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173-193,
1997
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Vol. 34,
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No. 2-4,
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1997
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S0898·1221(97)00123·5
Simple Bifurcations
Bifurcations
Leading
Simple
Leading
to
Hyperbolic
Attractors
to Hyperbolic Attractors
L.
P. SHIL’NIKOV
AND D.
L. P.
SHIL'NIKOV AND
D. V.
V. TURAEV
TURAEV
Institute
Mathematics
&& Cybernetics
Cybernetics
Institute for
for Applied
Applied Mathematics
Ul’janova
10, 603005,
Novgorod, Russia
Ul'janova st.
st. 10,
603005, Nizhny
Nizhny Novgorod,
Russia
Abstract-we
prove
the existence
of a new stability
boundary
of periodic
orbits
in a highAbstract-We prove the existence of a new stability boundary of periodic orbits in a highdimensional
case, thereby
resolving
the problem
on a “blue
sky catastrophe”
in a general
onedimensional case, thereby resolving the problem on a "blue sky catastrophe" in a general oneparameter
family.
We additionally
establish
the existence
of a codimension-one
boundary
which
parameter family. We additionally establish the existence of a codimension-one boundary which
separates
the MorseSmale
systems
from systems
with hyperbolic
SmaleWilliams-like
attractors.
separates the Morse-Smale systems from systems with hyperbolic Smale-Williams-like attractors.
The route across this boundary
is accomplished
by the disappearance
of a saddle-node
periodic
orbit.
The route across this boundary is accomplished by the disappearance of a saddle-node periodic orbit.
We also study
the principal
bifurcations
of a torus breakdown
which lead to Anosov
attractors
and
We also study the principal bifurcations of a torus breakdown which lead to Anosov attra.ctors and
to multidimensional
solenoids.
to multidimensional solenoids.
Keywords---Global
bifurcations,
Hyperbolic
attractors,
Saddle-node
bifurcation.
Keywords--Global bifurcations, Hyperbolic attractors, Saddle-node bifurcation.
INTRODUCTION
INTRODUCTION
The problems discussed in this paper are in a close relation to one of the main questions of
The problems discussed in this paper are in a close relation to one of the main questions of
nonlinear dynamics about the structure of boundaries of stability regions of periodic orbits, as
nonlinear dynamics about the structure of boundaries of stability regions of periodic orbits, as
well as to the question of finding simple global bifurcations leading to appearance of hyperbolic
well as to the question of finding simple global bifurcations leading to appearance of hyperbolic
attractors.
attractors.
The notion of a stability region is introduced in the following way. Consider a /r-parameter
The notion of a stability region is introduced in the following way. Consider a k-parameter
family of n-dimensional dynamical systems given by the system of ODE
family of n-dimensional dynamical systems given by the system of ODE
x = X(x,J.l),
where 2 = (XI,..., 2,) is the vector of phase variables and p = (~1,. . . ,pk) is the vector of
where x
(Xl, ... ,x ) is the vector of phase variables and J.l
(J.ll! ... ,J.lk) is the vector of
parameters. Suppose nthat, at p = p*, the system possesses a periodic orbit z = cp(t, CL*), all
parameters.of Suppose
J.l
J.l*, the system possesses a periodic orbit X = <p(t, J.l*), all
multipliers
which liethat,
strictlyat inside
the unit circle; i.e., the periodic orbit is stable (attractive).
multipliers
of
which
lie
strictly
inside
unit
the periodic
orbit
is stable
As it is well known, for all p sufficiently the
close
to circle;
p*, theLe.,
periodic
orbit will
exist
and the(attractive).
multipliers
As
it
is
well
known,
for
all
J.l
sufficiently
close
to
J.l*
,
the
periodic
orbit
will
exist
will remain inside the unit circle. Thus, one can consider the maximal region Dand
in the
the multipliers
parameter
will
the unitorbit
circle.
Thus,
can all
consider
the maximal
region
in theinparameter
spaceremain
where inside
the periodic
exists
andone
where
the multipliers
are less
than'0 unity
absolute
space
where
the
periodic
orbit
exists
and
where
all
the
multipliers
are
less
than
unity
in
absolute
value (this is analogous, in a sense, to analytical continuation). The region V is called the stability
value
is analogous,
in a (note
sense,that
to analytical
continuation).
region V is called
the stability
region (this
of the
periodic orbit
the stability
region can The
be many-sheeted
in some
cases).
region
of
the
periodic
orbit
(note
that
the
stability
region
can
be
many-sheeted
in
some
cases).
There exist two types of the boundaries 62) of stability regions:
There exist two types of the boundaries
a'D of stability regions:
(1) such that the periodic orbit exists for p E XV,
(1) such
such that
that the
periodic orbit
orbit disappears
exists for J.l atE av,
(2)
the periodic
/J E dD.
(2) such that the periodic orbit disappears at J.l E av.
For the first case, the principal1 boundaries of stability of periodic orbits can be easily enumerated.
For the
I boundaries
of stability
of periodic
orbits can
easily
enumerated.
Let
x(X,first
p) =case,
0 bethe
theprincipal
characteristic
equation
of the linearized
Poincare
mapbe (the
roots
X of this
Let X(A, J.l) = 0 be the characteristic equation of the linearized Poincare map (the roots A of this
This work was supported
by EC-Russia
Collaborative
Project
ESPRIT-P9282-ACTCS,
by the INTAS
Grant
No.
ext. supported
and by theby
Russian
Foundation
of Fundamental
Researches,
Grant
No. 96-01-01135.
This93-0570
work was
EC-Russia
Collaborative
Project ESPRIT-P9282-ACTCS,
by the INTAS Grant
li.e., 93-0570
those which
are by
typicalthe Russian
for generalFoundation
one-parameterof FUndamental
families
No.
ext. and
Researches, Grant No. 96-01-01135.
Ii-e., those which are typical for general one-parameter families
173
173
Typeset by
.A.MS-'lEX
L.
P.
L. P.
174
SHIL'NIKOV
D. V.
V. TURAEY
TURAEV
8HIL'NlKOY AND
AND D.
Figure 1. The phase portrait near a saddle-nodeperiodic orbit L. The strong-stable
Figure
The phase
portrait near
a saddle-node
periodic
orbittwo
L. The
strong-stable
manifold1. Wds
of L separates
a small
neighborhood
of it onto
regions:
the node
manifold
WSs of L separates a small neighborhood of it onto two regions: the node
regions all orbits of which tend to L as t -t +oo, and the saddle region where the
regions all orbits of which tend to Last ..... +00, and the saddle region where the
two-dimensional
unstable manifold WU lies.
two-dimensional unstable manifold W" lies.
\
\
o
~.
~.
Figure 2. The phase portrait for the Poincar6 map of somecross-sectionto a saddleFigureperiodic
2. The orbit.
phase portrait for the Poincare map of some cross-section to a saddlenode
node periodic orbit.
equation are the multipliers of the periodic orbit). The loss of stability of the periodic orbit
equation when
are the
multipliers
of the periodic
orbit).
of stability
of the
happens
some
of the multipliers
are lying
on theThe
unitloss
circle.
Therefore,
the perkldic
boundariesorbit
of
happens
when
some
of
the
multipliers
are
lying
on
the
unit
circle.
Therefore,
the
boundaries
of
stability are given by the following equations:
stability are given by the following equations:
X(&P)
X(l, J.t)
xc-LP)
X(-l,J.t)
x (efiw,p)
= 0
0
= 0
=
0
= 0,
O<W<?T.
x (e±iw,J.t) 0, O<W<1r.
Each equation defines, in general, a codimension one surface in the parameter space. The first
Each equation
defines,
in general,
surface
in theFor
parameter
The first
equation
corresponds
to the
presencea ofcodimension
a multiplier one
equal
to unity.
a general space.
one-parameter
equation
presence
of a multiplier
equal
For a general
one-parameter
family thatcorresponds
intersects to
thethesurface
transversely,
the rest
of to
theunity.
multipliers
lie strictly
inside the
family
thatand
intersects
surface transversely,
the equal
rest ofto the
lie strictly
inside the
unit circle
the firsttheLyapunov
value does not
zeromultipliers
at the moment
of bifurcation.
unit
circle
and
the
first
Lyapunov
value
does
not
equal
to
zero
at
the
moment
of
bifurcation.
Such periodic orbit is called a simple saddle-node. For this case, the phase portrait is shown in
Such periodic orbit is called a simple saddle-node. For this case, the phase portrait is shown in
Simple
Simple Bifurcations
Bifurcations
Figure
3. A homoclinic
loop
Figure 3. A homoclinic loop
Figure
4. A homoclinic
rr
loop
Figure 4. A homoclinic loop
to a saddle-node
equilibrium
175
175
state
0 on a plane.
to a saddle-node equilibrium state 0 on a plane.
r to a saddle
r
equilibrium
state
0 on a plane.
to a saddle equilibrium state 0 on a plane.
Figures 1 and 2. Note that the saddle-node periodic orbit disappears when the parameter value
Figures the
1 and
2. Note
that the saddle-node periodic orbit disappears when the parameter value
crosses
stability
boundary.
crosses
the
stability
boundary.
The second equation corresponds to the presence of one multiplier equal to -1. In general, the
equationremain
corresponds
to the
of one
equalvalue
to -1.L does
In general,
the
restThe
of second
the multipliers
inside the
unitpresence
circle and
the multiplier
first Lyapunov
not equal
restzero.
of the
multipliers
inside
unit circle
andcrossing
the firstthis
Lyapunov
L does not
to
The
periodic remain
orbit does
not the
disappear
when
stabilityvalue
boundary.
If Lequal
< 0,
to zero.
periodic orbit
does not takes
disappear
L < 0,
then
the The
period-doubling
bifurcation
place.when
If L crossing
> 0, thenthis
thestability
periodic boundary.
orbit losesIfstability
If L > 0,
thenof the
periodic
orbitmerges
loses stability
then
period-doubling
bifurcation
place.periodic
at
thethe
moment
of bifurcation
when antakes
unstable
orbit
doubled
period
into it.
at The
the third
moment
of bifurcation
whento an
of doubled periodmultipliers
merges into
it.
equation
corresponds
theunstable
presenceperiodic
of a pair orbit
of complex-conjugate
efiw.
The
third
equation
to multipliers
the presencelieofstrictly
a pair of
complex-conjugate
multipliers
One
may
assume
that corresponds
the rest of the
inside
the unit circle, that
w # 9 e±i"'.
and
thatfirst
the Lyapunov
rest of the value
multipliers
strictly
the The
unit periodic
circle, that
w ::j:.does and
One
w
# may
5 andassume
that the
L doeslienot
equal inside
to zero.
orbit
not
w ::j:. ~ andwhen
that crossing
the first such
Lyapunov
value and
L does
notthe
equal
zero.
The periodic
orbit doestorus
not
disappear
boundary,
either
birthto of
a stable
two-dimensional
disappear when crossing such boundary, and either the birth of a stable two-dimensional torus
2;
176
176
L.L. P.P. SHIL’NIKOV
SHIL'NIKOV AND
AND D.
D. V.V. TURAEV
TURAEV
accompanies
accompanies the
the loss
loss of
of stability
stability of
of the
the periodic
periodic orbit
orbit (L
(L << 0)
0) or
or an
an unstable
unstable two-dimensional
two-dimensional
torus
torus merges
merges into
into the
the periodic
periodic orbit
orbit at
at the
the moment
moment ofof losing
losing stability
stability (L
(L >> 0).
0).
These
These are
are the
the three
three principal
principal bifurcations
bifurcations of
of the
the loss
loss ofof stability
stability of
of the
the periodic
periodic orbit
orbit for
for the
the
case
where
the
periodic
orbit
does
exist
on
the
stability
boundary.
As
we
mentioned,
there
case where the periodic orbit does exist on the stability boundary. As we mentioned, there are
are
stability
stability boundaries
boundaries of
of another
another type
type on
on which
which the
the periodic
periodic orbit
orbit does
does not
not exist.
exist. The
The existence
e>dstence of
of
such
such boundaries
boundaries wss
was discovered
discovered by
by Andronov
Andronov and
and Leontovich.
Leontovich. They
They found
found for
for two-dimensional
two-dimensional
systems
systems of
of ODE
ODE that
that there
there are
are exactly
exactly two
two principal
principal stability
stability boundaries
boundaries of
of the
the given
given kind:
kind: on
on
the
first
boundary
the
stable
periodic
orbit
disappears
merging
into
a
homoclinic
loop
of
the first boundary the stable periodic orbit disappears merging into a homoclinic loop of aa simple
simple
saddle-node
saddle-node equilibrium
equilibrium state
state (Figure
(Figure 3),
3), and
and on
on the
the second
second boundary
boundary the
the stable
stable periodic
periodic orbit
orbit
merges
merges into
into aa homoclinic
homoclinic loop
loop of
of aa saddle
saddle equilibrium
equilibrium state
state (Figure
(Figure 4)
4) with
with negative
negative saddle
saddle value.2
value. 2
Analogous
Analogous stability
stability boundaries
boundaries were
were found
found in
in [l]
[1] for
for the
the multidimensional
multidimensional case:
case:
(1)
(1) on
on the
the first
first boundary,
boundary, the
the system
system has
has aa simple
simple saddle-node
saddle-node equilibrium
equilibrium state3
state 3 and
and the
the
only
orbit
that
leaves
the
saddle-node
at
t
=
--oo
returns
to
the
saddle-node
as
t
only orbit that leaves the saddle-node at t = -00 returns to the saddle-node as t +--+ +oo,
+00,
forming
forming aa homoclinic
homo clinic loop;
loop; itit is
is assumed
assumed also
also that
that the
the homoclinic
homoclinic orbit
orbit does
does not
not; lie
lie in
in the
the
strong-stable
manifold
(Figure
5);
strong-stable manifold (Figure 5);
(2)
(2) on
on the
the second
second boundary,
boundary, the
the system
system has
has aa saddle
saddle equilibrium
equilibrium state
state of
of the
the type
type (n
(n -- 1,l)
1,1)
X,-l
have
negative
real
parts
and
X,
(i.e.,
the
characteristic
exponents
X1,.
.
.
,
(Le., the characteristic exponents AI,.'" An-1 have negative real parts and An >> 0)
0) and
and
one
one of
of the
the two
two orbits
orbits leaving
leaving the
the saddle
saddle at
at tt = -oo
-00 returns
returns to
to the
the saddle
saddle as
as tt +--+ +oo,
+00,
forming
forming aa homoclinic
homo clinic loop
loop (Figure
(Figure 6);
6); the
the saddle
saddle value
value
n=X,+
(j = An
max
+ i=l,...,n-1
max %Xi
~eAi
i=l, ... ,n-l
is
is negative.
negative.
In [l],
stable periodic
In
[1], there
there was
was shown
shown that
that aa stable
periodic orbit
orbit merges
merges into
into the
the homoclinic
homoclinic loop
loop in
in both
both cases.
cases.
When approaching the bifurcational moment, the length of the periodic orbit remains bounded
When approaching the bifurcational moment, the length of the periodic orbit remains bounded
whereas the period tends to infinity.
whereas the period tends to infinity.
In this connection the following question arises: can there be some other types of stability
In this connection the following question arises: can there be some other types of stability
boundaries of codimension one? In the present paper we show that the answer is positive. We
boundaries of co dimension one? In the present paper we show that the answer is positive. We
find a new stability boundary which is an open subset of a codimension one bifurcational surface
find a new stability boundary which is an open subset of a codimension one bifurcational surface
corresponding to the presence of a saddle-node periodic orbit. This open set is distinguished
corresponding to the presence of a saddle-node periodic orbit. This open set is distinguished
by some qualitative conditions determining the geometry of unstable set of the saddle-node (see
by some qualitative conditions determining the geometry of unstable set of the saddle-node (see
Figure 7) and also by some quantitative restrictions (some value should be less than unity, see
Figure 7) and also by some quantitative restrictions (some value should be less than unity, see
below). We will show under these conditions that when the saddle-node disappears a new stable
below). We will show under these conditions that when the saddle-node disappears a llew stable
periodic orbit arises, and both the period and the length of this periodic orbit tend to infinity
periodic orbit arises, and both the period and the length of this periodic orbit tend to infinity
when approaching the bifurcation moment (Theorem 1).
when approaching the bifurcation moment (Theorem 1).
Actually, this is one of the possible variants of the global bifurcation of disappearance of a
Actually, this is one of the possible variants of the global bifurcation of disappearance of a
saddle-node periodic orbit such that all the orbits of its unstable set return to the saddle-node
saddle-node periodic orbit such that all the orbits of its unstable set return to the sf~ddle-node
as t -+ +CXJ.~ Without loss of generality, we may assume that all these orbits do not. lie in the
as t --+ +00. 4 Without loss of generality, we may assume that all these orbits do not lie in the
strong-stable manifold of the saddle-node.
strong-stable manifold of the saddle-node.
The study of this global bifurcation has a long history. Originally, this problem appeared in
The study of this global bifurcation has a long history. Originally, this problem appeared in
the 20’s in connection with the study of the phenomenon of the transition from the synchronizathe
in connection
with the study
of the
of the
transition from the synchronization 20's
to the
amplitude modulation
regime
in phenomenon
the van der Pol
equation
tion to the amplitude modulation regime in the van der Pol equation
With the assumption that p is a small parameter and that the resonance 1 : 1 takes place (i.e.,
With
thatand
fL isVitt
a small
parameter
that the from
resonance
1 : 1 takes placeto (Le.,
w-WQ theN assumption
Jo), Andronov
showed
that theand
transition
the synchronization
the
W
Wo "" fL), Andronov and Vitt showed that the transition from the synchronization to the
2the
sum
of characteristic
exponents
3i.e., sum
an equilibrium
state exponents
whose characteristic
exponents
lie strictly
to the left of the imaginary
axis except
for
2the
of characteristic
3Le.,
an equilibrium
state whose
characteristic
lie strictly
thenotleftequal
of theto imaginary
axis except for
one simple
zero characteristic
exponent,
and the exponents
first Lyapunov
value to
does
zero
one
exponent,
Lyapunov
value does
not saddle-node
equa.! to zeroas t + --cm as well.
4Sincesimple
these zero
orbitscharacteristic
belong to the
unstable and
set,the
theyfirsttend,
by definition,
to the
4Since these orbits belong to the unstable set, they tend, by definition, to the saddle-node as t .... -eXl as well.
Simple Bifurcations
Bifurcations
Simple
177
177
rP
Figure
5. A homoclinic
loop r to a saddle-node
Figure
6. A homoclinic
loop r to a saddle
equilibrium
state
0 for the dimension
Figure 5. A homoclinic loop r to a saddle-node equilibrium state 0 for the dimension
of the phase space more than two.
equilibrium
state
0 in the multidimensional
Figure 6. A homoclinic loop r to a saddle equilibrium state 0 in the multidimensional
case.
CaSe.
amplitude modulation regime is connected with the bifurcation of the birth of a stable limit
amplitude
connected with
the bifurcation
the birth
a stable limit
cycle
from amodulation
homoclinic regime
loop of isa saddle-node
equilibrium
state (seeofFigure
3) in of
a time-averaged
cycle from
a homoclinic
of a equation,
saddle-node
Figure 3)
in a time-averaged
system.
Returning
to theloop
initial
oneequilibrium
can see thatstate
the (see
analogous
picture
holds for the
system.
Returning
to
the
initial
equation,
one
can
see
that
the
analogous
picture
holdsand
for the
the
two-dimensional Poincare map where the saddle-node is now the fixed point of the map
two-dimensional
Poincare
map
where
the
saddle-node
is
now
the
fixed
point
of
the
map
and
the
homoclinic loop is not a single orbit, but is a continuum of orbits which compose the unstable
homoclinic
loop is not In
a single
orbit,such
butkind
is a of
continuum
of orbits
which out.
compose the unstable
set
of the saddle-node.
that time
analysis was
not carried
set of the saddle-node. In that time such kind of analysis was not carried out.
178
178
L.
L. P.
P. SHIL’NIKOV
SHIL'NIKOV AND
AND D.
D. V:
V: TURAEV
TURAEV
Figure
7. The
configuration of
unstable manifold of asaddle-node periodic orbit L
the
Figure 7. The configuration of the unstable manifold of a saddle-node periodic orbit L
the closure w is not a manifold.
which may lead to the blue sky catastrophe: the closure W" is not a manifold.
which may lead to the blue sky catastrophe:
The systematical study of this bifurcation was begun with the paper [2] by Afraimovich and
The systematical study of this bifurcation was begun with the paper [2] by Afrairnovich and
Shil’nikov under the assumption that the dynamical system that undertakes the bifurcation is
Shil'nikov under the assumption that the dynamical system that undertakes the bifurcation is
either nonautonomous and periodically depending on time, or autonomous but. possessing a global
either nonautonomous and periodically depending on time, or autonomous but possessing a global
cross-section (at least, at that part of the phase space which is studied). Essentially, the problem
cross-section (at least at that part of the phase space which is studied). Essentially, the problem
was reduced to the study of a one-parameter family of C’-diffeomorphisms
(T- I 2) which has,
was reduced to the study of a one-parameter family of C r -diffeomorphisms (r 2:: 2) which has,
at p = 0, a fixed point of saddle-node type such that all orbits of the unstable set of the saddleat J1, = 0, a fixed point of saddle-node type such that all orbits of the I.}nstable set of the saddlenode return to its small neighborhood and tend to it, as the number of iterations of the map tends
node return to its small neighborhood and tend to it as the number of iterations of the map tends
to +oo.
to +00.
Recall that the saddle-node point has one multiplier equal to unity and all the other multipliers
Recall that the saddle-node point has one multiplier equal to unity and all the other multipliers
are less than unity in absolute value. Near the saddle-node, the diffeomorphism has the form
are less than unity in absolute value. Near the saddle-node, the diffeomorphism has the form
5 = AY+ WY,4
= Ay + H(y, z)
fj
f=z+G(y,a),
z = z + G(y,z),
(1)
(1)
where z E R’, y E Rn,n A is a matrix whose eigenvalues lie strictly inside the unit circle, LI(0, 0) = 0,
where z E Rl, YE R , A is a matrix whose
eigenvalues lie strictly inside the unit circle, H(O, 0) = 0,
H&, (O,O) = 0, WA 0) = 0, G&J 0,O) = 0. Here, the fixed point is in the origin. As it is well
H(t/,z)
(0,
0)
=
0,
G(O,O)
=
0,
G(t/,z)
(0,0)
O. Here,
the fixed
point
is iny the
origin.
As ~(0)
it is =well
known, there exists a CT-smooth
invariant =center
manifold
of the
form
= Q(E),
where
0,
known,
there
exists
a
Cr-smooth
invariant
center
manifold
of
the
form
y
=
1](z),
where
1](0)
= 0,
q’(O) = 0. Restricted onto the center manifold, the map takes the form
1]'(0) = O. Restricted onto the center manifold, the map takes the form
E = 2 + g(z),
z = z + g(z),
(2)
(2)
where g(z) 3 G(v(.z),z) E C’, g(0) = 0, g’(0) = 0.
where
== G(1](z),z)
E C rsimple
0, g'(O)
, g(O) =
The g(z)
saddle-node
is called
if g”(0)
# =
0. O.In this case equation (2) takes the form
The saddle-node is called simple if gl/(O)
1= O. In this case equation (2) takes the form
E = z + 1222 + * * * ,
where 12 = g”(O)/2 # 0. Without loss of generality, one can assume 12 > 0.
where l2 = g"(0)/2 1= O. Without loss of generality, one can assume 12 > O.
(3)
(3)
Simple
Simple Bifurcations
Bifurcations
179
179
Figure
8. When
the saddle-node
disappears
(fi > 0), all orbits
leave its small neighFigure 8. When the saddle-node disappears (It > 0), all orbits leave its small neighborhood
borhood.
l
• •
) \•
\
./
JII/f"
./ '
-0
t
'x.---.....
l
i
1’
a
Figure 9. When ~1 < 0, the saddlenode
disintegrates
onto two fixed points:
saddle 01
Figure
9. When
It < 0, the saddle-node disintegrates onto two fixed points: saddle 01
and
stable
02.
and stable 02.
One can see (Figure 2) that a small neighborhood of the origin is split by the strong-stable
One can
see (Figure
that(c(O)
a small
theregions:
origin isthesplit
by region
the strong-stable
invariant
manifold
{z = 2)
t(y)}
= 0, neighborhood
t’(O) = 0) intooftwo
node
{z < t(y)}
invariant
manifold
{z =
(€(O) All 0,orbits
e'(0) from
= 0) the
intonode
two regions:
the node
< e(y)}
and
the saddle
region
{.ze(y)}
> c(y)}.
region tend
to theregion
origin {xalong
the
and
the
saddle
region
{z
>
€(y)}.
All
orbits
from
the
node
region
tend
to
the
origin
along
z-axis. In the saddle region, the one-dimensional unstable manifold {y = n(z), z > 0) lies, the
all
z-axis. ofInwhich
the saddle
region,
the one-dimensional
unstable
= 1J(z),
z >other
O} lies,
all
orbits
tend to
the origin
with the iterations
of themanifold
inverse {y
map.
All the
orbits,
orbits
of which
the origin
with the iterations
of thewith
inverse
map. All of
theboth
other
orbits
from the
saddle tend
regiontoleave
the neighborhood
of the origin
the iterations
map
(1)
from the
region leave the neighborhood of the origin with the iterations of both map (1)
itself
and saddle
its inverse.
itself
and its
To take
intoinverse.
account the dependence on the parameter CL,one must consider the functions H
To
take
into
the dependence
on the
parameter
p., one must
the tofunctions
H'
and G from (1) account
to be functions
on p also.
Moreover,
the parameter
,u consider
is supposed
be chosen
and G from (1) to be functions on p. also. Moreover, the parameter p. is supposed to be chosen
180
180
L.
L. P.
P. SHIL'NIKOV
SHIL'NIKOV AND
AND D.
D. V.
V. TURAEV
TURAEV
Figure
10. The
case where
the set w
is smooth.
Figure 10. The case where the set W" is smooth.
Figure
11. The case where the set w
is a nonsmooth
circle (the tangent
vector does
the setonW"
a nonsmooth
tangent vector does
Figure
not
have 11.a The
limit case
when where
the point
w is approaches
0 circle
from (the
the left).
not have a limit when the point on W" approaches 0 from the left).
such that the saddle-node disappears when 1-1> 0 (Figure 8), and when 1-1< 0,
such two
that fixed
the saddle-node
disappears
when
J.t > 09).(Figure
8), and when
< 0,
onto
points: saddle
and stable
(Figure
The restriction
of theJ.t map
onto two is
fixed
points:assaddle and stable (Figure 9). The restriction of the map
manifold
rewritten
manifold is rewritten as
E = 2 + p + 1$2 + * f. .
it disintegrates
it disintegrates
onto
the center
onto the center
(4
(4)
As we mentioned, the orbits of all points of the local unstable manifold are supposed to return
we node
mentioned,
all points
of the
unstable
manifold
are supposed
return
into.Ai?,the
region the
and orbits
tend toofthe
origin at
p= local
0. The
union of
these orbits
is denotedto as
w”
into
the
node
region
and
tend
to
the
origin
at
J.t
=
O.
The
union
of
these
orbits
is
denoted
as
WU
and this set is here homeomorphic to a circle. It occurred that the set w” may be a smooth
and
this
set
is
here
homeomorphic
to
a
circle.
It
occurred
that
the
set
WU
may
be
a
smooth
circle (Figure 10) or it may be nonsmooth (Figure 11).
circle
(Figure
10) case
or it there
may be
11). the saddle-node disappears, an attractive
For the
smooth
wasnonsmooth
found in [2](Figure
that when
For theinvariant
smooth curve
case there
wastofound
saddle-node disappears,
an attractive
smooth
inherits
w. inIf [2]
thethat
mapwhen
undertheconsideration
is the Poincare
map of a
smoothcross-section
invariant curve
inherits
to WU.
If the then
map the
under
consideration
the line
Poincare
map of a
global
for some
system
of ODE,
invariant
curve isis the
of intersection
global
cross-section
for
some
system
of
ODE,
then
the
invariant
curve
is
the
line
of
intersection
of an invariant two-dimensional torus with the cross-section. The Poincare rotation number on
of antorus
invariant
torus with the cross-section. The Poincare rotation number on
the
tends two-dimensional
to zero as p + +O.
the torus tends to zero as J.t ~ +0.
Simple Bifurcations
Simple
Bifurcations
181
181
This
of the
transition from
from the
synchronization to
This result
result gave
gave aa rigorous
rigorous explanation
explanation of
the transition
the synchronization
to the
the an+
amplitude
forced nonlinear
the only
plitude modulation
modulation in
in periodically
periodically forced
nonlinear systems:
systems: when
when pJ.L << 0,
0, the
only stable
stable regime
regime
is
periodic orbit
synchronization, and
that
is the
the stable
stable periodic
orbit which
which corresponds
corresponds to
to synchronization,
and the
the invariant
invariant torus
torus that
exists
corresponds to
exists when
when pJ.L >> 00 corresponds
to modulation
modulation regime.
regime.
The case
case where
nonsmooth manifold
considered in
series of
of papers.
The
where rWU is
is aa nonsmooth
manifold was
was considered
in aa series
papers. In
In the
the
aforementioned paper
[2], there
(under the
so-called “big
lobe” condition)
aforementioned
paper [2),
there was
was established
established (under
the so-called
"big lobe"
condition) the
the
presence
of
the
sequence
of
intervals
(pi,&)
accumulating
at
p
=
+0
such
that
the
system
has
presence of the sequence of intervals (J.Li,J.LD accumulating at J.L = +0 such that the system has
nontrivial
sets at
nontrivial hyperbolic
hyperbolic sets
at /J
J.L E
E (pi,
(J.Li, pi).
J.L~). Without
Without the
the big
big lobe
lobe condition
condition (but
(but for
for one-parameter
one-parameter
families
of
a
special
kind),
this
result
was
proved
in
[3]
by
Newhouse,
Palis
and
They
families of a special kind), this result was proved in [3] by Newhouse, Palis and Takens.
Takens. They
used
essentially aa theorem
on existence
used essentially
theorem by
by Block
Block on
existence of
of periodic
periodic orbits
orbits for
for endomorphisms
endomorphisms of
of aa
circle.
In
(41,
the
results
of
[2,3]
were
extended
onto
the
general
case;
there
was
also
shown
circle. In [4], the results of [2,3] were extended onto the general case; there was also shown
that
for aa “sufficiently
parameter values
to complex
that for
"sufficiently small
small lobe”
lobe" the
the intervals
intervals of
of parameter
values corresponding
corresponding to
complex
dynamics
(hyperbolic
sets)
and
simple
dynamics
(a
continuous
invariant
curve
with
dynamics (hyperbolic sets) and simple dynamics (a continuous invariant curve with rational
rational
Poincare
Poincare rotation
rotation number)
number) alternate
alternate accumulating
accumulating at
at ~1
J.L == +O.
+0.
The
important feature
of the
parameter values
The important
feature of
the nonsmooth
nonsmooth case
case is
is the
the existence
existence [3,4]
[3,4] of
of parameter
values in
in an
an
arbitrary
closeness
to
p
=
+0
which
correspond
to
the
presence
of
saddle
periodic
orbits
arbitrary closeness to J.L = +0 which correspond to the presence of saddle periodic orbits with
with
s&ucturally
orbits. According
modern knowledge
[5]), this
structurally Imsta6le
unstable homoclinic
homoclinic orbits.
According to
to the
the modern
knowledge (see
(see [5]),
this leads
leads
to
extremely
complicated
dynamics:
to
the
Newhouse
phenomenon
(persistence
of
homoclinic
to extremely complicated dynamics: to the Newhouse phenomenon (persistence of homoclinic
tangencies,
tangencies, coexistence
coexistence of
of infinitely
infinitely many
many sinks)
sinks) [6-111,
[6-11], to
to HBnon-like
Henon-like attractors
attractors [12,13],
[12,13), and
and
to infinite degenerations [14] which make it impossible to give a complete description of all
to infinite degenerations [14] which make it impossible to give a complete description of all
bifurcations that may occur in this case.
bifurcations that may occur in this case.
/u
<o
f
>0
For /L < 0, there exist two periodic
Figure
12. The local saddle-node
bifurcation.
Figure stable
12. The
saddle-node
p, <latter
0, there
exist two periodic
orbits:
and loca.l
saddle;
the unstablebifurca.tion.
manifold For
of the
is homeomorphic
to a
of the latter
is homeomorphic
to a
orbits: stable andcylinder.
saddle; the
two-dimensional
Whenunstable
p > 0,manifold
the saddle-node
disappears
and all orbits
two-dimensional
cylinder. When p, > 0, the saddle-node disappears and all orbits
leave
its small neighborhood.
leave its small neighborhood.
In the present paper we point out that if an autonomous system with a saddle-node does not
In the
present
paper we point
if an autonomous
systempossible
with a saddle-node
have
a global
cross-section
there out
may that
be essentially
more different
cases. Let does not
have a global cross-section there may be essentially more different possible cases. Let
i = X(x, p)
x = X(X,J.L)
be a one-parameter family of n-dimensional CT-smooth (T 2 2) dynamical systems which has, at
one-parameter
family
of n-dimensional
or -smooth
~ 2)local
dynamical
systems
which
at
pbe=a 0,
a periodic orbit
of saddle-node
type (i.e.,
such that(r the
Poincare
map has
formhas,
(1)).
J.When
L = 0, ap periodic
orbit
of
saddle-node
type
(Le.,
such
that
the
local
Poincare
map
has
form
(I)).
varies, the local bifurcations go as follows (Figure 12). For p < 0, there exist two
When J.L orbits:
varies, the
as follows
(Figureof 12).
For J.Lis<homeomorphic
0, there exist totwoa
periodic
stablelocal
andbifurcations
saddle; the gounstable
manifold
the latter
periodic
orbits:
stable
and
saddle;
the
unstable
manifold
of
the
latter
is
homeomorphic
to a
0,
these
periodic
orbits
unite
in
one
structurally
unstable
two-dimensional cylinder. At /J =
two-dimensional
cylinder.
At
J.
L
::::
0,
these
periodic
orbits
unite
in
one
structurally
unstable
periodic orbit L of the simple saddle-node type. The local unstable set IV, is homeomorphic to
periodic orbit L of the simple saddle-node type. The local unstable set Wl is homeomorphic to
182
182
L.
SHIL’NIKOV AND
L. P.
P. SHIL'NIKOV
AND D.
D. V.
V. TURAEV
TURAEV
aa half-cylinder
has also
half-cylinder R+
R+ xx 5”.
8 1 . The
The orbit
orbit LL has
also the
the strong-stable
strong-stable manifold
manifold Wr
Wl8 which
whkh divide
divide aa
neighborhood
of
L
into
two
regions:
saddle
and
node.
When
p
>
0,
the
saddlenode
&appears
neighborhood of L into two regions: saddle and node. When J.L > 0, the saddle-node disappears
and
leave its
starting point
and all
all orbits
orbits leave
its small
small neighborhood.
neighborhood. Note
Note that
that ifif the
the starting
point of
of some
som.;) orbit
orbit lies
lies
in
the
node
region,
then
the
time
which
the
orbit
spends
in
a
small
fixed
neighborh.ood
of the
the
in the node region, then the time which the orbit spends in a small fixed neighborhood of
saddle-node
infinity as
ss pp. +- +O,
as the
the length
length of
of the
saddle-node tends
tends to
to infinity
+0, as
as well
well as
the corresponding
corresponding piece
piece of
of the
the
orbit.
orbit.
Suppose
all the
orbits of
Wi return
the node
Suppose that,
that, at
at /.J
p. == 0,
0, all
the orbits
of Wl
return to
to the
node region
region and
and tend
tend to
to LL es
as tt -+
_ +oo,
+00,
8
not
lying
in
WY.
The
union
w’
of
these
orbits
may,
for
instance,
be
a
smooth
two-dimensional
not lying in W1 • The union WU of these orbits may, for instance, be a smooth two-dimensional
surface:
or aa Klein
Klein bottle
is nonorientable
if the
surface: aa torus,
torus, or
bottle (the
(the latter
latter may
may be
be ifif the
the phase
phase space
space is
nonorientable or
or if
the
dimension
[2], the
dimension nn of
of the
the phase
phase space
space is
is not
not less
less than
than four).
four). Analogously
Analogously to
to [2],
the smooth
smooth invariant
invariant
two-dimensional
for /.L
set rWU is
nonsmooth torus,
is preserved
preserved for
p. >> 0.
O. As
As above,
above, ifif the
the set
is aa nonsmooth
torus,
two-dimensional surface
surface is
then
saddle
periodic
orbits
with
homoclinic
curves
may
appear
at
/.L
>
0;
the
same
then saddle periodic orbits with homoclinic curves may appear at p. > 0; the same may
may be
be in
in the
the
case
where w”
conditions [4].
case where
W isis aa nonsmooth
nonsmooth Klein
Klein bottle
bottle under
under some
some additional
additional conditions
[4].
Essentially
earlier unknown
set WU
w” is
Essentially different
different situation
situation was
was earlier
unknown where
where the
the set
is not
not aa manifold.
manifold. First,
First,
consider
the
following
example.
Let
a
two-parameter
family
of
three-dimensional
vector
consider the following example. Let a two-parameter family of three-dimensional vector fields
fields
have,
at some
some parameter
values, aa saddle-node
L and
parameter values,
saddle-node periodic
periodic orbit
orbit L
and aa saddle-node
saddle-node equilibrium
equilibrium
have, at
state
(Figure 13).
13). Suppose
that all
all orbits
of Wl
Wz tend
to 00 as
and that
state 00 (Figure
Suppose that
orbits of
tend to
as tt +- +oo
+00 and
that the
the oneonedimensional
separatrix
of
0
tends
to
L.
If
one
of
parameters
is
varied
so
that
0
disappears
dimensional separatrix of 0 tends to L. If one of parameters is varied so that 0 disappears and
and
L does not, then the set w” will have the form shown in Figure 7. The intersection of w” with
L does not, then the set WU will have the form shown in Figure 7. The intersection of W with
a local cross-section S to L will be a union of a countable set of circles accumulating at the point
a local cross-section S to L will be a union of a countable set of circles accumulating at the point
Sn L (Figure 14). Evidently, any neighborhood of this point in the set w is not homeomorphic
S n L (Figure 14). Evidently, any neighborhood of this point in the set WU is not homeomorphic
to a ball. Therefore, in this situation the set 3
is not a manifold.
to a ball. Therefore, in this situation the set WU is not a manifold.
Systems having a saddle-node periodic orbit with the set w” being as shown in Figure 7
Systems having a saddle-node periodic orbit with the set WU being as shown in Figure 7
compose codimension one surfaces in the space of smooth flows in Rn (n 2 3). Below, we will
compose codimension one surfaces in the space of smooth flows in Rn (n ;::: 3). Below, we will
show (see Theorem 1) how open subsets are distinguished on these surfaces such that for any
show (see Theorem 1) how open subsets are distinguished on these surfaces such that for any
one-parameter family X, that intersects such subset transversely at p = 0, the system X, has an
one-parameter family XIJ. that intersects such subset transversely at J.L = 0, the system XIJ. has an
attractive periodic orbit for all small ~1> 0,5 whose period and length tend to infinity as p + +O.
attractive periodic orbit for all smallp. > 0,5 whose period and length tend to infinity as p.- +0.
Note the connection of this result to the problem on “the blue sky catastrophe” [15]. The
Note the connection of this result to the problem on "the blue sky catastrophe" [15]. The
original formulation was as follows. Does there exist a continuous one-parameter family
of smooth
original formulation was as follows. Does there exist a continuous one-parameter family of smooth
vector fields A, on a compact manifold such that A, has a closed orbit L, for all ,u :a 0, and as
vector fields AIJ. on a compact manifold such that AIJ. has a closed orbit LIJ. for all J.L > 0, and as
p + +O, the period of L, tends to infinity, and at p = +O, the orbit L, disappears on a finite
J.L - +0, the period of LIJ. tends to infinity, and at p.
+0, the orbit LIJ. disappears on a finite
distance of equilibrium states?6
distance of equilibrium states?6
Virtual bifurcations of such kind were called blue sky catastrophes by Abraham.
The first
Virtual bifurcations of such kind were called blue sky catastrophes by Abraham. The first
example of such catastrophe was constructed by Medvedev [16] for a one-parameter family of
example
of such
by Medvedev
for aat one-parameter
family of
vector
fields
on a catastrophe
Klein bottle was
whichconstructed
has a saddlenode
periodic[16]orbit
~1= 0. The Medvedev’s
vector fields on a Klein bottle which has a saddle-node periodic orbit at p. = O. The Medvedev's
family was of a rather special kind: the system that corresponds to p = 0 is also embedded in a
family was of a family
rather ofspecial
kind: the
system
thatallcorresponds
to J.L =
also embedded
in a
one-parameter
conservative
vector
fields,
orbits of which
are0 is
closed.
The Poincare
one-parameter
family
of
conservative
vector
fields,
all
orbits
of
which
are
closed.
The
Poincare
map for this conservative family has the form
map for this conservative family has the form
CF= -‘p + w(p) mod 1,
<p = -<p + w(p.) mod 1,
where w + 00 as p + +O. This map has two fixed points, all other points are of period two.
where
00 as J.L used
- +0.
mapthis
hasfamily
two fixed
points,
all other
period
two.
Actually,w -Medvedev
the This
fact that
can be
perturbed
so aspoints
to haveare
onlyoftwo
periodic
Actually,
Medvedev
used
the
fact
that
this
family
can
be
perturbed
so
as
to
have
only
two
periodic
points: one stable and one unstable fixed point; the stable fixed point corresponds to a stable
points: one
one unstable
the stable
periodic
orbitstable
whoseand
period
and lengthfixed
tendpoint;
to infinity
as p +fixed
+O.point corresponds to a stable
periodic
orbit
whose
period
and
length
tend
to
infinity
as
p.
+0.
In the general one-parameter family, both the fixed points will bifurcate infmitely rnany times
general
one-parameter
family,
fixed in
points
will bifurcate
many
as In
/.J +the+O,
changing
their stability
(thisboth
wasthe
noticed
[17] and
studied ininfinitely
more detail
in times
[18]).
as
J.L
+0,
changing
their
stability
(this
was
noticed
in
[17]
and
studied
in
more
detail
Formally, the blue sky catastrophe also takes place here because the structural stability inof[18]).
the
Formally,orbit
the blue
catastrophewas
alsonottakes
place inhere
structural stability of the
periodic
undersky
consideration
required
the because
original the
formulation.
periodic orbit under consideration was not required in the original formulation.
5We assume that the region ~1> 0 corresponds to the disappearance of the saddlenode L.
6The
latter implies
thatregion
the length
of L, also tends
infinity.
5We assume
that the
J.t > 0 corresponds
to thetodisappearance
of the saddle-node L.
6The latter implies that the length of L,.. also tends to infinity.
Simple Bifurcations
Bifurcations
Simple
W”
WU
Figure
13. An example
of bifurcation
creating
the configuration
shown
in Figure
7.
Figureall13.orbits
An example
of bifurcation
configuration
shown orbit
in Figure
Here,
of the unstable
manifold creating
WU of the
a saddle-node
periodic
L tend 7.
Here,
a.\l orbits ofequilibrium
the unstable
saddle-node
periodic orbit
L tend
to
a saddle-node
statemanifold
0 ss t -+W"fm,of aand
the one-dimensional
separatrix
to 0a saddle-node
state 0 asis tvaried
-> +00,
one-dimensional
of
tends to L. equilibrium
If one of parameters
so and
that the
0 disappears
and L separatrix
does not,
of parameters
is varied
so that7. 0 disappears and L does not,
of 0 tends
then
the set tov L. If
willone
have
the form shown
in Figure
then the set W" will have the form shown in Figure 7.
0):1
\ //~~~<:>~..
SnL
..........1___...____
Figure
14. For the configuration
shown in Figure
7, the intersection
of v
with a
local
cross-section
to L is a union shown
of a countable
of circles
accumulating
at the a
Figure
14. For theS configuration
in Figure set
7, the
intersection
of W" with
point
S rl L.
S to L is a union of a countable set of circles accumulating at the
local cross-section
point
sn L.
183
183
184
184
L.
L. P.P. SHIL’NIKOV
SHIL'NIKOV AND
AND D.
D. V.V. TURAEV
TURAEV
Evidently,
Evidently, the
the construction
construction proposed
proposed in
in the
the present
present paper
paper gives
gives also
also aa solution
solution to
to the
the blue
blue
sky
sky problem.
problem. At
At the
the same
same time,
time, itit seems to
to be
be more
more adequate
adequate because
because the
the periodic
periodic orbit
orbit in
in
Theorem
1
is
attractive
and
structurally
stable
for
all
p
>
0,
and
this
property
holds
for
an
open
Theorem is attractive and structurally stable for all /.L > 0, and this property holds for an open
set
set of
of one-parameter
one-parameter families.
families.
The
The second
second main
main result
result of
of the
the paper
paper is
is given
given by
by Theorem
Theorem 22 which
which establishes
establishes that
that in
in the
the space
space
of
smooth
flows
in
Rn
(n
2
4),
there
exist
codimension
one
bifurcation
surfaces
sepamking
Morseof smooth flows in Rn (n ;::: 4), there exist codimension one bifurcation surfaces separat:.~ng MorseSmale
of the
the SmaleSmale- Williams
Williams solenoid
solenoid type
t~,pe [19,20].
[19,201.
Smale systems
systems and
and systems
systems with
with hyperbolic
hyperbolic attractors
attractors of
We
will
show
that
the
solenoid
does
not
undergo
bifurcations
when
approaching
the
We will show that the solenoid does not undergo bifurcations when approaching the boundary
boundary
and
and period
period and
and length
length of
of any
any periodic
periodic orbit
orbit of
of the
the solenoid
solenoid tend
tend to
to infinity;
infinity; i.e.,
i.e., we
we also
also deal
deal with
with
aa variant
variant of
of the
the blue
blue sky
sky catastrophe
catastrophe here.
here.
Another
is known
known [21,22]
[21,22] which
which separates
separates Morse-Smale
Morse-Smale systems
systems and
and syssysAnother attainable
attainable boundary
boundary is
tems
with
nontrivial
hyperbolic
sets.
This
boundary
corresponds
to
the
existence
of
a
structurally
tems with nontrivial hyperbolic sets. This boundary corresponds to the existence of a structurally
unstable
unstable equilibrium
equilibrium state
state of
of the
the saddle-saddle
saddle-saddle type
type which
which has
has two
two or
or more
more homoclinic
homoclinic orbits.
orbits.
When
When approaching
approaching the
the boundary,
boundary, the
the hyperbolic
hyperbolic set
set also
also does
does not
not undergo
undergo bifurcations;
bifurcations; the
the main
main
difference
with
the
cases
considered
in
the
present
paper
is
that
the
hyperbolic
set
(the
solenoid)
difference with the cases considered in the present paper is that the hyperbolic set (the solenoid)
is
is now
now attractive.
attractive.
ItIt is
is well
well known
known that
that the
the hyperbolic
hyperbolic attractors
attractors have
have not
not ever
ever been
been found
found in
in dynamical
dynamkal models
models
arising
in
natural
applications
and
that
the
chaotic
dynamics
demonstrating
by
the large
large variety
variety
arising in natural applications and that the chaotic dynamics demonstrating by the
of
systems
has
a
nonhyperbolic
character;
i.e.,
it
is
connected,
for
instance,
with
the
Lorenz-like
of systems has a nonhyperbolic character; i.e., it is connected, for instance, with the Lorenz-like
attractor
attractor [23-261,
[23-26], the
the H&on-like
Henon-like attractor
attractor [12],
[12], or
or with
with quasiattractors
quasiattractors [5,26-281.
[5,26-28]. The
The fact
fact we
we
establish
here
that
a
hyperbolic
attractor
can
appear
in
a
relatively
simple
bifurcation
allows
establish here that a hyperbolic attractor can appear in a relatively simple bifurcation allows us
us
to suppose the hyperbolic attractors can be found in dynamical systems of natural origination if
to suppose the hyperbolic attractors can be found in dynamical systems of natural origination if
the dimension of phase space is four or more. In this connection, we propose (in the Concluding
the dimension of phase space is four or more. In this connection, we propose (in the Concluding
Remarks) more examples of bifurcations leading to hyperbolic attractors. In the next section, we
Remarks) more examples of bifurcations leading to hyperbolic attractors. In the next section, we
give the precise formulation of Theorems 1 and 2 and the proof for the case where the smoothness
give the precise formulation of Theorems 1 and 2 and the proof for the case where the smoothness
of the system with respect to the phase variables and the parameter is sufficiently high (the case
of the system with respect to the phase variables and the parameter is sufficiently high (the case
of low smoothness requires more delicate calculations [29]).
of low smoothness requires more delicate calculations [29]).
MAIN RESULTS
MAIN RESULTS
We consider a F-smooth
(r > 2) family of dynamical systems X, in Rn (n 2 3) which has
We consider a Cr-smooth (r ;::: 2) family of dynamical systems XI-' in ~ (n ;::: 3) which has
a periodic orbit Lo of the simple saddle-node type at ,a = 0. All the orbits of the unstable
a periodic orbit Lo of the simple saddle-node type at /.L = O. All the orbits of the unstable
manifold W” are supposed to return to Lo and to come into the node region as t + +oo.
manifold W" are supposed to return to Lo and to come into the node region as t - +00.
Herewith, any orbit of W” is bi-asymptotic to LO and w’ is compact.
Herewith, any orbit of W" is bi-asymptotic to Lo and W'" is compact.
Let U be a small neighborhood of w’ and UC, be a small neighborhood of Lo, U,J E U. Let
Let U be a small neighborhood of W'" and U be a small neighborhood of Lo, Uo E U. Let
us introduce coordinates in a neighborhood Us oof LO. We cut Uo along some cross-section
S
us introduce coordinates in a neighborhood Uo of Lo. We cut Uo along some cross-section S
and consider the coordinates (y, z, cp) where cp E [0, l] is the angular variable and (y:, .z) are the
and consider the coordinates (y, z, cp) where cP E [0,1] is the angular variable and (y" z) are the
normal coordinates; z E R1 is a coordinate on the center manifold, y E Rnm2 is a vector of
normal coordinates; z E R1 is a coordinate on the center manifold, y E Rn-2 is a vector of
coordinates corresponding to the multipliers less than unity in absolute value; values cp = 0 and
coordinates corresponding to the multipliers less than unity in absolute value; values cP = 0 and
cp = 1 correspond to the points lying on S.
cP =
correspond
to the points
We1 chose
the coordinates
suchlying
that onLOS.is the curve (y = 0, z = 0); we also locally straighten
chosemanifold
the coordinates
thatthe
Lo form
is the{ycurve
= 0, z = 0); we also locally straighten
theWe
center
(so that such
it takes
= 0))(yand
the strong-stable manifold (so that
the
center
manifold
(so
that
it
takes
the
form
{y
=
O})
and
the strong-stable manifold (so that
it takes the form {z = 0)).
it takes
the
form
{z
=
O}).
Consider two cross-sections SO and Si to the flow X, which are close enough to LO and which
Consider
twoz cross-sections
the flow XI-' which are close enough to Lo and which
have
the form
= --E < 0 and So
z =and
E >S10,torespectively.
have
-e; <at0 all
andsmall
z CL)
e; >all0,orbits
respectively.
At the
p = form
0 (andz =
hence,
of W” return to the node region U+ = {z < 0) in
At
f.L
=
0
(and
hence,
at
all
small
f.L)
all
orbits
of
W" return to2’1the
U+neighborhood
= {z < O} in
a finite time. Therefore, the flow X, defines a diffeomorphism
bynode
whichregion
a small
aof finite
time.
Therefore,
the
flow
XI'defines
a
diffeomorphism
T1
by
which
a
small
neighborhood
the intersection line I- : {y = 0) = W” I-I S1 is mapped into SO. This map has the form
of the intersection line l- : {y = O} = W" n S1 is mapped into So. This map has the form
Yo = P(cpl,Yl;d
9
CPO = q(vl,yl;p)
modl,
(5)
Yo =p(CP1,Y1;/.L) ,
CPo =q(CP1,Y1;/.L) modI,
(5)
where the coordinates on S’s and Si are denoted as (cps,ys) and (cpi,yi), respectively; smooth
and S1 are indenoted
as (CPo,Yo) and (CP1,yt), respectively; smooth
where
functionsthepcoordinates
and q mod 1onareSol-periodical
‘p.
functions p and q mod I are I-periodical in cpo
Simple
Simple Bifurcations
Bifurcations
185
185
The curve
{ YO == p(cpi,
0; 0), cpo
q(cpi,O;O) mod
1) is the
The
curve Z+
l+ == TlZT1I- :: {yO
P(<Pl,OjO),
<Po == q(<Pl,OjO)
modI}
the intersection
intersection of
of WU
and
the function
function q can be written
in the
and SO.
So. Note
Note that
that the
written in
the form:
form:
q(cp,Y;P) = mcp+
qo(cp,
Y; ~1,
(6)
where
qo is
The integer
defines the
class of
in So
Se (the
sign of
where qo
is periodic
periodic in
in cp.
<po The
integer m
m defines
the homotopy
homotopy class
of Z+
l+ in
(the sign
of m
m
defines
orientation
of
I+
with
respect
to
l-).
defines orientation of l+ with respect to l-).
If the
dimension nn of
If
the dimension
of the
the phase
phase space
space is
is greater
greater than
than three,
three, then
then the
the solid
solid torus
torus SO
So is
is at
at least
least
three-dimensional
and
the
integer
m
may
be
of
arbitrary
value
(see
Figure
15
for
the
csse
m
=
three-dimensional and the integer m may be of arbitrary value (see Figure 15 for the case m 2).
2).
At
3, the
the cross-section
is aa two-dimensional
Therefore, in
in this
cross-section SO
So is
two-dimensional annulus.
annulus. Therefore,
this case,
case, there
there may
may
At nn == 3,
be
only m
14) and
(Figures 77 and
and 14)
and m
m == 1.
1.
be only
m == 0 (Figures
°
Figure
15. If the dimension
of the phase space is more than 3, then the crossFigure 15.
If the5’1 dimension
of the
phase space is more
thansituation
3, then isthepossible
crosssections
So and
are, at least,
three-dimensional,
and the
sections
three-dimensional,
and the situation
is possible
when
the So
curveandl- Sl= are,
IV” at
n Slleast,
is mapped
onto the double-round
curve.
when the curve 1- = W" n S1 is mapped onto the double-round curve.
Note that the structure of the set W” is completely determined by the way W” adjoins to LO
Note that the structure of the set W" is completely determined by the way WU adjoins to Lo
from the side of the node region U +. It is not hard to see that the intersection of W” n U+ with
from
the side of the node region U+. It is not hard to see that the intersection of W" n U+ with
any cross-section of the kind {‘p = const} consists, at m # 0, of Irnl pieces glued at the point
any
cross-section
{<p const}
consists,
m I 0,ofofW”Iml corresponding
pieces glued atto the
point
cp - const}.
It is clear
that atsamples
different
{y=O,
z=O}=Lr-l{of the kind
{y = Q,of Zm=are
O} mutually
= L n {<pnonhomeomorphic.
= const}. It is clear
W" iscorresponding
different
values
It isthat
also samples
clear thatof W”
a topological to
manifold
if
It
is
also
clear
that
W"
is
a
topological
manifold
if
values
of
m
are
mutually
nonhomeomorphic.
and only if m = fl (a torus or a Klein bottle, respectively).
and
only
if
m
±1
(a
torus
or
a
Klein
bottle,
respectively).
Consider the function
Consider the function
f(v) = w + q0(cp, 0; 0).
(7)
(7)
Note that f is not defined uniquely because it can be changed by a transformation of coordinate cp
Note
f is not defined
uniquely
because
it cangive
be an
changed
by a transformation
of coordinate
on
thethat
cross-sections
So and
Si. Below
we will
algorithm
how to choose the
coordinates<P
on the
cross-sections
SI' Below
we : will
give
algorithm
choose
on
Se and
Si so that So
theand
through
map TO
Se -+
Si an
defined
by thehow
flowto X,
for pthe
> coordinates
0, acts as a
So
and
SI
so
that
the
through
map
To
:
So
t
SI
defined
by
the
flow
XI-'
for
p,
>
0, acts as a
on
pure rotation in q-coordinate
pure rotation in <p-coordinate
91 = 90 + GL),
(8)
(8)
<PI = <Po + t(p,),
where t(p) is the flight time from Sc to S1 for the orbits of X,, t(p) + fco as ~1+ +O. It can
where
t(p,)[29]
is the
time fromf So
to S1 modulo
for the orbits
of XI-" additive
t(jt) - t +00
as p, and
- t +0.
It can
be
shown
thatflight
the function
defined
an arbitrary
constant
a shift
of
be shown [29] that the function
~J4-2/4-<;
f defined modulo an arbitrary additive constant and a shift of
L.
TURAEV
L. P.
P. SHIL'NIKOV
SHIL'NIKOV AND
AND D.
D. V.
V. TURAEV
186
186
the
the origin
origin
is
of smooth
is independent
independent of
of the
the choice
choice of
of E
e and
and of
smooth
of
the
through
map.
of the through map.
Now we
the
the main
main results
results of
of the
Now
we can
can formulate
formulate the
°
coordinate
(8)
coordinate transformations
transformations preserving
preserving form
form (8)
paper.7
paper. 7
THEOREM
1. Let
m == 0 and
and If’(v)1
all <po
‘p. Then,
Then, for
~1>
> 0,
THEOREM 1.
Let m
1/'(<p)1 << 11 for
for all
for a11
all smitll
small f..l
0, the
the system
system X,
XIJ. has
has
&11
(nonhomotopic to
in U)
U tend.
tend.
an attractive
attractive periodic
periodic orbit
orbit L,
LIJ. (nonhomotopic
to Lo
Lo in
U) to
to which
which all
all orbits
orbits of
ofU
THEOREM
If/((~)1 >> 1
1 for
for all
~1>> 0,
U tend
tend
THEOREM 2.
2. Let
Let Irnl
Iml 2;::- 22 and
and 11'(<p)1
all cp.
<po Then,
Then, for
for all
all small
small f..l
0, all
all orbits
orbits ‘of
ofU
to
equivaJent to
to aa hyperbolic
hyperbolic attractor
attractor R,
n/.L topologically
topologically equivalent
to the
the suspension
suspension over
over the
the inverse
inverse spectrum
spectrum
limit
expanded map
of the
the expanded
map of
of aa circle
circle
limit of
@=mmcpmodl.
r:p = m<p mod 1.
(9)
(9)
As we
above, systems
systems close
periodic orbit
As
we mentioned
mentioned above,
close to
to X0
Xo and
and having
having aa saddle-node
saddle-node periodic
orbit close
close to
to Lo
Lo
form
a
codimension
one
bifurcational
surface
in
the
space
of
dynamical
systems.
It
can
also
be
form a codimension one bifurcational surface in the space of dynamical systems. It can also be
shown that the function f depends continuously on the system on the bifurcational surface. Thus,
shown that the function 1 depends continuously on the system on the bifurcational surface. Thus,
if the conditions of either Theorem 1 or 2 are fulfilled for some system X0, they are also fulfilled
if the conditions of either Theorem 1 or 2 are fulfilled for some system X o, they are also fulfilled
for all close systems on the bifurcational surface. This implies that Theorem 1 or 2 (d.epending
for all close systems on the bifurcational surface. This implies that Theorem 1 or 2 (depending
on the value of m) is valid for any one-parameter family which intersects the surface transversely
on the value of m) is valid for anyone-parameter family which intersects the surface transversely
near X0. In other words, our blue sky catastrophes may occur in general one-parameter families.
near Xo. In other words, our blue sky catastrophes may occur in general one-parameter families.
The proof of Theorems 1 and 2 is baaed on the calculation of the through map TO : (~0, cpo) H
The proof of Theorems 1 and 2 is based on the calculation of the through map To : (Vo, <Po) 1-+
(~1, cpl) from SO into S1 which is defined by the orbits of X, for all small p > 0.
(Yb <pt) from 80 into 81 which is defined by the orbits of X/.L for all small f..l > 0.
In suitable coordinates, the through map can be written in the form (see Lemma 1). Since the
In suitable coordinates, the through map can be written in the form (see Lemma 1). Since the
last two equations in (5) are independent of y, the map TO is written in the form
last two equations in (5) are independent of Y, the map To is written in the form
Yl = Y(cpo,Yo,P)
Yl
Y(<Po,Yo,f..l)
cpl =CPO+T((PO,P)
= <Po + T (<Po,f..l)
modl,
(10)
(10)
mod 1,
where Y is a smooth function l-periodic in cpo. The function T is the flight time from So to S1.
where Y is a smooth function I-periodic in <Po. The function T is the flight time from 8 0 to 8 1 ,
Note that the flight time does not depend here on yo. This is connected with the existence
Note
that the flight
time does
not depend
Yo. This is connected with the existence
of the strong-stable
invariant
foliation
in UO.here
Thisonfoliation
can be locally straightened; i.e.,
of the strong-stable invariant foliation in Uo. This foliation can be locally straightened; i.e.,
the coordinates in UO can be chosen such that the leaves of the foliations would have the form
the
Uo canThe
be invariance
chosen such
the leaves
of thethat
foliations
have on
the aform
{z =coordinates
const, cp = inconst}.
of that
the foliation
means
for any would
two points
leaf
{z
const,
<p
const}.
The
invariance
of
the
foliation
means
that
for
any
two
points
on a leaf
the orbits of these points will intersect the same leaves simultaneously; i.e., at each moment
of
the
orbits
of
these
points
will
intersect
the
same
leaves
simultaneously;
i.e.,
at
each
moment
time the coordinates (z, ‘p) of one point coincide with the coordinates (z, ‘p) of the other points,of
time
the coordinates
<p) of
with the coordinates
(z, <p)
of the other
independently
of the (z,
value
of one
the point
startingcoincide
y-coordinates.
Thus, in such
coordinates,
thepoints,
flight
independently
of
the
value
of
the
starting
y-coordinates.
Thus,
in
such
coordinates,
time from the cross-section {z = -E} to the cross-section {z = E} is independent of y,-,,the flight
time
cross-section
= -e} periodic
to the cross-section
{z =~(90,~)
e} is independent
of SO.
Yo
Thefrom
flightthetime
is smooth {z
function
in cpo. Clearly,
+ 00 as ~1-+
The flight time is smooth function periodic in <Po. Clearly, T(<PO,f..l) ---> 00 as f..l---> +0.
THEOREM
3. The coordinates can be chosen such that the flight time ~(90, CL) is independent
THEOREM
of
cpo; i.e., 3. The coordinates can be chosen such that the flight time T(<PO, f..l) is independent
of <Poi i.e.,
cpOut = cp’” + T(P) mod 1
(11)
<pout <pin + T(f..l) mod 1
(11)
in formula (10). Besides, the following estimate
in formula (10). Besides, the following estimate
IIY IIcv-~ 5 Ke-a7(p)
02)
<P1
(12)
holds, where K and cy are same positive constants.
holds, where K and a are same positive constants.
‘We
need the invariance
of function
f with respect to coordinate
transformations
to be sure that
the values
max 1fneed
‘I andthemininvariance
(f ‘I are well-defined
7We
of function objects.
I with respect to coordinate transforma.tions to be sure that the values
max 1/'1 and min 1f'1 are well-defined objects.
Simple
Bifurcations
Simple Bifurcations
187
187
Inequality
an obvious
obvious consequence
consequence of
of the
fact that
that the
flow near
Inequality (12)
(12) is
is an
the fact
the flow
near the
the saddle-node
saddle-node is
is
exponentially
strictly inside
exponentially contracting
contracting in
in y-variables
y.variables which
which correspond
correspond to
to the
the multipliers
multipliers lying
lying strictly
inside
the
circle. The
cy must
of these
multipliers lies
the unit
unit circle.
The constant
constant a
must be
be chosen
chosen such
such that
that the
the spectrum
spectrum of
these multipliers
lies
inside
the
circle
{](.)I
=
eVQ}
in
the
complex
plane.
inside the circle {lOI =: e-(X} in the complex plane.
On the
the other
other hand,
the first
On
hand, the
first statement
statement of
of the
the theorem
theorem is,
is, technically,
technically, the
the most
most complicated
complicated
part
the work.
we mentioned,
the evolution
(z, ‘p)
independent of
part of
of the
work. As
As we
mentioned, the
evolution of
of the
the coordinates
coordinates (z,
V') is
is independent
of
y-coordinates.
corresponding differential
y-coordinates. The
The corresponding
differential equation
equation is
is
z == g(z, V', jJ)
(13)
(13)
rp == 1,
where
would be
evidently fulfilled
g(O, cp,
V', 0)
0) E
== 0,
0, g(O,
g(O, cp,
V', cl)
jJ) >> 00 for
for pjJ >> 0.
O. Note
Note that
that equality
equality (11)
(11) would
be evidently
fulfilled
where ?j(O,
ifif the
second
equation
in
(13)
were
autonomous;
i.e.,
if
fi
were
independent
of
cp.
the second equation in (13) were autonomous; Le., if g were independent of V'.
We prove
where 9
j is
v, and
p.
We
prove here
here Theorem
Theorem 33 for
for the
the case
case where
is smooth
smooth enough
enough with
with respect
respect to
to x,
x, cp,
and p.
Actually,
case, the
nonautonomous vector
made "almost"
“almost” autonomous
in aa
Actually, in
in this
this case,
the nonautonomous
vector field
field can
can be
be made
autonomous in
small
neighborhood
of
Lc
by
a
smooth
transformation
of
coordinates,
namely,
by
reduction
to
small neighborhood of Lo by a smooth transformation of coordinates, namely, by reduction to aa
normal
normal form.
form.
First, note that if j is sufficiently smooth, then one can make it independent of ‘p at ~1 = 0
First, note that if 9 is sufficiently smooth, then one can make it independent of V' at jJ 0
(see (31) by a smooth transformation of variables. Besides, using the standard normalizing pro(see [3]) by a smooth transformation of variables. Besides, using the standard normalizing procedure, any finite segment of the Taylor expansion of 3 in powers of z and p can also be made
cedure, any finite segment of the Taylor expansion of 9 in powers of z and jJ can also be made
independent of cp. This implies that, for an arbitrarily large k, we can write
independent of cp. This implies that, for an arbitrarily large k, we can write
&T
&
og = 0 (8)
(-k)
-=0 g
8V'
(14
(14)
as p + +0 (this estimate holds in a small neighborhood of Lo and the larger k we choose, the
as jJ -+ +0 (this estimate holds in a smalilleighborhood of Lo and the larger k we choose, the
smaller the size of the neighborhood may be). We can also write
smaller the size of the neighborhood may be). We can also write
$
a3j
= 0 (J”) , g-$ = 0 (6:“), . . ,
(15)
(15)
for an arbitrary fixed number of derivatives.
for an arbitrary fixed number of derivatives.
We consider the case where Lo is a simple saddle-node. Then
We consider the case where Lo is a simple saddle-node. Then
3 = z2z2+ o(z)2,
12
# 0
at p = 0. Without loss of generality, we assume 12 = 1. Suppose the family X, is transverse to
at jJbifurcational
= O. Without
loss ofofgenerality,
we assume
= 1. Suppose
theallows
familyusXI-'to isassume
transverse to
the
surface
systems with
a simple l2saddle-node.
This
the bifurcational surface of systems with a simple saddle-node. This allows us to assume
~! =P o.
Resealing, if necessary, the parameter p we get
Rescaling, if necessary, the parameter jJ we get
Thus, we can write
Thus, we can write
3 = p f z2 + 0 (JpJJz]+ lz13 + p2) .
9 = p + Z2 + 0 (ljJllzl + Izl 3 + Ii) .
Consider an orbit {z(t;cpe,p),cp(t; cpo,~)} of system (13) starting at
{Z(tiV'o,jJ),cp(t;cpo,jJ)}
system
(13) starting
Since g (Z j(z,cp,p)) of
does
not vanish
at p > at
0,
(cpConsider
= cpo, z an
= orbit
4).
(cp = V'o,of zz,==~0,-e:).
(== we
g(z,cp,jJ»
does not vanish at jJ > 0,
function
and Since
CL. By ~(13),
get
function of z, V'o, and /J. By (13), we get
V' = V'o
+ t (z, V'o, /J)
t = 0 with the point
twe==can
0 with
the t point
express
as a
we can express t as a
188
188
L.
SHIL’NIKOV
AND D.
L. P.
P. SHIL'NIKOV
AND
D. V.
V. TUF~AEV
TUMEV
and
and
t (Z, CPo, p,)
=
j
ds
z
_(
(
).
-e g S,CPO + t S,CPO,J.L ,J.L)
(16)
Denote
(16) with
respect to
Denote uu == e.
~. By
By differentiating
differentiating (16)
with respect
to cpc,
CPo, we
we find
find
[1 + u (s, CPo, p,)) ds.
u (z, CPo, p,)
(17)
(17)
By
(14), we
l$$/g2/ +-+ 00 as
we have
have 1~/iPI
as pJ.L +-+ +0,
+0, so
so the
the function
function uu can
can be
be found
found by
by the
the successive
successive
By (14),
approximation
method
as
the
unique
continuous
solution
of
integral
equation
(17).
approximation method as the unique continuous solution of integral equation (17).
Note
function uu at
> 0.
right-hand side
Note that
that we
we defined
defined the
the function
at /.J
p, >
O. Nevertheless,
Nevertheless, the
the right-hand
side of
of th’e
the integral
integral
equation
has
a
limit
as
1-14
+0
(it
vanishes).
Therefore,
the
solution
u
has
also
a
limit,
equation has a limit as J.L -+ +0 (it vanishes). Therefore, the solution u has also a limit, namely,
namely,
u -+
-+ 00 as
+O. Using estimates (14),(15), we can repeat these arguments for a number of
u
as pJ.L -+
-+ +0. Using estimates (14),{15), we can repeat these arguments for a number of
derivatives
cpc and
(this number
the larger
derivatives of
of ‘II
u with
with respect
respect to
to CPo
and pp, (this
number is
is the
the larger,
larger, the
larger the
the value
value of
of k).
k).
Thus, the value of ‘11tends to zero along with the arbitrary given number of derivatives as p + +O.
Thus, the value of u tends to zero along with the arbitrary given number of derivatives as p, -+ +0.
By
By definition,
definition, the
the flight
flight time
time r((pe,p)
r{cpo,p,) equals
equals to
to
90
t(z=E,'Po,II)-t(&,O,II)+
4,cl)
db
+J
J 4~
90
cpo
=cpo
+
4,CL)
d4
r
+J
Jo 4~
t (z = c, CPo, J.L) == t (c, 0, J.L)
{'PO
0
o
u(c, ¢, J.L) d¢.
We see that the coordinate transformation
We see that the coordinate transformation
O
CPo = CPo
0
u{c, ¢, J.L) d¢
brings the second equation of (10) to required form (11). The theorem is proved.
brings the second equation of (10) to required form (11). The theorem is proved.
As it follows from Theorem 3 and from formulas (5)-(7), if p is sufficiently small and positive,
As it follows from Theorem 3 and from formulas (5)-(7), if J.L is sufficiently small and positive,
then on a small neighborhood of l- = W”U n Si in 5’1, there is defined the map T s TOTI :
then on a small neighborhood of l- = W n 8 1 in 8 1 , there is defined the map T == ToTl :
(cpr, yr) H (ai, ~1) by the orbits of the flow X, which can be represented in the following form
(CP1, Yl) 1-+ (<P1I til) by the orbits of the flow X,." which can be represented in the following form
(we omit the indices “1”):
(we omit the indices "I"):
fi = p«({J, Y; p,)
<p :;; r(J.L) + f{({J)
+ ij{cp, Y; p,) mod 1,
(18)
where jj and < @are smooth functions periodic in (p; moreover, there exist C’-l-limits
l:im,,+ep
where
P and <g ij are smooth functions periodic in ({J; moreover, there exist Cr-l-limits lim,.,,->+op
and lim,,+e
and lim,.".... +o ij
ixP,Y;P=o)-o
p«({J,y;p, = 0) == 0
fj(‘P,y=O;p=O)rO.
ij(cp,y OiP, == 0)
=
== O.
(1%
(19)
(20)
(20)
These formulas mean that, as p -+ +O, the map T becomes arbitrarily close (in CT-‘-topology)
formulas mean map
that, as J.L -+ +0, the map T becomes arbitrarily close (in cr-I-topology)
to These
the one-dimensional
to the one-dimensional map
p = T(P) + mcp + m(cp, O,O) = t(p) + f(v) mod 1.
<p r{p,) + m({J + qo( ({J, 0, 0) == t(p,) + f{({J) mod 1.
(21)
(21)
If m = 0 and If’(v)1 < 1, then for any r‘, map (21) has an attractive fixed point to which
If
m = tend.
0 and The
If'{({J)same
I < 1,is then
any for
r, map
(21) maps,
has aninattractive
to which
all orbits
clearlyforvalid
all close
particular, fixed
for point
the map
T at
all
orbits
tend.
The
same
is
clearly
valid
for
all
close
maps,
in
particular,
for
the
map
T at
small p > 0. Since the map T is defined by the orbits of the flow X,, the fixed point corresponds
small
> O. Sinceperiodic
the maporbit
T is L,defined
the orbits
of Theorem
the flow X,.",
fixed be
point
corresponds
to
the p,attractive
of XP;bythis
gives us
1. Itthe
should
noted
that the
to
the attractive
periodic
of X,.,,;
this gives
1. Ittheshould
noted
thatdoes
the
period
of L, is about
T(P)orbit
and L,."
tends
to infinity
as /Lus-t Theorem
+cxx Since
vectorbefield
of X,
period
of L,."
r{p,) that
and the
tends
to infinity
as p, -+
not
vanish
in isU,about
it follows
length
of L, tends
to +00.
infinitySince
also.the vector field of X,." does
not vanish in U, it follows that the length of L,." tends to infinity also.
Simple
Simple Bifurcations
Bifurcations
189
189
To
To prove
prove Theorem
Theorem 2, we
we note
note that
that ifif ]f’((~)]
If'(<p)1 >> 1 for
for all
all cp,
<p, then,
then, by
by virtue
virtue of
of (19),(20),
(19),(20), map
map (18)
(18)
is
<p and
and strongly
strongly contracting
contracting in
in yy for
for all
all small
small yy and
and pJL 2~ 0.
O. This
This allows
allows us
us with
with
is expanding
expanding in
in cp
the
the standard
standard technique
technique [25,30]
[25,30] to
to establish
establish the
the existence
existence of
of a continuous
continuous contracting
contracting invariant
invariant
foliation
with
the
leaves
of
the
form
foliation with the leaves of the form
{cp
{<p = Q*
cI>* (Y;
(Yi 4,
<p', PII,
JL)} ,
(22)
where
where cp’
<p' is
is the
the coordinate
coordinate of
of the
the intersection
intersection of
of the
the leaf
leaf with
with the
the line
line {y
{y = 0);
O}; the
the function
function a*
cI>*
is
Lipschitz
with
respect
to
y
and
continuous
on
(cp’,
cl)
at
all
small
p
2
0.
is Lipschitz with respect to y and continuous on (<p', JL) at all small JL ~ O.
Indeed,
. . be
Indeed, take
take any
any point
point PP on
on Sc.
So. Let
Let PO
Po == P,
P, PI,
P1 , Pz
P2 .••.
be the
the iterations
iterations of
of PP by
by the
the map
map T:
T:
Note
that
for
p
and
y
small
enough,
there
exists
X
>
0
such
that
for
a
smooth
TPi
=
Pi+l.
TPi = PH1 . Note that for JL and y small enough, there exists A > 0 such that for a smooth
surface
C :: {‘p
{<p == G(y)}
cI>(y)} satisfying
satisfying the
the Lipschitz
Lipschitz condition
condition
surface .C
E II<A
/III~:II
$ A
dY
(23)
(23)
-
and
T-l(L)
that
and containing
containing the
the point
point Pi,
Pi, the
the connected
connected component
component k£ of
of its
its preimage
preimage T-1(C)
that contains
contains
Pi-1
is
the
surface
of
the
similar
form
{cp
=
6(y)},
w
h
ere
6
is
defined
for
all
small
Pi - 1 is the surface of the similar form {<p = ~(y)}, where ~ is defined for all small yy and
and itit
satisfies
Lipschitz
condition
(23)
with
the
same
X.8
S
satisfies Lipschitz condition (23) with the same A.
Furthermore, it is also easily checked that for any two curves Li and Cs satisfying (23) the
Furthermore, it is also easily checked that for any two curves C 1 and C2 satisfying (23) the
CO-distance between ,?r and 2s is less than the CO-distance between Li and &.
CO-distance between £1 and £2 is less than the CO-distance between C 1 and C2 •
Thus, if Xi is the space of the smooth surfaces .C containing Pi and having the form cp = a(y),
Thus, if 'J-li is the space of the smooth surfaces C containing Pi and having the form <p cI>(y) ,
where Q satisfies (23), then the map Fi : JZ++ 2 is contracting in CO-metric and ‘I;i(7&) c F&-i.
where cI> satisfies (23), then the map Ti : C ~ £ is contracting in CO-metric and Ti('J-l i ) ~ 'J-li-l'
Applying a lemma from [31] ( “on the fixed pont of a contracting operator in an infinite product
Applying a lemma from [31] ("on the fixed pont of a contracting operator in an infinite product
of complete metric spaces”) to the spectrum of the spaces and the maps
of complete metric spaces") to the spectrum of the spaces and the maps
where the bars means Co-closure, we find that there exists a unique sequence of surfaces fZz :
where the bars means CO-closure, we find that there exists a unique sequence of surfaces Ci
{up = @f(y)} such that Pi E L;, TLf C Lz+,, and Cf E 7?i (i.e., the value X is the Lipschits;
{<p
cI>i(y)} such that Pi E Ct, TCi C C:+1' and Ci E Hi (Le., the value A is the Lipschitz;
constant for !I$).
constant for cI>T).
In other words, for any point P, there exists a unique Lipschitz surface L*(P) E LY, containIn other words, for any point P, there exists a unique Lipschitz surface C(P) ;;
contain·,
ing P and such that all forward iterations of C*(P) by the map T remain Lipschitz with the
ing P and such that all forward iterations of C*(P) by the map T remain Lipschitz with the
Lipschitz constant X. By uniqueness, the surface L*(P) depends continuously on the point F’
Lipschitz constant A. By uniqueness, the surface C*(P) depends continuously on the point P
and on the parameter p, and if two such surfaces have an intersection point, they coincide (see
and on the parameter JL, and if two such surfaces have an intersection pOint, they coincide (see
details in [25] where analogous considerations were carried out for the Lorenz-type maps).
details in [25] where analogous considerations were carried out for the Lorenz-type maps).
Thus, the surfaces L* compose a continuous invariant foliation of required form (22). The
Thus,
the surfaces
continuous
foliation of
required map
formof(22).
The
map
T factorized
with C*
thecompose
leaves ofathe
foliation invariant
is an exponentially
expanding
the circle
map
factorized
with
the leavestheof factorized
the foliation
exponentially
expanding
map of the
for allT small
,U > O.g
Therefore,
mapis an
is conjugate
[32] to
linear expanding
mapcirciEl
(9)
9
for
all
small
JL
>
0.
Therefore,
the
factorized
map
is
conjugate
[32]
to
linear
expanding
map
(9)
for all small p > 0, what gives Theorem 2 (analogously to [20,33]).
for all small JL > 0, what gives Theorem 2 (analogously to [20,33]).
.co
sThis
assertion
is easily
verified
and we omit
the calculations
because
this
is rather
standard
and repeats
analogous
SThis
assertion inis easily
verified
we omitbethe
calculations
because
is rather
and(=repeats
analogous
considerations
[25]. The
value and
X should
taken
greater
than
min this
].f’((p)]/
mm standard
l4L(rp, 0; O)l
W-(20)).
considerations
in [25J.
should
be taken
greater
11'(11')11
Iq~('P, one
0; 0)1can(see
(18)-(20».
QTo prove, consider
a liftThe
F value.>.
of the map
T onto
the strip
-oc than
< cpmin
< +co.
By max
(18)-(20),
easily
check that
9To
prove,
lift P'
T ofwiththeequal
map T
onto the strip
< V' < +00.
(18)-(20), oneofcan
check that
for any
two consider
points P aand
y-coordinates,
the -00
difference
between By y-coordinates
the easily
ith iterations
Pi
for any
points P andPpiandwith
difference
of thecpcoordinates
ith iterations for
Pi
and
Pi two
of, respectively,
P'equal
by they-coordinates,
lift F is muchtheless
than thebetween
differencey-coordinates
between
their
all positive
i. Then,
it isP not
to the
see that
distance lessbetween
and Pi grows
exponentially
as i - +oc;for
and
PI of, respectively,
andhard
pi by
lift Ttheis much
than thePldifference
between
their <p-Coordinates
i.e., positive
faster
than
K”dist(P,P’)
where to K
1 canthebedistance
chosen between
arbitrarily PI close
to grows
min If’(v)]
if p and as
a iare- small
all
i. Then,
it is not hard
see> that
and Pi
exponentially
+oc;
I
enough.
leaves of the
foliation K >
are1 uniformly
Lipschitz,arbitrarily
the distance
the iterations
any small
two
i.e., fasterSince
thantheK'dist(P,P
can be chosen
close tobetween
minlf'(V')1
if iJ. and'llof are
) where
leaves by Since
the lifttheT leaves
grows of
asymptotically
distance Lipschitz,
between
thetheiterations
any twothe
points
lying of
one any
on tw.)
one
the foliation ssaretheuniformly
distance ofbetween
iterations
enough.
of the by
leaves
and Tthegrows
other asymptotically
on the other. asThus,
the distance
between
the iterations
any points
two leaves
also on
grows
leaves
the lift
the distance
between
the iterations
of anyoftwo
lying one
one
exponentially.
the leaves and the other on the other. Thus, the distance between the iterations of any two leaves also grows
of
exponentially.
190
190
L. P.
P. SHIL'NIKOV
SHIL’NIKOV
AND D.
D. V.
V. TURAEV
TURAEV
L.
AND
CONCLUDING
REMARKS:
CONCLUDING
REMARKS:
THE DISAPPEARANCE
DISAPPEARANCE
OF A
A SADDLE-NODE
SADDLE-NODE
TORUS
THE
OF
TORUS
The idea
idea of
of the
the use
use of
of the
the saddle-node
saddle-node bifurcation
bifurcation to
to produce
produce hyperbolic
hyperbolic attractors
attractors gives
gives many
many
The
more new
new examples
examples for
for the
the case
case of
of bifurcations
bifurcations of
of invariant
invariant tori.
tori. We
We do
do not
not develop
develop here
here the
more
the
theory
of
the
global
bifurcations
connected
with
the
disappearance
of
a
torus
and
restrict
ourself
theory of the global bifurcations connected with the disappearance of a torus and restrict ourself
by
the study
study of
of aa model
situation.
by the
model situation.
Consider
a
one-parameter
family of
of smooth
smooth dynamical
dynamical systems
systems
Consider a one-parameter family
x=
which has,
has, at
at p
p
which
assume
that
the
assume that the
X(x, p),
= 0,
0, an
invariant m-dimensional
m-dimensional torus
torus 'T7”’m filled
filled by
quasi-periodic orbits.
orbits. We
We
=
an invariant
by quasi-periodic
vector
field
takes
the
form
vector field takes the form
yL == 414~
A(p)y
i=p+z2
i = p + z2
9<p =
= f-G)
rl(p)
in
neighborhood of
rm. Here,
y E Rn+“-l,
‘p E Y. The matrix A(p) is stable; i.e.,
in aa neighborhood
of'Tm.
Here, .z
zE
E R1,
Rl, Y E Rn-m-l, 'P E 'Tm. The matrix A(p) is stable; Le.,
its eigenvalues lie strictly to the left of the imaginary axis.
its eigenvalues lie strictly to the left of the imaginary axis.
At ~1 = 0, the torus is given by the equation y = 0. When ~1 > 0, the torus disappears.
At p = 0, the torus is given by the equation y = O. When p > 0, the torus disappears.
Analogously to the case of disappearance of a saddle-node periodic orbit which we considered in
Analogously to the case of disappearance of a saddle-node periodic orbit which we considered in
the previous section, we construct two cross-sections SO and SI: {y = fs} where E is a small
the previous section, we construct two cross-sections 8 0 and 8 1 : {y = ±€} where € is a small
quantity. The map TO : SO -+ Sr will be defined for p > 0 by the orbits of the system. For our
quantity. The map To : 8 0 ....... 8 1 will be defined for p > 0 by the orbits of the system. For our
model case, it can be easily calculated:
model case, it can be easily calculated:
Yl
= eA(/.L)t(J.L)yO
'Pl = 'Po
+ rl(p)t(p),
where t(p) N l/,,$ + . . . . Suppose that, at h = 0, all orbits of the unstable set W,U,, : {y = 0,
where t(p) "'" 1/Vii + .. '. Suppose that, at p 0, all orbits of the unstable set Wl~c : {y 0,
z > 0) of the torus 9 m return in a small neighborhood of ? m and tend to it as t + +oo. All
z > O} of the torus 'T return in a small neighborhood of 'T and tend to it as t ....... +00. All
these orbits will intersect SO, defining thereby the map Tl : Sr 4 Se. We write this map in the
these orbits will intersect 8 0 , defining thereby the map Tl : 8 1 ....... 8 0 , We write this map in the
form
form
Yo =P(Yl,cpl,PL)
Yo P (Yl, 'PI, p)
PO = q(Y17(PlrP)>
'Po = q (Yb 'PI, p) ,
where p and q are smooth for all small CL. The image of the torus W“ n Se by the map TI is also
where p and q are smooth for all small p. The image of the torus WU n 8 0 by the map Tl is also
a torus given by the equation
a torus given by the equation
Yo =P(o,cpl,o)
Yo = P (0, 'PI. 0)
cpo = Q (0, cpl,
'Po
0) .
q (0, 'Pl> 0) .
The Poincare map T = TOTI : 5’1 -+ 5’1 is written as
The Poincare map T = ToTI : 8 1 ....... 8 1 is written as
fj = eAcpJtcp)p(y, cp,p)
fi = eA(J.L)t(J.L)p(y, 'P, p)
P = fYP)tb> + dG 93 PI*
cp = rl(p)t(p) + q(z, 'P, p).
Since t(p) -+ +oo as p + +0 and since the spectrum of A(p) lies strictly
Since t(p)
....... +00
as p .......
and since
the spectrum
strictly
imaginary
axis,
the norm
of +0
eA(p)t(fi)
is extremely
small offorA(p)
smalllies1-1
> 0.
imaginary
axis,
the
norm
of
eA(J.L)t(J.L)
is
extremely
small
for
small
p
> O.
map (24) is very close to the “shortened” map of a torus
(24)
(24)
to the left of the
to themeans
left ofthat
the
This
This means that
map (24) is very close to the "shortened" map of a torus
8 = 4~) + Bv + qo(vL
cp = w(p) + B'P + qo('P),
(25)
(25)
Simple
Simple Bifurcations
Bifurcations
191
191
where
where
W(P) = a)+)
09 CL)
W(p)
O(p)t(p) ++ do7
q(O, 0,
p)
Bv
+ qo(cp)
= cd%
cp, P) -- q(O,O,p).
4% 0, ~1.
Bcp +
qo(cp) =
q(O,cp,p)
Here, B
integer matrix
is aa periodic
the mean
mean value
Here,
B is
is an
an integer
matrix and
and qo
qo is
periodic function
function of
of cp
cp with
with the
value equal
equal t,cI
to
zero.
Denote
zero. Denote
f(v)
+ qo(cp>.
/(cp) == BP
Bcp +
qo(cp).
°
IfIf IIf’
for all
all cp
and if
if qo
qo is
the shortened
shortened map
is
III'(cp) II << 11 for
cp (for
(for instance,
instance, ifif BB == 0 and
is small),
small), then
then the
map is
contracting
for
all
p
>
0,
and
the
Poincarb
map
T
is
contracting
also.
Since
a
contracting
map
contracting for all p > 0, and the Poincare map T is contracting also. Since a contracting map
has
fixed point,
point, we
at the
statement.
has only
only one
one fixed
we arrive
arrive at
the following
following statement.
1. If IIf’
< 1 for all cp, then the Aow X, has a unique attractive periodic orbit;
PROPOSITION 1. IfII/'(cp)1I < 1 for all cp, tben tbe flow XJ1. bas a unique attractive periodic orbi/;
for
aJJ
small
p
>
0.
for all small p > 0.
This result
to Theorem
Theorem 11 and
it gives
example of
catastrophe,,
This
result is
is analogous
analogous to
and it
gives us
us aa new
new example
of the
the blue
blue sky
sky catastrophe"
An analogue of the theorems of [2,4,17] on the birth of a smooth invariant manifold at a saddleAn analogue of the theorems of [2,4,17] on the birth of a smooth invariant manifold at a saddle-,
node bifurcation is given by the following statement.
PROPOSITION
node bifurcation is given by the following statement.
°
2. If IdetBl = 1 and dety(cp) # 0 f or all cp (i.e., if the shortened map is a
PROPOSITION 2. If IdetBI = 1 and detl'(cp) -:j:.
for all cp (i.e., if tbe sbortened map is ,a
diffeomorphism),
invariant attractive
m-dimensional torus
aJJ
diffeomorphism), then
tben the
tbe Poincare’
Poincare map
map has
bas an
an invariant
attractive m-dimensional
torus for
for alJ
p > 0 (correspondingly, the ffow has an invariant attractive manifold homeomorphic to a skew
p > (correspondingly, tbe flow bas an invariant attractive manifold bomeomorpbic to a skew
product of the torus on a circle).
PROPOSITION
°
product of tbe torus on a circle).
The proof of this statement is based on the fact that if for a sequence p,, -+ +O, the sequence
The proof of this statement is based on the fact that if for a sequence Pn -+ +0, the sequencEi
w(pn) mod 1 tends to some point w*, then the Poincark map that corresponds to p = p,, has, EC;
w(Pn) mod 1 tends to some point W*, then the Poincare map that corresponds to p = Pn has, as
a limit, the map
a limit, the map
g=o
p = w * +t(cp).
This map has a stable invariant smooth attractive torus {y = 0). The standard fact of the
This map has a stable invariant smooth attractive torus {y = O}. The standard fact of the
theory of normal hyperbolicity is that the invariant manifold is preserved for all close maps i:f
theory of normal hyperbolicity is that the invariant manifold is preserved for all close maps if
the restriction of the map on the invariant manifold is a diffeomorphism and if the contraction
the
restriction of the map on the invariant manifold is a diffeomorphism and if the contraction
in normal directions (y-directions in our case) is stronger than that which may take place along
in normal directions (y-directions in our case) is stronger than that which may take place along
the directions tangential to the manifold. The latter requirement is trivially fulfilled in our case.
theNote
directions
tangential
thethe
manifold.
requirement
trivially
fulfilled
our case.
that the
restrictionto of
PoincareThe
maplatter
on the
invariant is
torus
is close
to theinshortened
Note that the restriction of the Poincare map on the invariant torus is close to the shortened
map
p = w * +j(cp).
<j5=W*+/(CP)·
map
This implies, in particular, that if the shortened map is Anosov for all w* (for instance, if the
This Bimplies,
in particular,
that ifonthetheshortened
map
Anosov
for all
(forrestriction
instance, ofif the
the
matrix
does not
have eigenvalues
unit circle
andis qo
is small),
thenw*the
matrix
does isnot
have
eigenvalues
unit circle and qo is small), then the restriction of thl~
PoincarhB map
also
Anosov
for all on
(p the
> O).”
Poincare
map
is
also
Anosov
for
all
(p
>
0).10
We arrive, hence, at the following result.
We arrive, hence, at the following result.
3. If the shortened
tbethesbortened
flow on
aAnosov
byperbolic
attractor
diffeomorphism. tbe flow on
PROPOSITION
3. If
aPROPOSITION
hyperbolic attractor
map is Anosov for all w*, then for all ~1 > 0 there exists
map isisAnosov
for all conjugate
W*, tben toforthe
all suspension
p >
tbere
which
topoJogicaJJy
overexists
the
°
wbich is topologically conjugate to tbe suspension over tbe
Anosov
diffeomorpbism.
The birth of hyperbolic attractors can be proved not only in the case where the shortened map
birth of hyperbolic
attractors
can holds
be proved
in the case
the shortened
is aThe
diffeomorphism.
Namely,
this result
true not
if theonly
shortened
mapwhere
is expanding
or if it map
is a
is a diffeomorphism. Namely, this result holds true if the shortened map is expanding or if it is a
‘ORecall
the definition:
a diffeomorphism
of a torus is called Anosov
if the tangent
space is decomposed
into a
direct
sumtheof definition:
two subspacesa diffeomorphism
(stable
and unstable)
suchis called
that thisAnosov
decomposition
is continuous
and invariant into
with a
l0Recall
of a torus
if the tangent
space is decomposed
respect sum
to the
differential
of the
map, and
andunstable)
the differential
of this
the decomposition
map contracts
(exponentially)
vectors
of stable
is
continuous and
invariant
with.
direct
of two
subspaces
(stable
such that
subspaces
vectors
unstableand subspaces.
Among of the
other map
significant
properties,
Anosov vectors
map areof known
respect to and
the expands
differential
of theof map,
the differential
contracts
(exponentially)
stable
to
be structurally
stable. vectors of unstable subspaces. Among other significant properties, Anosov map are known
subspaces
and expands
to be structurally stable.
192
192
L. P.
D. V.
L.
P. SHIL’NIKOV
SHIL'NIKOV AND
AND D.
V. TURAEV
TURAEV
so-called Anosov
map is
expanding if
if the
so-called
Anosov covering.
covering. A
A map
is called
called expanding
the length
length of
of any
any tangent
tangent vector
vector grows
grows
exponentially
of the
the differential
differential of
exponentially under
under the
the action
action of
of the
the map.
map. An
An example
example is
is the
the algebraic
algebraic map
map
rp
where
where the
the
close
close to
to itit
B~mod
1,
spectrum
of the
integer matrix
B lies
lies strictly
spectrum of
the integer
matrix B
strictly outside
outside the
the unit
unit circle,
circle, and
and any
any map
map
is
also
expanding.
If
]](F’((p))-l]]
<
1,
then
the
shortened
map
is also expanding. If II(F'(~))-Ill < 1, then the shortened map
P = W(P)+ f(p) = 4~) + Bv + CJO((P)
is
expanding for
is expanding
for all
all pJ.L >> 0.
O.
Shub
established
stable [32].
[32]. The
The study
study of
Shub established that
that expanding
expanding maps
maps are
are structurally
structurally stable
of expanding
expanding
maps
and
their
connection
with
smooth
diffeomorphisms
was
continued
by
Williams
maps and their connection with smooth diffeomorphisms was continued by Williams [33].
[33]. Using
Using
this
work,
we
obtain
the
following
result
which
is
analogous
to
our
Theorem
2
proved
for
the
case
this work, we obtain the following result which is analogous to our Theorem 2 proved for the case
of
disappearance
of
a
saddle-node
periodic
orbit.
of disappearance of a saddle-node periodic orbit.
4. Jf ]](f’(cp))-l]]
< 1, then for all ~1 > 0 a hyperbolic attractor exists locally
PROPOSITION 4. If 11(f'(~))-111 < 1, then for all J.L > 0 a hyperbolic attractor exists locally
homeomorphic
to
the
direct
product
R*+’ on
homeomorphic to the direct product of
of Rm+l
on aa Cantor
Cantor set.
set.
An endomorphism
is called
un Anosov
covering ifif the
invariant decompoAn
endomorphism of
of aa torus
torus is
called an
Anosov covering
the continuous
continuous invariant
decompo-sition of the tangent space into the direct sum of stable and unstable subspaces exists, as for an
sition of the tangent space into the direct sum of stable and unstable subspaces exists, as for an
Anosov map (the difference is that the Anosov covering is not a on&o-one map, and therefore,
Anosov map (the difference is that the Anosov covering is not a one-to--one map, and therefore,
it is not a diffeomorphism).
The map
it is not a diffeomorphism). The map
PROPOSITION
(P= w*+ Bv+qo(cp)
will be Anosov covering for all w* if, for instance, ]detB] > 1 and if qo is small enough. The
will be Anosov covering for all w· if, for instance, IdetBI > 1 and if qo is small enough. The
following result is analogous to Proposition 4.
following result is analogous to Proposition 4.
PROPOSITION
5. If the shortened map is an Anosov covering for all w*, then the system has,
PROPOSITION 5. If the shortened map is an Anosov covering for all w·, then the system has,
for all ,u > 0, a hyperbolic attractor locally homeomorphic to the direct product of R’“‘+l on a
for all J.L > 0, a hyperbolic attractor locally homeomorphic to the direct product of Rm+! on a
Cantor set.
Cantor set.
To complete our list of examples of bifurcations connected with the disappearance of a torus,
To complete our list of examples of bifurcations connected with the disappearance of a torus,
consider the following case. Suppose that the variables cp can be separated into two groups:
consider
case.
Suppose that and
the 92
variables
be separatedSuppose
into twothat
groups:
cp
= (cpi,the
cpz),following
where cpi
is k-dimensional
is (m ~
- can
Ic)-dimensional.
the
~
(~1. ~2)' where ~1 is k-dimensional and ~2 is (m - k)-dimensional. Suppose that the
shortened map takes the form
shortened map takes the form
R = w1+ Bw + 41(~1,cp2)
= wI + B~I + ql (~1l ~2)
f472 =w2+42(‘p1,(p2),
2
CP2 = w + q2 (~I' ~2)'
CPl
where q1 and qz are l-periodic in cp. Let q1 and q2 be sufficiently small along with their first
where ql andAnalogously
q2 are I-periodic
in!.p. Let
qi and
sufficiently small along with their first
derivatives.
to Proposition
2, we
have q2thebefollowing.
derivatives. Analogously to Proposition 2, we have the following.
PROPOSITION
6. If ]detB] = 1, then the Poincard map has an invariant attractive Jc-dimensional
PROPOSITION 6. If IdetBI = 1, then the Poincare map has an invariant attractive k-dimensional
torus.
torus.
This gives us an example of a blue sky catastrophe not for a periodic orbit but for a torus of
This
gives (in
us an
exampledimension.
of a blue sky
not matrix,
for a periodic
orbit
torus of
an arbitrary
principle)
If B catastrophe
is a hyperbolic
then the
flowbut
on for
the a.newborn
an
arbitrary
(in
principle)
dimension.
If
B
is
a
hyperbolic
matrix,
then
the
flow
on
the
newborn
torus is Anosov, and we have, thus, a hyperbolic attractor for all p > 0. If B is a hyperbolic
torus
Anosov,
wethen
have,analogously
thus, a hyperbolic
attractor
for all
J.L > O.
is a hyperbolic
matrix isbut
]detB] and
> 1,
to Propositions
4 and
5 there
canIfbeBestabliished
that
but IdetBI
> 1, homeomorphic
then analogously
to Propositions
5 there
be established
amatrix
hyperbolic
attractor
to the
direct product4 and
of Rk+’
to can
a Cantor
set existsthat
for
a hyperbolic
attractor homeomorphic to the direct product of Rk+1 to a Cantor set exists for
all
p > 0.
all J.L > O.
Simple
Simple Bifurcations
Bifurcations
193
193
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