Table Of ContentUniversal S
aling in Saddle-Node Bifur
ation
Cas
ades (II)
Intermitten
y Cas
ade
5
0 a,b
0
Jesús San-Martín
2
a
n
Departamento de Matemáti
a Apli
ada, E.U.I.T.I, Universidad Polité
ni
a de
a
J Madrid. Ronda de Valen
ia 3, 28012 Madrid Spain
b
8
Departamento de Físi
a Matemáti
a y Fluidos, U.N.E.D. Senda del Rey 9, 28040
1
Madrid Spain
]
D
C
.
n Abstra
t
i
l
n
Thepresen
eofsaddle-nodebifur
ation
as
adeinthelogisti
equationisasso
iated
[
with an intermitten
y
as
ade; in a similar way as a saddle-node bifur
ation is
1
asso
iated with an intermitten
y. We merge the
on
epts of bifur
ation
as
ade and
v
6 intermitten
y. The mathemati
al tools ne
essary for this pro
ess will des
ribe the
3 stru
ture of the Myrberg-Feigenbaum point.
0
1
0
Key words: Intermitten
y Cas
ade. Saddle-Node bifur
ation
as
ade. Attra
tor of
5
0 attra
tors. Stru
ture of Myrberg-Feigenbaum points.
/
n
i
l
n
:
v
i
X
1 Introdu
tion
r
a
Period-doubling [1,2℄, Quasi-periodi
ity [3,4℄ and Intermitten
y [5,6℄ are well
known routes of transitionfrom periodi
to
haoti
behavior, and whose origin
isinlo
albifur
ations.Initially,the system has a stable limit
y
le, fora range
r r
c
ontrol parameter . As this parameter is in
reased beyond a
riti
al value
r
c
system behavior
hanges a
ording to a lo
al bifur
ation that o
urs at .
Inordertostudythegenesisofthetransitionwe resorttothePoin
arese
tion.
In this se
tion, the original stable limit
y
le of the system generates a (cid:28)xed
point, whose evolution is studied in parallel to the
ontrol parameter
hanges.
Email address: jsmdfmf.uned.es (Jesús San-Martín).
Preprint submitted to Elsevier S
ien
e 8 February 2008
Paying attention to the di(cid:27)erent lo
al bifur
ation kinds we will have di(cid:27)erent
transitions to
haos. So, if the (cid:28)xed point shows su
essive pit
hfork bifur
a-
tions,whi
hdoublerepeatedlytheperiodoftheoriginalorbit,theFeigenbaum
period doubling
as
ade is obtained. The (cid:28)nal ending is a period-∞ orbit, a
haoti
attra
tor.
Quasi-periodi
ity o
urs as a new Hopf bifur
ation generates a se
ond fre-
quen
y in the system, whi
h is in
ommensurate with the original system fre-
quen
y. If an irrational winding number is (cid:28)xed it goes through
haos.
At last, the intermitten
y is a
haoti
regime
hara
terized by an apparently
regular behavior, whi
h undergoes irregular bursts from time to time. This
intermitten
y between two behaviors names this kind of
haos.
The regular behaviororlaminarregime
orresponds toan evolutionof the sys-
tem in a narrow region or a
hannel in the phase spa
e. Su
h regular behavior
stems from the fa
t that the system maintains a (cid:16)ghost(cid:17) of laminar regime
Whereasintheothertransitionsto
haos,Period-doublingandQuasi-periodi
ity,
the system is totally regular before the transition and
haoti
later. This does
not happen with the intermitten
y. The intermitten
y shows a
ontinue tran-
ε = r r
c
sition from regular behavior to a
haoti
one. The smaller the value −
is the longer the laminar regime will be and the lesser it will be altered by
a
haoti
behavior, due to the fa
t that the average time of laminar regime
l ε ε−β β > 0 r
c
h i for small is , being . Therefore, beyond the laminar behavior
ε
alternates with irregular bursts, the smaller (bigger) the value is the more
l
(less) the laminar regime h i is.
The intermitten
y transition was dis
ussed by Pomeau and Manneville [5,6℄.
They pointed out three kinds of intermitten
y. The system is periodi
for
r < r
c
parameter values smallerthan the
riti
alpoint . This periodi
behavior
generatesastable(cid:28)xedpointinthePoin
arese
tion.Astheparameterrea
hes
r = r
c
the
riti
al value the (cid:28)xed point losses its stability. The loss of stability
is
ausedwhentheeigenvaluemodulusofthelinearizedPoin
aremapbe
omes
larger than unit. This may happen is in three di(cid:27)erent ways.
i) There is a real eigenvalue
rossing the unit
ir
le by plus one. A saddle-
node bifur
ation is generated, whi
h has asso
iated tylpe-I εin−t1ermitten
y. In
2
this
ase the average length of laminar regime time is h i ∼ .
ii) A
ouple of
omplex
onjugate eigenvalues
rosses the unit
ir
le. This
ir
umstan
e is asso
iated with the birth of a Hopf bifur
ation and it involves
l ε−1
a type-II intermitten
y with h i ∼
iii) A real eigenvalue
rosses the unit
ir
le by minus one, generating a (cid:29)ip
l
bifur
ation. This time the type-III intermitten
y o
urs and we obtain h i ∼
2
ε−1
.
Other kinds of intermitten
ies have been studied su
h as type-X [7℄ one whi
h
shows a transition with hysteresis and type-V one [8℄ for dis
ontinuous maps.
Intermitten
ies have also been studied whose laminar regime o
urs alterna-
tively between two
hannels, as a result of symmetry in the problem [9℄.
The generalization and logi
al development of this latter problem is to (cid:28)nd
intermitten
ies with an arbitrary number of
hannels. Furthermore it is de-
sirable to have a di(cid:27)erent number of
hannels for di(cid:27)erent values of
ontrol
parameters in the same system, and not to look for di(cid:27)erent systems ad ho
with the appropriate symmetries whi
h show a (cid:28)xed number of
hannels. We
say that su
h behavior is desirable in an unique system be
ause, if a
hange
of the
ontrol parameter implies an in
rease of the number of
hannel, what is
obtained is an intermitten
y
as
ade; in a similar way a
hange of the
ontrol
parameter in the logisti
map generates a period doubling
as
ade.
In this paperwe aregoingtoshow thatsu
h phenomenono
urs inthelogisti
map,bene(cid:28)tingfromthefa
tthatthismapshows saddle-nodebifur
ation
as-
ades [10℄and that the type-I intermitten
yis asso
iatedwith the saddle-node
bifur
ation. We will
hara
terize
ontrol parameter values at whi
h su
essive
intermitten
ies are generated, how some intermitten
ies are related to others,
the number of
hannels, whi
h is the average time of laminar regime of inter-
mitten
ies, and what relationship there is between su
h regimes for di(cid:27)erent
intermitten
ies. To answer these questions we will use the universal proper-
ties of the logisti
map [1,2℄ and the saddle-node bifur
ation
as
ade this map
shows [10℄.
The saddle-node bifur
ation
as
ade is a sequen
e of saddle-node bifur
ations
in whi
h the number of (cid:28)xed points showing this kind of bifur
ation is du-
pli
ated. The su
essive elements of the sequen
e are given by an equation
identi
al to the one that Feigenbaum found for period doubling
as
ade [10℄.
The way of a
ting is as follows. As we mentioned above type-I intermitten
y
is asso
iated to one saddle-node bifur
ation. Therefore, ea
h element of the
sequen
e of the saddle-node bifur
ation
as
ade has asso
iated a type-I in-
termitten
y. The number of
hannels of this intermitten
y
oin
ides with the
number of (cid:28)xed points that simultaneously show a saddle-node bifur
ation.
3Fo3r in2s3tan2
2e,3th2e3sadd3le-2nqode bifur
ation
as
ade, sym3bolized by the sequen
e
, · , · , · ,..., · , points out that there are (cid:28)xed points at a (cid:28)rst
r = r 3 2 r = r
3 3·2
parameter value , there are · at and so on. Ea
h saddle-node
(cid:28)xed point
ontributes to the intermitten
y with one
hannel, a
ordingly in
3th3is 2in3ter2m2i3tte2n3
y
3as
2aqde there will be a sequen
e of intermitten
ies with
, · , · , · ,..., · ,....
hannels.
The
hannels responsible for laminar regime are
lose to
riti
al points of lo-
3
gisti
map(Fig..2).The way,how theneighborhoodofthese
riti
alpointsare
ontra
ted in the su
essive iterated of the map, determines how the
hannels
are
ontra
ted and it allows us to look for the
onne
tion between them. The
s
aling of the neighborhood of
riti
al points for iterated map was
al
ulated
near a pit
hfork bifur
ation by Feigenbaum [1℄. We will follow this work to
al
ulate the s
aling near a saddle-node bifur
ation, be
ause it is here where
intermitten
y o
urs.
As many iterated one-dimensional maps are nearly quadrati
under renor-
malization [11℄, we have to expe
t that intermitten
ies
as
ade is a
ommon
phenomenon in many natural pro
esses.
x = f(x ) = rx (1 x ) f3
n+1 n n n
Let be the logisti
equation − . The graph of
fn = f f x = x r =
n+1 n
( ◦ ◦ )isshown inFig..1,where itistangenttoline ,at
r = 1+√8 3
c
,whi
h meansa period- orbitexists .There arethree saddle-node
r > r
c
(cid:28)xed points, and we are atft3he genesis of a saddle-rn=odre bifur
ation. For
c
the valleysand the hillsof are sharper than at , and ea
h saddle-node
point has generated two points: one saddle and one node. If we de
reased from
r > r r < r
c c
to we would observe that the saddle point approa
hes the node
r = r r < r
c c
one, tou
hing at , as the saddle-node bifur
ation o
urs. For
the valleys and the hills are pulled away from the diagonal and saddle-node
(cid:28)xed points disappear. After the bifufr
3ations have disappeared three narrow
hannels, delimited by the graph of and the diagonal, remain, whi
h are
responsible for laminar regime of intermitten
y. This is what we meant when
we said earlier that ea
h S-N point would be responsible for generating a
hannel in the intermitten
y
as
ade.
Ea
h iteratedoflogisti
mapwilltake alongtimetogothroughthese
hannels
(slee Fεi−g.1 .2). The average number of iterated inside a
hannel is given by
2
h i ∼ [6℄, and so it is for the set of three
hannels.
Tq hqe 2saqddl2e2-nqod2e3bifuqr
a2tnionq
=as2
amde involves saddle-node bifufrq
aft2iqons wfiqt2hn
, · , · , · ,..., · , 6 (cid:28)xed points for the maps , ,....,
respe
tively, and the same number of
hfa3n·2ne=lsff6or thfe3i·2n2te=rmfi1t2ten
y. Fig. .3
and .4 show saddle-node bifur
ations for and respe
tively.
l
We want to
onne
t the length of laminar regime h i of one intermitten
y with
the laminarregimeof other inte1r,m1 itten
ies present inthe
as
ade. If we noft3i
e
in the neighborhood of porin=tr(cid:16)2 2(cid:17)in Fig. .3 we will see1 thαa>t t1he graph of is
reprodu
ed in Fig. .1 at c, es
alated by a fa
tor α, . The same
an
1, 1 fq2n
2 2
bne said in the neighborhood of point (cid:16) α (cid:17)in Fig. .4. As the iterated with
−→ ∞ are
onsidered the
onstant we get is the Feigenbaum
onstant.
(see appendix)
f3·2
If we
ome ba
k to Fig. .3 we will noti
e that reprodu
es again the graph
4
f3 f(1),f(1)
2 2
of inthe neighborhoodof (cid:16) (cid:17),and thatthis one isnotes
alated by
1, 1
the same fa
tor as in the neighborhood of the point (cid:16)2 2(cid:17). From one iterated
f3·2n f3·2n+1
to the following one , that is, from to 1 , half of the neighborh1ood of
riti
al points es
alate approximately with α and the other hfa3lf with α2 (see
appendix). We will be able to answer ourfq3u·2enstions be
ause is reprodu
ed
in the neighborhood of
riti
al points of and also be
ause we know how
these neighborhoods s
ale in the su
essive elements of the
as
ade
2 Intermitten
y Cas
ade
r fq·2n q = 2m
q·2n,SN
Let be the pfaqr2anmeter value at whi
h , 6q 2n has a saddle-node
bifur
ation, that is, has a saddle-node orbit wifthq2n · points. The points
of this orbit are lo
ated right where the fun
tion is tangent to the line
y = x q 2n 2n
q . The ·2npoints
an be
lassi(cid:28)2end into sets, ea
hf2onne of them having
points. The sets
orrespond to
riti
al points of
losest to the line
y = x q
. The saddle-node points are
aptured in a neighborhood of ea
h one
fq
of these
riti
al points, in other words, thef3g·r2a2ph of is
aptured in evefry3
one of su
h neighborhoods; for instan
e,2in2 we noti
e howf2t2he graph of
is
aptured in the neighborhoods of the
riti
al points of (Fig. .4).
If we
hoose
r = r ε 0 < ε 1
q·2n,SN
− ≪ (1)
then the sqad2dnle-node bifur
atfiqo2nn will be about to o
ur. In theyse=
xonditions,
there are · pqoin2nts where is almost tangent to the line fq2n. In su
h
points there are ·
hannels, whi
h are formed by the graph of and the
y = x ε
line . These
hannels are the narrower the smaller is in Eq. (1), and in
ε
them the laminar regime o
urs. The time to
ross the
hannel depends on .
l x = f(x )
Letqh i2nnbe the average time thatftqh2ne iterates of n+1 n spend to
ross
the ·
hannelfsqg·2enn+e1rated by . If we
onsider the laminar regime of an
intermittfeqn·2
ny+1of= fq·2n thfeqn·2nthe number of
hannels will be
foqme dupli
ated
be
ause ◦ ,fi2nnother words, the graph of is dupli
ated
lose to the f
rqiti
al points of . But in the do1ubling pro
ess the rep1li
as
of graph of are
ontra
ted, half of them as α and the others as α2 as
n α
→ε =∞,rwhere isrthe Feigenbaum
onstfaqn·2tn+(1see appendix). A
ordingly,
q·2n+1,SN
for f−q·2n the intermitten
y of shows the
hann1els of the
intermitten
y of 1dupli
ated, but half of them
ontra
ted as α anfdq·2tnhe
other
ontra
ted as α2. As the average time for the intermitten
y of is
l fq·2n+1
h in it turns out that the average time for the intermitten
y of will be
hli 1 hli
n n
α , whi
h
omes from the
hannels
ontra
ted by α, plus α2 , whi
h are given
5
1
by the
hannels
ontra
ted by α2. In
on
lusion, the average time of laminar
fq·2n+1 l (1 + 1 ) l
regime of the intermitten
y of is h in α α2 , where h inis the average
fq·2n fq·2n fq·2n+1
time of laminar regime of , and both and are at the same
ε
distan
e from the
orresponding saddle-node bifur
ation in the parameter
r = r ε r = r ε
q·2n+1,SN q·2n,SN
spa
e, that is, − and − .
Let's noti
e that ina saddle-node bifur
ation
as
ade the laminarregime from
(an1 +int1er)mitten
y to the next one in the sequen
e is de
reafsqe·d2n+bmy a fa
tor
α α2 . A
ordingly, the average time for intermitten
y of is
1 1
l = l ( + )m
h in+m h in α α2 (2)
fq·2n fq·2n+1 fq·2n+m
Given the saddle-node biffuqr·2
nationr
as
ade of , ,...., ,.. if the
q·2n,SN
bifur
ation parameter of is at then the other bifur
ation param-
eters are given by [10℄
1 1
r = r +(1 )r
q·2n+1,SN δ q·2n,SN − δ ∞ (3)
δ r
∞
where is the Feigenbaum
onstant; and is the Myrberg-Feigenbaum point
of a
anoni
al window where Feigenbaum
as
ade (cid:28)nishes and where also all
q = 2m
saddle-node bifur
ation
as
ades (cid:28)nish, whatever 6 is.
Eqs. (2) and (3) determine the intermitten
y
as
ade, be
ause the parameter
values at whi
h it o
urs and the average time of its
orresponding laminar
regimes are known.
j
The former results are valid if a intermitten
y
as
ade o
urs in a period-
window(i1n+stea1d) of a
anoni
al window. Be
ause bjoth the s
aling of laminar
regime α α2 and Eq. (3) are valid in a period- window, although the Eq.
(3) turns into
1 1
r = r +(1 )r
q·2n+1,SN δ q·2n,SN − δ ∞,j
to indi
ate that the
onvergen
e is the one to the Myrberg-Feigenbaum point
r j
∞,j
of the period- window.
There is a se
ond way of
hanging the
ontrol parameter in the intermitten
y
as
ade, whi
h is more important to the experimenters.
The Eq. (2) gives the length of average times to an ifnqt·e2rnmfiqt·t2en+n1
y
afsq
·a2nd+emas-
so
iated with the saddle-node bifur
ation
as
ade of , ,...., ,..,
ε
if the value of is
onstant. Su
h value gives the distan
e from the
ontrol
6
parameter to the saddle-node bifur
ation parameter. Nonetheless, it is possi-
ε
ble to
hange in the su
essive saddle-node bifur
ations in su
h a way that
l
the average time of laminar regimen is kept
onstant, and equal to a h in, for
the whεole intermitten
y
as
ade. For that purpose, all we need is(t1o+ha1ve)the
value , taken in the (cid:28)rst intermitten
y, is res
aled by a fa
tor α α2 for
ea
h one of the su
essive intermitten
ies of the
as
ade, that is, the values
1 1 1 1
ε,ε( + ),...,ε( + )m,...
α α2 α α2 (4)
m = 0,1,2,3,...
for . This is so be
ause
1
l
h i ∝ ε (5)
for the type-I intermitten
y, whi
h is present in the logisti
equation. If the
values of (4) are introdu
ed in Eq. (5) then the laminar regimen is in
reased
in a fa
tor whi
h is the same as the one that
ontra
ts a
ording to Eq. (2).
The result is that the average time of laminar regimen stays
onstant.
ε
It is ne
essary1 to
hange ε in this way. As Eq. (3) shows a geomre=trir
progresε-
sion,ofratio δ,ifwe held
onstant thenveryqui
klythevalue q·2n,SN−
[r ,r ]
q·2n+m+1,SN q·2n+m,SN
would not be within the parameter interval and we
would not observe
hannels
orresponding to two su
essive saddle-node bi-
r r
q·2n+m+1,SN
fur
ation of the
as
ade. The result would be that ≪ and the
intermitten
y
as
ade would not be observed. Obviously, this is vital for the
experimenters and for the development of numeri
al experiments as well.
ε ε
Bear in mind that as
hanges(a1s+in E1q). (4) it turns out that also
hanges as
geometri
progression of ratio α α2 . This geometri
progression
onverges
r
faster than the progression Eq. (3). A
ordingly, the parameter value
an
r [r ,r ]
q·2n+m+1,SN q·2n+m,SN
be held su
h that ∈ , and therefore the
hannel as-
r
q·2n+m,SN
so
iated with the saddle-node bifur
ation at
an be observed. It is
ne
essary for the experimenter to
hange the parameter as in Eq. (4), in order
to stay
lose to su
essive saddle-node bifur
ations of the
as
ade and get a
l
onstant value of h i. It is easy to get this variation be
ause the bifur
ation
parameters are given by Eq. (3).
3 Myrberg-Feigenbaum point stru
ture
f3
As shown in Fig. .1 we
an see a saddle-node bifur
ation of . This same
(cid:28)gure appears twi
e in Fig. .f2.3·T2nhey are the twno (cid:28)rst elements of the saddle-
node bifur
ation
as
ade of . The bigger is the more times Fig. .1 is
7
f3·2n y = x
repli
ated in the graph of along the line .
As shown in the appendix, Fig. .1 appears twi
e more every time we move
on one stage1 in a saddle-node bifur
a1tion
as
ade, half of the (cid:28)gures are
on-
tra
ted by α and the other half by α2. The out
ome is that in a saddle-node
y = x
bifur
ation
as
ade the points, that are tangent to the line , dupli
ate
at the same time as the region they are pla
ed in
ontra
ts.
We would hope to (cid:28)nd any kind of Cantor set and, what is worse, one Cantor
q 2n q = 2m
set for ea
h period- · , 6 saddle-node bifur
ation
as
ade, be
ause all
r
∞
saddle-node bifur
ation
as
ades approa
h the Myrberg-Feigenbaum point
n
as → ∞. Nonetheless the solution is extraordinary simple at the limiting
r
∞
value .
q,q 2,...,q 2n,... q = 2m
If we
onsider the
as
ade · · , 6 it will turn out that
fq
thefg2rnaph of will be reprodu
ed in the neighborhood of the
riti
al points
of1,f(1,)w,fh2i(
h1).
.o..r,rfe2snp−o1n(d1)to the points of the restri
ted-super
y
le given by
n2 2 2 1 2 o(see appendix1). Half of these neighborhoods are
ontra
ted by α and the other half by α2, ea
h time we move on one stage
in saddle-node bifur
ation
as
ade, that is, the saddle-node points dupli
ate.
n
T1h,efre(f1o)r,ef,2i(n1)t.h..e.,lfim2ni−t1(1→) ∞ea
h neighborhoo1d,hfa(s1)
,ofl2la(p1)s.e.d..,tfo2na−p1(o1in)t of
2 2 2 2 2 2 2 2
n on→∞.Thepointsofn on→∞
n
oin
ide with the period doubling orbit as → ∞. The out
ome is that the
2n n q 2n q = 2m n
period- ( → ∞)
haoti
orbit
oin
ides with period- · , 6 , → ∞
orbit (cid:22)in the sense of limit(cid:22), that is, the limits
annot tell from ea
h other.
We have the same limitorbit, both the one whi
h
omes from period doubling
r < r
∞
as
ade at , and the one whi
h
omes from saddle-node bifur
ation
r > r
∞
as
ades at .
q 2n 2n
The fa
t that the period- · saddle-node orbit tends to the period- as
n q 2n
→ ∞hides another fa
t: the
ollapse of period- · window to a point at
r q 2n
∞
Myrberg-Feigenbaum point . This is so be
ause the period- · window
q 2n q 2n+1
starts withthebirthoftheperiod- · saddleq-no2dne+o1rbitandtheperiod- ·
window starts with the birth of the period- · saddle-node. As the birth
r n
∞
of both saddle-node orbits tends to , as → ∞, then the window length
tends to zero. This result is
aptured in the expression (see [10℄)
L
n
= δ
L
n+1
whi
h shows that the length of two su
essive windows of a saddle-node bi-
δ δ
fur
ation
as
ade are
ontra
ted by a fa
tor , being Feigenbaum
onstant,
that is, the windows length tends to zero.
q 2n
Ifwe
onsidertheperiod- · windowitwillhave aMyrberg-Feigenbaumpoint
8
r n
∞,q·2n
, at whi
h its
orresponding Feigenbaum
as
ade will (cid:28)nish. As → ∞,
the window length tends to zero and it brings the following
onsequen
es and
q 2n
interpretation, relative to period- · window.
i) The whole period doubling pro
ess also
ollapses to a point, and the same
happens to the rest of the saddle-node bifur
ation
as
ades present in the
q 2n
period- · window, be
ause the pro
ess o
urs in a window whose length
tends (
ollapses) to zero.
r
∞
ii) The Myrberg-Feigenbaum point of
anoni
al window and the Myrberg-
r q 2n
∞,q·2n
Feigenbaum point of the period- · window are the
loser to ea
h
n n
other the bigger is. The distan
e tends tozero as → ∞. The same happens
q 2n
with every one of saddle-node bifur
ation
as
ades lo
ated in the period- ·
window. A
ordingly the a
umulating point of every saddle-node bifur
ation
r
∞,q·2n
as
ade (a new Myrberg-Feigenbaum) tends to , and therefore it tends
r
∞
to . The pro
ess is applied again to the new windows whi
h are born from
a saddle-node bifur
ation
as
ade and so forth.
ThisexplainswhyaMyrberg-Feigenbaumpointisanattra
torofotherMyrberg-
Feigenbaum points, whi
h are attra
tor of other Myrberg-Feigenbaum points
and so forth (see [10℄)
The former
onvergen
e pro
ess has been expounded for a (cid:28)xed saddle-node
q = 2m q
bifur
ation
as
ade with 6 , but it is valid for any value of . Therefore
there are in(cid:28)nite sequen
es whi
h mimi
the formerpro
ess, onefor ea
h value
q = 2m
of 6 .
The approa
hing to the Myrberg-Feigenbaum point, and the multipli
ity of
onvergent sequen
es, explain
ompletely the me
hanism of attra
tor of at-
tra
tor introdu
ed in [10℄
4 CONCLUSIONS
Thepresen
eofsaddle-nodebifur
ation
as
adeinthelogisti
equationassures
the genesis of a intermitten
y
as
ade. Ea
h saddle-node bifur
ation of the
bifur
ation
as
ade is asso
iated with an intermitten
y. As the lo
ation of
the saddle-node bifur
ation is known it brings that so is the genesis of the
intermitten
y
as
ade. The knowledge, a priori, of the length of the laminar
regime in the type-I intermitten
y, and of the s
aling the peaks and valleys of
the su
essive iterated of logisti
map, allows us to establish the length of the
laminar regime in the intermitten
y
as
ade.
The intermitten
y
as
ade is a phenomenon that takes pla
e in all windows
9
of the logisti
map, and not only in windows asso
iated to (cid:28)rst-o
urren
e
orbits.
It isproved that the windows
ollapsetothe Myrberg-Feigenbaum points, this
me
hanismbeing responsible forthe fa
tthat Myrberg-Feigenbaumpoints are
attra
tors of attra
tors.
A
knowledgments
The author wishes to thank Daniel Rodríguez-Pérez for helpful dis
ussions
and help in the preparation of the manus
ript.
APPENDIX
Let's (cid:28)nd the s
aling law of high-order
y
les in the saddle-node bifur
ation
as
ade. To do so we will follow the Feigenbaum work [12℄ , whi
h is
arried
out
lose to pit
hfork bifur
ation.
The s
aling law is not determined by xth=e l1o
ation oxf t=he0elements on the x-
2
axis, but by their orderxas=it1eratexs o=f 0 ( or of after a
oordinate
translation that moves 2 to ). This is the point whi
h ne
essarily
belongs to any super
y
le. Feigenbaum denotes the distan
e from the m-th
2n
element of a -super
y
le to its nearest neighbor by
d (m) = x f2n−1(x )
n m − Rn m
R
n
where is the
ontrol parameter value at whi
h super
y
le o
urs.
q 2n
To generalize the latter de(cid:28)nition to the period- · saddle-node orbit, whi
h
isinthesaddle-nodebifur
ation
as
ade, we arenotgoingtotake intoa
ount
all its points, but only a few very parti
ular ones.
q 2n
As a period- · saddle-node orbit undergoes a period doubling pro
ess there
will be a
ontrol parameter value, prior to the dupli
ation , at whi
h the orbit
q 2n R
n,q
will be a period- · super
y
le. Let be su
h parameter, and let be
1 1 1
,f( ),....,fq·2n−1( )
(cid:26)2 2 2 (cid:27) (.1)
the super
y
le in question.
10
