic attractor gives rise to an interesting structure of the power spectrum (see e.g. Fig. lc), some aspects of which are reminiscent of the var- iance spectra of the paleoclimatic record. For in- stance, the peaks have a fine structure associated
with the splitting of the central line to satellite lines centred on nearby frequencies (Pestiaux 1984). Some indications of more complex aper- iodic dynamics likely to be associated with a chaotic attractor (see Fig. 4b) were also found in the numerical simulations far from bifurcation.
The point of view adopted in the present work is that one way of modelling complex behaviour is to localize in parameter space singular situa- tions corresponding to coalescent bifurcations, and subsequently to cast the dynamics into a nor- mal form of general validity. We believe that the method is promising for a variety of reasons.
First, in m a n y instances the detailed parameter values and even the very form of feedbacks is poorly known; under these conditions it becomes important to make qualitative predictions that do not depend in a sensitive way on the specific form of the evolution equations. Second, in a typical problem of geosciences one deals with a multivar-
rl 1
-2
variables o b e y i n g the normal form equations.
This p r o v i d e s o n e with a dynamical a n a l o g u e o f
"diagnostic" a n d " p r o g n o s t i c " e q u a t i o n s em- p l o y e d a b u n d a n t l y in m e t e o r o l o g y and climatolo- gY.
Acknowledgements. J. M. Salazar acknowledges a fellowship from the Universidad Nacional A u t o n o m a de Mexico. This work is supported, in part, by the E u r o p e a n Economic Com- munity u n d e r contracts ST2J-0079-1-B (EDB) and ST2J-0079- 2-B.
A p p en d i x A
20 40 t 60 80
q2 I
0
-I
20 g0 60 80
t
02
_t,
' I ' I ' [ ' [ ' I , I ' I
C
I I I I i I 1 I i I I I I I
-6 -2 01 2 6
Fig. 5. a Time ev_olution of 7h. b Time evolution of r/2. c Two- dimensional projection o f the periodic attractor in the G, 02 plane. Parameters used: as in Fig. 4 except a 2 = 3 . 4 , d: =0.01 a n d d2 = 0.05
iate system. Focussing on the class o f b e h a v i o u r c o n d i t i o n e d b y a particular kind o f b i f u r c a t i o n al- lows one to r e d u c e drastically the n u m b e r o f de- gress o f f r e e d o m by retaining a set o f privileged
The first step to transform (4) into a normal form is to diagon- alize the linear operator R Eq. (7). This is achieved by the lin- ear transformation Q, which in the present case is a 4 x 4 ma- trix whose columns are the four right eigenvectors of R Eq.
(12).
Q --1RQ = diag {21... 2.4} (A 1)
Some elements of Q - i are q ~ l =
qi~ ~ --
q ~ l
q ~ =
d(1 +22)
(~,1-22)(21 - 2,)(& -21)
(1+21)
a2(,L --&)(& --24)(23 --&) X X [--24(22+2223-[" 1 - a 2 - 2 3 ) - 2 2 ( 1 - 22 +23) +2.3 ( a 2 - 1 ) - (2a2 + 1 + a2s)]
d(1 +23) (23 --24)(/].3 --22)(23 --21)
(1 +23)
(a2 (23 - 24)(23 - )].2) (23 - 21 )) X [21 (24 ~-2224 -~-22 "q- 1 +a2) +24(1 + a2 +22) +22(1 -- a2) + 2 ( a 2 + s ) - 1]
(A2)
with s = b z c - d.
A p p en d i x B
The linear transformation given in A p p e n d i x A leads to a sys- tem of equations for the (complex-valued) variables zi, Eq.
(13). This new system of equations contains two types of non- linearities, namely resonant nonlinearities of the type z* Izjl 2 and n o n - r e s o n a n t nonlinearities of the type z/3, (z*) 3, (z/2) zj, zi (zj) , zi (z*) z, ziz*z~ and z*~z*z~ with i , j , k = 1,2. The second step is to eliminate the latter through the following change of coordinates (Arnold 1980).
Z = x + P[x] (B1)
with x =Ix1 =y~, X2 =y]', x3=yz, xa=y~] and P a polynomial of third order which is used to eliminate all the n o n - r e s o n a n t terms from (13). Substituting (B1) into (13) we obtain
= [ I + DP(x)I - ~f[x + P(x)l (B2)
where D P is the Jacobian matrix of P
114 Salazar and Nicolis: Self-generated aperiodic behaviour in a simple climate model The expansion of this last equation, retaining only the
terms up to third order, is given by
j=l 0xj ~J (B3)
where the k3(x) denote the terms of k~ of third order. There- fore, the polynomial P should satisfy
~.~e,[xl- ~, ee'Z.x.=-/#Ixl
j=l {)xj JJ (B4)
and (B4) gives the nonlinear transformation k which elimi- nates the non-resonant terms from (13).
Appendix C
Effect of the parameters in the eigenvalues of the linearized operator
A small change in the control parameters, bi=bg~+btj for L j = 1,2, will lead to a small change in the eigenvalues A~ = ~ +/_t,~] for i = 1,2 where ,~ is complex valued and given by
~] = Aa[Af]3 + B][A~]2 + C,[A~] + D1 2[A~][2[&~] 2 + B~]
with
A I = - [bl 1 "}- b21]
B1 = - 2 [bll q- b21] +,[.bl lb2c q blcb21 ] - d[bn + bzd Cl = - [bll + b21] + 2 [bl lb2c -}- bl cb21] - 2d[bn + b21]
-- [blla2 -t- b21al]
D1 = bz~b21 +bnb2~ - d [ b n +bad -[bna2 + b21ad
Normal form coefficients
References
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Applying (B4) to (13) we obtain the coefficients of the normal form given by
A2]oo=qfi][q]2q~] + 2qllq21q22]
+ qi-31 [q34q21 + 2q31q41q42]
Aim1 = q fi12[qnqz3q24 ] + q i)12[q3 Iq43q44 + q33q41q44 + q34q41q43]
Boo2] = q~31 [q34q423 -I- 2q33q43q~]
+ q~il [qiaq23 + 2q13qz3q24]
Ba110 = q~i12 [qllq22q23 + q12q21q23 + q13qzaq22]
+ q3312[q3aq42q43 + q32qalq43 "~ q33qalqaz]
The numbers qij and ql)-I for i = 1... 4 represent the elements
of Q and Q -1, respectively. Received December 17, 1987/Accepted May 24, 1988
