Bifurcations and aggregation in large scale systems
Abstract
This paper deals with the following problem: assume that a qualitative analysis (behaviour modes, bifurcation points, type of atractors .•• ) of a nonlinear dynamical system has been carried out and that afterwards this dynamical system is transformed into a large scale system through a disaggregating process of some (or all) of its variables. The problem at stake is to analyze whether the disagaregation gives rise to new behaviour modes, as a consequence of the appearance of new bifurcations in the disaggregated dynamical system. That leads us to study whether the original system and the disaggregated one are "equivalents" or whether the second one is richer in behaviours than the first one.The paper develops general results for standard disaggregation forms. Furthermore, practical applications of the proposed methodology to urban dynamics models is included .
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a ge
Sc
ale
51'S e
n
s:
Th
eo
y and App
li
ca
i
ons 1986. Zu ich.
Swi ze land. 1986
BIFURCATIONS
AND
AGGREGATION
IN
LARGE
SCALE SYSTEMS
M.
To o
and
J.
A acil
Depa a
l/l
ll u de A u u
lll
i
ic
a. Escuela Supe iu de i
llg llinoJ
ill
du
s ial
es.
AI'
d
a.
Reil/a "'·
e a
d
es
5/
11.
Se 'ilia, Spaill
ABSTRACT
This
pape
deals
wi h
he
ollowing
p oblem:
assume
ha
a
quali a i e
analysis
(beha iou
modes,
bi u ca ion
poin s,
ype
o
a ac o s
.••
)
o
a
nonlinea
dynamical
sys em
has
been
ca ied
ou
and
ha
a e wa ds
his
dynamical
sys em
is
ans o med
in o
a
la ge
scale
sys em
h ough
a
disagg ega ing
p ocess
o
some
(o
all)
o
i s
a iables.
The
p oblem
a
s ake
is
o
analyze
whe he
he
disaga ega ion
gi es
ise
o
new
beha iou
modes,
as
a
consequence
o
he
appea ance
o
new
bi u ca ions
in
he
disagg ega ed
dynamical
sys em.
Tha
leads
us
o
s udy
whe he
he
o iginal
sys em
and
he
disagg ega ed
one
a e
"equi alen s"
o
whe he
he
second
one
is
iche
in
beha iou s
han
he
i s
one
.
The
pape
de elops
gene al
esul s
o
s anda d
disagg ega ion
o ms.
Fu he mo e,
p ac ical
applica ions
o
he
p oposed
me hodology
o
u ban
dynamics
models
is
included
.
INTRODUCTION
Conside
a
dynamical
sys em
gi en
by
he
equa ions:
whe e
ha
a
o
(1),
1, )
is
wi h
•
z
'(z,q)
(1)
z E R
and
q E
R .
I
is
assumed
disagg ega ion
p ocess
is
applied
in
such
a
way
ha
e e y
zi
(i
=
deco~posed
in
pa s
Xj
such
ha :
z =
!.
x
(2)
i ... j
i-I
0(
=
~
n
k=l
k
0(
+n
i
being
ni
he
numbe
o
pa s
in
which
z~
has
been
decomposed.
The
se
{xll.<j<~)
will
be
called
module
i ,
associa ed
o
zi.
A e
disaga ega ion,
he
dynamical
sys em
(1)
will
lead
o
a
new
one,
o
he
o m
:
.
x =
(x,p)
(3)
whe e
x
ERn,
p E RS
being
n = n
I
should
be
no ed
ha
p
is
di e en!
om
q,
due
o
he
g ea e
ichness
in
he
desc ip ion
o
(3)
ela i e
o
(1).
The
model
(3)
will
be
conside ed
a
model
e inemen
o
he
model
(1).
The
p oblem
is
o
s udy
i
he
sys em
(3)
will
show
beha iou
modes
no
shown
by
(1)
.
F om
a
quali a i e
poin
o
iew
ha
means
o
s udy
i
(3)
will
exhibi
bi u ca ions
no
appea
i
ng
in
(1).
The
answe
o
hose
ques ions
will
be
ound
h ough
he
quali a i e
analysis
o
(1)
and
(3).
Howe e ,
sys em
(3)
is
a
la ge
scale
sys em
and,
he e o e,
he
quali a i e
analysis
can
be
a
e y
di icul
ask.
In
his
pape
we
conside
only
dynamical
135
sys ems
wi h
poin
a ac o
s;
ha
is,
we
a e
es ic ed
o
dynamical
sys ems
wi h
s a ic
bi u ca ions.
Fo
hese
sys sms
we
p opose
a
me hod
wich
allows
o
analyze
i
he
beha iou
modes
o
(3)
a e
he
same
as
hose
o
(1);
ha
is,
i
,
as
a
consequence
o
he
disagg ega ion
p ocess,
he e
appea
bi u ca ions
in
(3)
no
shown
by
(1)
.
This
kind
o
esul s
has
p ac ical
in e es
because
he
p ocess
s a s
no mally
wi h
dimension
model
(Rande s
1980)
disagg ega ed
la e
on
.
a
lo
o
modellina
a
small
which
is
The
pape
p oposed
models.
ends
wi h
applica ions
o
hs
me hod
o
some
u ban
dynamics
BIFURCATION ANALYSIS
Conside ing
only
s a ic
bi u ca ion
analysis
o
educed
o
he
s udy
o
he
equa ion:
' '(z,q)
= 0
b i
u ca ions,
he
he
model
(1)
is
he
solu ions
o
(4)
when
he
pa ame e s
q
a e
a ied,
and
o
he
s abili y
s udy
o
each
one
o
hose
solu
i
ons.
The
g aphical
ep esen a ion
o
hese
solu ions
e sus
e e y
pa ame e
q
gi es
ise
o
he
bi u ca ion
diag am
o
(4).
These
d i
ag ams
can
be
ob ained
nume ically
wi h
he
help
o
con inua ion
me hods
(~ubi~ek
1976).
Le
(zo,qO)
be
a
solu ion
o
Eq.
(4).
I
a ying
q
a ound
qo
he
numbe
o
solu ions
o
(4),
o
jus
he
s abili y
o
any
o
hem,
a e
chanaed,
hen
i
is
said
ha
(zo
,
qO)
is
a
bi u ca ion
poin .
These
poin s
a e
undamen al
in
he
quali a i e
analysis
o
sys em
(1)
,
since
hey
supply
all
he
in o ma
i
on
needed
o
de e mine
he
quali a i e
shape
o
he
bi u ca ion
136 M.
To o
and
J.
A acil
diag am.
Fo
s a ic
bi u ca ions
he
bi u ca ion
poin s
a e
gi en
by
Eq.
(4)
and
(5)
since
in
he
bi u ca ion
poin s
an
eigen alue
o
Jacobian
ma ix
Dz'(z,q)
is
ze o.
REDUCIBLE DTNAnICAL STSTE S
Conside
ha
we
a e
in e es ed
on
he
bi u ca ion
analysis
o
a
la ge
scale
dynamical
sys em.
This
p oblem
could
be
g ea ly
simpli ied
i
we
can
ind
a
subsys em
o
he
la ge
scale
one
ha
would
"concen a e"
all
he
bi u ca ions.
Theo ems
1
and
2
below
help
o
cope
wi h
ha
p oblem.
Suppose
we
a e
gi en
a
dynamical
sys em:
·
u =
h(u,a)
la ge
scale
ha
can
be
pa i ioned
in o
he
o m:
•
u h
(u
,u
,
a)
1 1 1 2
•
u h
(u
,u
,a)
2 2 1 2
whe e
u =
(u
1
,u
2
).
Equilib ia
solu ions
o
[he
equa ions:
h(u,u,a)
0
112
h(u,u,a)
0
212
Then,
he
s a ed.
Tb.o e.
1
ollowing
heo ems
(6.1
)
(6.2)
o
(6)
a e
(7.1)
(7.2)
can
be
I
Eq.
(7)
can
be
ans o med
in o
he
o m
h
(u
,u
,a)
0
(8.1)
112
u =
F(a)
2
(8.2)
hen
sys em
(6)
has
he
same
bi u ca ion
diag am
han
he
associa ed
educed
sys em
•
u = h
(u
,u
,a)
(9)
1 2
whe e
u2
is
now
a
(cons an )
pa ame e ,
bu
ela ed
o
pa ame e s
a
by
Eq.
(8.2).
Tbeo e.
2
I
he
hypo heses
o
heo em
a e
ull illed
and,
u he mo e,
he
same
condi ions
ha
gua an ee
he
s abili y
in
e e y
b anch
o
he
bi u ca ion
diag am
o
(9),
can
gua an ee
he
s abili y
o
he
co esponding
b anches
in
he
bi u ca ion
diag am
o
(6),
hen
sys em
(6)
is
educible
o
(9).
These
heo ems
will
be
p o ed
gene alized
in
a
o hcoming
pape .
and
APPLICATION
TO
THE
DISAGGREGATION PROCESS
Take
Eqs.
(3)
and
eo de
hem
in
such
a
way
ha
he
ollowing
pa i ion
could
be
made:
x
(x
,x
,w
)
1 1 1 2 1
.
(10)
x =
(x
,x
,w
)
2 2 1 2 2
whe e
xl
E R
and
x2 E Rn- .
The
ec o
x
is
o med
by
a
" ep esen a i e"
componeJ
o
e e y
module
i.
The
componen s
o
ec o
x2
a e
so ed
in
blocks
coming
om
he
di e en
modules
ob ained
by
disagg ega ion
o
he
a iables
zi
I
is
con enien
o
ans o m
ec o
(xl'
x2)
in o
ec o
(y,k),
whe e
Yi
( ha
is,
Yi
=
zi)
will
be
he
addi ion
o
all
he
x.
a iables
belonging
o
module
i
and
k
He
a e
o
a iables
x
21
ela i e
o
y
~
This
ans o ma ion
is
ca ied
ou
by:
i
whe e,
being
B
an
(
x
i
x2.
belongs
o he loli
se,
and
ma ix
wi h
c
JJ
module
i.
(11
)
(12)
n- )
ma ix,
wi h
b
iJ
= 1
o
module
i
and
b
ij
=
0,
whe e
C
is
a
d agonal
=
l/Yi
i
x
2j
belongs
o
T ans o ma ion
(11)
has
an
in e se,
which
is
meaning ul
o
s udying
he
equilib ia
o
(10)
whe he
Yi
~
0,
o
whe he
Yi
=
0,
and
he
o m
o
he
equa ions
causes
y
disappea
om
he
denomina o .
Thl~
happens
when
Eq.
(3)
has
he
o m:
•
x i
(x,w)x
(13
)
i i
A e
applying
ans o ma ions
(11)
o
Eq.
(10)
we
ge :
y
(x
,x
,w)
) +
B
(x
,x
,w)
1 1 2 2 1 2
(14)
k
C
(x
,x
,w)
2 1 2
I
Eq.
(3
)
akes
he
o m
(13
)
hen
Eq.
(14.2)
will
ake
he
o m:
k
(x
,x
,w)k
2 1 2 i
Bi u
ca ions
and
Agg ega ion in La
g
e Scale Sys ems 137
In
Eqs.
(14)
xl
and
x2
a e
unc ions
o
y
and
I<.i'
and
a e
gI en
by
Eq.
(11).
Reo de ing
pa ame e s
w,
Eqs.
(14)
can
be
w i en:
y g
(y,l<.,p
)
1 1
•
(15)
k g
(y,k,p
)
2 2
The
ans o ma ion
o
pa ame e s
w
in o
(Pl,P2)
should
be
made
in
o de
o
1001<.
o
a
co espondence
be ween
he
a iables
and
pa ame e s
o
Eq.
(16)
below
and
he
ones
o
Eq.
(1).
I
he
hypo eses
o
heo ems
1
and
2
a e
ull illed,
hen
he
dynamical
sys em
(15)
is
educed
o:
y g
(y,l<.,p
)
(16)
1 1
whe e
k
is
now a
cons an
pa ame e .
SOnE
SPECIAL
CASES
P e ious
esul s
can
be
kinds
o
disagg ega ion,
used.
applied
o
wo
which
a e
widely
a)
Linea
dl
...
a e.a lon.
Suppose
ha
unc ions
21
appea ing
in
Eq.
(10)
a e
linea
unc ions
o
a iables
x
beloging
o
he
same
module
as
x
21
•
T ien
he
disaggega ion
is
called
linea .
In
such
a
case,
unc ions
g2
o
(15)
do
no
depend
on
y,
due
o
[he
o m
o
ans o ma ion
(11).
Indeed,
ans o ma ion
(11)
can
be
conside ed
as
an
applica ion
o
wo
successi e
ans o ma ions.
The
i s
one
ans o ms
(x
1, x 2)
in o
(y,xi)
h ough
ma ix
T.
T ie
second
one,
ans o ms
(y,x
)
in o
(y,k)
by
means
o
ma ix
T1•
Func ion
2i
is
ans o med
in o
l
h oueh
T2•
This
las
unc ion
2i
s
linea
in
y
and
in
a iables
x
2j
'
whe e
he
la e
6elong
o
module
i.
Th ough
Tl
unc ions
e
2j
al<.e
he
o m
=
2
/Yi'
Since
2j
is
linea
in
Yi
x
2j
,jex
p
essions
2j
/Yi
only
depend
on
I
apa
om
i
beine
a
lin.a
disaeg ega ion,
ma ix
D
g2
in
(15)
is
s able
hen
he
abo e
~eo ems
can
be
applied
and
he
disaeg ega ion
does
no
add
new
bi u ca ions
(new
beha iou
modes)
.
In
nex
sec ion
an
example
o
his
case
will
be
p esen ed.
b)
Dl
....
••
a lon
wl b
ke nel
This
disagg ega ion
occu s
when
he
non
linea i ies
ha
appea
in
l
and
2
ha e
as
he
only
a gumen
he
a iables
Yi
( ha
is,
he
addi ions
o
all
he
a iables
x
belonging
o
module
i)
and,
u he mo e~
when,
a e
ans o ma ion
(11)
,
heo ems
1
and
2
can
be
applied.
APPLICATIONS
TO
URBAN
DYNA ICS
In
u ban
dynamics
(Al eld
and
G aham,
1976)
he
e olu ion
o
he
housing,
o
o
he
business
s uc u es,
is
desc ibed
by
a
model
o
he
o m
z =
z(qlT(hz)-q2)
(17)
I
he
case
o
he
housing
e olu ion
is
conside ed,
hen
z
s ands
o
housing,
ql
o
he
a e
o
housing
demoli ion
and
qlT(hz)
o
he
a e
o
housing
cons uc ion.
Func ion
T(hz)
ep esen s
he
housing-land
mul iplie
and
i s
shape
is
show
in
Fig.
1.
15
~--
----~~----
--------,
10
05
(J)
0' 06 o.
10
Fig.
1
The
quali a i e
analysis
o
his
model
can
be
ound
elsewhe e
(A acil,
1981),
and
some
ela ed
ma e ial
in
(A acil,
1984).
Model
(17)
desc ibes
he
housing
e olu ion.
Howe e ,
a
disagg ega ion
o
he
housing
sec o ,
al<.ing
in o
accoun
he
connec ion
be ween
housine
uni s
and
he
socioeconomic
s a us
o
hei
occupan s,
can
lead
o
a
model
e inemen .
This
is
done
in
(Al eld
and
G aham,
chap.
9)
whe e
he
ollowing
model
is
p oposed
as
a
disagg eea ion
o
(17).
~
n
(x
+n
x
> (x
+x +x
)-n
x
12122
123
5 1
.
(18)
x = n x
-x
2 5 1 2
x = n x
-x
362
3
whe e
he
o al
numbe
o
houses
z
has
been
disagg ega ed
in o
a iables
x1
,x
2
,x
3
co espondine
o
uppe
income,
middle
income
and
lowe
income
houses.
The
disagg ega ion
om
(17)
o
(18)
is
o
he
same
ype
as
he
one
om
(1)
o
(3).
A
ans o ma ion
o
ype
(11)
can
be
applied
o
his
model,
gi ing:
•
y n
(1
-I<.
-I<.
+n
I<.
)~(y)y-n
I<.
y
2 2 3 2 2 7 3
I<.
n
(1
-I<.
-k
)-n
I<.
2 5 2 3 6 2
I<.
= n
I<.
-n
k
36273
I
should
be
no iced
disagg ega ion
is
o
linea
(19)
is
educible
p o ided
D g
is
s abl
•.
In
his
case,
I<.
2
(19)
ha
he
ype.
Sys em
ha
ma ix
we
ha e:
138 M. To
o
a
nd
J.
A acil
D g =
I<.
2 [
-n
5
-n
6
n 6
whose
s abili y
is
gua an ed
p o ided
ha
ni>O.
Consequen ly,
sys em
(19)
is
educible
o:
y n
(l
-I<. -I<.
+n
I<.
)T(y)y-n
I<.
y
(20)
2 2 3 2 2 7 3
I
should
be
no iced
ha
(20)
is
equi alen
o
(l
7)
so
ha
a
co espondence
be ween
he
pa ame e s
o
(l8)
he
and
hose
o
(17
)
can
be
o m:
q
~
n
(1-1<.
-I<.
+n
I<.
)
1
223
2 2
q~nl<.
273
s a ed
in
( 21 )
whe e
I<.
and
1<.3
can
be
exp essed
as
unc iona
o
pa ame e s
n
om
equilib ium
equa ions
o
sys em
(i9).
The
disagg ega ion
p ocess
has
no
eupplied
new
bi u ca ions,
bu
i
has
ai en
a
mo e
de ailed
way
o
compu ing
he
pa ame e e
o
he
agg ega ed
model.
I
now
exp esion
(17)
models
he
e olu ion
o
business
s uc u es,
hen
in
(Al eld
and
G aham
1976
,
chap.8)
a
di e en
o m
o
disaga ega ion
is
conside ed.
Tal<.ing
in o
accoun
he
aging
and
obsolesc~nce
o
he
business
s uc u es
a
disagg ega ed
model
o
he
ollowing
o m
is
p oposed
•
x = x
(n
,(hex
+x +x
)-
n )
(22)
,I
1 1 1 2 3 4
x = n x
+x
(n
,(~hex
+x +x
)-n
-n
)
2 4 1 2 6 1 2 3 7 8
·
x = n x + x
(n
"(h(x
+x +x
)-n
)
3 7 2 3 9 2 3
10
whe e
now
z
in
Eq .
(17)
s ands
o
he
numbe
o
business,
and
x2
,x
1
and
x3
in
Eq.
(22)
o
new
bus ness,
ma u e
business,
and
de e io a ing
business.
Reo de ing
(22)
and
applying
ob ain:
in o
he
o m
(X
1
,X
2
,X
3)
ans o ma ion
(1),
we
I
we
a e
gi en
ni
hen
Eq.
(24)
a e
~~:~: issy: ~:~c ~~~
~~'
=a~~~)
~:~:~~ :~
equi ed
by
heo em
1.
Eq .
(23.1)
can
be
w i en:
(25)
The
condi ion
o
an
equilib ium
y
~
0
o
equa ion
(25)
o
be
s able
is
T'(hy)(O.
This
condi ion
gua an ees
he
s abili y
o
he
co esponding
b anch
in
he
bi u ca ion
diag am
o
(23),
wi h
y
~
0
and
I<.
~O,
since
he
jacobian
ma ix
o
his
as
sys em
wo l<.s
ou
o
be:
1 •
c
"C(hy)
2 •
c 4
o
[
C
'('(
hy)
c
'
(hy)
3 - c 6
whe e
all
pa ame e s
ci
a e
posi i e
i
n
a e
wi hin
he
ange
o
alues
meaning u
o
he
model.
The
condi ion
o
(26)
o
be
s able
is
T'(hy)<O
as
i
is
easily
shown.
Compa ing
Eq.
(23.1)
easy
o
deduce
he
om
Eq.
(22)
which
pa ame e s
om
Eq.
wi h
Eq,
(17)
i
is
g ouping
o
pa ame e s
a e
equi alen
o
he
(17)
.
ACQlOVLEDG lENT
This
wo l<.
was
suppo ed
by
CAICYT
unde
p
ojec
1102/84
REFERENCES
Al eld,
L.
and
A.
G aham
(1976)
.
In oduc ion
o
u ban
dyn
..
ic
••
W igh -Allen
P ess.
A acil
,
J.
(1981).
S uc u al
s abili y
o
low-o de
sys em
dynamics
models.
In .
J.
Sy. e.
Science.
12,423-441.
A acll,
J,
(1984)
.
Quali a i e
analysis
and
bi u ca ions
in
sys em
dynamics
models.
IEEE-S"C-14,
4,
688-696,
Kubicel<.,
H.
(1976).
Algo i hm
502.
Dependence
o
solu ions
o
Nonlinea
Sys ems
on
a
pa ame e .
AC"
T an
••
"- h.
So wa e
2,
98-107
Rande s,
J.
(1980),
Ele
..
n .
o
be
.y.
••
dyn
..
ic
...
hod.
HIT
P ess,
y
(I<.
n +(1-1<.
-I<.
)n
+1<.
n
)"C(hy)y-(n
(1-1<.
-I<.
)+n
I<.)y
(23.1)
.
I<.
..
• 1
I<.
=
3
1 1 1 3 6 3 9 8 1 3
10
3
(n
"C(hy)-n
)1<.
1 4 1
n
(1-
I<.
-I<.
)+1<.
(n
"C(hy)-n
)
7 1 3 3 9
10
Taking
in o
accoun
he
equilib ia
o
(23),
he
aluss
o
ki
a
he
equilib ium
poin
can
be
ob ained
om
pa ams e s
ni
h ouah
he
equa ions:
(I<.
n +(1-1<.
-k
)n
+1<.
n
)n
In
-en
(1-1<.
-I<.
)+n
1<.)=0
1 1 1 3 6 3 9 4 1 8 1 3
10
n (1-1<.
-I<.
)+1<.
(n
n
In
-n
)=0
7 1 3 3 9 4 1
10
(23.2)
(23.3)
(24
.
1)
(24.2)