Copy igh © IFAC L
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)