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Journal ArticleDOI

The periodic orbits of an area preserving twist map

S. B. Angenent1
01 Sep 1988-Communications in Mathematical Physics (Springer-Verlag)-Vol. 115, Iss: 3, pp 353-374
TL;DR: In this paper, the authors studied the oscillation properties of periodic orbits of an area-preserving twist map, inspired by the similarity between the gradient flow of the associated action-function, and a scalar parabolic PDE in one space dimension.
Abstract: We study the oscillation properties of periodic orbits of an area preserving twist map. The results are inspired by the similarity between the gradient flow of the associated action-function, and a scalar parabolic PDE in one space dimension. The Conley-Zehnder Morse theory is used to construct orbits with prescribed oscillatory behavior.

Summary (1 min read)

C F(C)

  • There are several structures on X which are relevant for their problem.
  • Hence all the off diagonal elements of the Jacobian of grad W are nonnegative, which implies (a).
  • For parabolic equations this has been generalized by Matano [9] .
  • The following result was proved by Smillie [13] : Proposition 2.3.

Proposition 3.2. The spectrum of L is given by

  • This set (with the appropriate topology) is naturally identified as a circle ("polar coordinates"), and therefore the authors may regard M(λ) as an orientation preserving diffeomorphism of the circle.
  • The orientation explains the occurrence of the +1 in the Jordan normal forms for some μ >.
  • In this way one verifies the classification of the M(A)'s given above.

Definition. τ(s) = -ρ(0).

  • As maps of the circle the integer part of their rotation numbers are not defined, and it is this integer part which the authors shall need.
  • A straightforward calculation shows that this flow is gradientlike with respect to.

V(ξ) = (ξ,Lξ).

  • For the particular Jacobi matrix one gets by putting ^ΞI and β { = 0, one can explicitly determine the I k and their homotopy indices.
  • The reader who wants to verify this should bear in mind that, since the flow is gradient like, all isolated invariant sets consist of fixed points and orbits connecting them.
  • The authors claim that the I k with \<2k<q contain at least two such pairs.
  • After some thought one concludes from this that: and if q is even, which proves the lemma.

The final step in computing h

  • Apply the construction (and notation) of Sect, three to this situation.
  • Using the fact that the periods of the solutions of X"(t) + ω 2 /πsmπX(ή = O increase with amplitude one sees that the matrix M(0) is conjugate to Ίwith α>0.

Corollary. h(M k ) = h k is the homotopy type of S 1

  • By the unstable manifold theorem the flow near M k is the product of that near a hyperbolic fixed point with index 2/c -1, and the trivial flow on a circle.
  • This corollary finishes the proof of Theorem 5.

Lemma 10.3. There is an orbit y in Y such that a<y<b and y -x has one or two sign changes

  • The authors can exclude the first alternative for the following reasons.
  • Thus the authors are left with the second alternative, which proves the lemma.

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Communications
in
Commun.
Math.
Phys.
115,
353-374
(1988)
Mathematical
Physics
©
Springer-Verlag
1988
The
Periodic
Orbits
of an
Area
Preserving
Twist
Map
S.
B.
Angenent*
Department
of
Mathematics,
University
of
Leiden,
Niels
Bohrweg
1,
Leiden,
The
Netherlands
Abstract.
We
study
the
oscillation properties
of
periodic orbits
of an
area
preserving twist map. The results are inspired
by
the similarity between
the
gradient
flow
of
the associated action-function, and
a
scalar parabolic PDE
in
one
space dimension. The Conley-Zehnder Morse theory
is
used
to
construct
orbits with prescribed oscillatory behavior.
1.
Introduction
We shall consider
a C
1
area preserving diffeomorphism
F of
the cylinder
S
1
xR
onto
itself.
Such
a
diffeomorphism can
be
described
by a
mapping
F:
R
2
-+R
2
(its
lift) given
by
F(x, y) = (f(x,
y\
g(x,y)\ where
x is the
angle coordinate.
The
components
of F
satisfy
the periodicity conditions
The
map
F is
said to be
a
twist
diffeomorphism
iϊf(x, y)
is
an increasing function
of y, and
in
fact
d
2
f{x,y)>0
(1.1)
holds
for all
(x, y)
in R
2
.
Here
d
k
denotes differentiation with respect
to
the fc-th
argument.
We shall consider twist diffeomorphisms which
satisfy
the
infinite
twist
condition,
i.e.
lim
f(x,
y) =
±
oo
y-*
± oo
for any xeR.
We
shall study the
set of
periodic orbits
of F.
The
main feature which distinguishes twist maps from other area preserving
maps
is
that they have
a
single valued generating function, i.e. there
is a
C
2
function
*
Current
address:
Department
of
Mathematics,
University
of
Wisconsin,
Van
Vleck
Hall,
480
Lincoln
Drive,
Madison,
WI
53706,
USA

354 S.B. Angenent
h(x,x')
on R
2
such that
(x'
5
y)
= F(x,y)
iff y
= d
ί
h(x, x')
and y
f
=
d
2
h(x,x').
This function also
satisfies
d
l
d
2
h(x,x')>0
for all x
and
x'. (1.2)
The
generating function
ft is
uniquely determined (up to
a
constant)
by
the map
F.
Its
construction
is
given
by
Mather [10].
See
also
Aubry
and
le
Daeron
[2].
The
existence
of
ft
implies that any orbit
of
the map
F is
completely determined
by
its
sequence
of x
coordinates {x
n
:neZ}. One
easily
verifies
that
a
sequence
{x
n
:neZ] can
only
be the
sequence
of x
coordinates
of
an orbit
of F if
d
i
W(...,x-
ί
,x
θ9
x
ί9
...)
= O
9
where
W(x)=
+
Σ
AfoXi+i)
ί
=
00
(this
sum
is not
well
defined, however
its
partial derivatives are).
Orbits
of F
are therefore critical points
of
the "function"
W.
This idea becomes
useful
if
we restrict
ourselves
to
periodic orbits
of
F.
Let X
Ptq
be
defined
by
Xp,q =
{(xdi
e
z-Xi
+
q
=
Xi
+
P
to
a11
iεZ},
where
p
and
q are
integers and
q>0.
For
any x in X
pq
we
define
W(x)=
Σ
h{x
i9
x
i
+
ί
).
i=l
Now
W
is a
well
defined
C
2
function
on X
pq
and
its
critical points
are
periodic
orbits
of F
with period q, which "go around the cylinder"
p
times after
q
iterations.
The
usual
way of
constructing such orbits
is to
prove that
W
takes
its
maximum
at
some point
in X
p
,
q
.
This maximum must
of
course
be a
critical point
of
W.
Most
of
the critical points
of
W are not maxima however. It
is
our intention to
study these other critical points.
Our
approach
is
based on the similarity between the present problem and that
of finding solutions
of
(V'(ί) +
f(t
9
x(ή) = 0,
-oD<t<+
oo,
U
\x{t + p) = x(t) +
q,
where f(t,x)
is a C
1
function
satisfying
/(ί+l,x)
= /(ί,x +
l)
= /(ί,x).
Indeed,
this problem
is
variational, with potential function
where
g(t,x)=ίf(t,ξ)dξ.
0

Periodic
Orbits 355
In
this case the gradient
flow
of W (with respect to the L
2
(0, q) innerproduct) is a
semilinear parabolic partial differential equation:
U
s
x
tt
+
f(t,x)
9
s>0,
-oo<ί<+oo,
\
The
flow
induced by (II) on some function space has a number of geometric
properties, the most prominent of which is the maximum principle. The gradient
flow
of W on X
pq
turns out to have similar properties, and in fact our main results
will
concern systems which contain (I) as a special case.
To
illustrate the analogy we have in mind we shall compute W, d
k
W, and F in
the
following
specific example. Let g(x) be a C
2
function such that g(x) = g(x+1)
holds for all real x, and define f(x) = g'{x)
9
and
Then
h(x
9
x') is the generating function of a
twist
map F :(x,y)-+(x\y
f
)
given
by
y = d
ί
h(x
9
x') = x'
x
9
i.e. x'
y=-d
2
h(x,x')
= x'-x-f{x'), i.e.
so F(x
9
y) = (x + y, y
f{x + y)) If one takes g(x) = (k/2π)cos(2πx) (fc>0), one
obtains a map known as the "standard map."
We see that the orbit {(x
k9
y
k
)}
keZ
is indeed completely determined by its
sequence of x coordinates, since we have y
k
= x
k + x
x
k
. Furthermore a sequence
{
χ
k}kez
is the sequence of x coordinates of an orbit of the map F if and only if it
satisfies
the second order difference equation
d
k
W=x
k
+
1
-2x
k
+ x
k
_
l
+f{x
k
) = 0
9
keZ,
where
Note
that this is the numerical analyst's (simplest) version of the differential
equation
occurring in I! The gradient
flow
of W is
given
by
X
k
= X
k+l~
2<X
k
+
χ
k
- 1
~t~
J (
X
k)
•>
which is the discretized version of II.
Seen in this way, it should not come as a surprise that the gradient
flow
of W
has much in common with the parabolic
PDEII.
2.
The
Gradient
Flow
of W
Let p and q be
given
integers, with
q>0
9
and let F
l9
...
9
F
q
be C
1
area preserving
twist
maps of R
2
with generating functions h
ί9
...,h
q
respectively. We define hj for
j<\ and j>q by requiring hj = h
j+q
for all;. We shall assume that the maps Fj
satisfy
the
following
condition:
The
no flux
condition.
Let C be a curve which winds around the cylinder once. Then
the
total area between C and F(C) vanishes, i.e.
\ydx=
J ydx.
C F(C)

356 S.B. Angenent
This condition
is
equivalent
to
the following condition on the generating
functions
hpc.x'):
(H)
h(x
+1,
x' +1) = h(x, x') for all x,
x
1
.
We recall that the twist property (1.1) of the maps
F
}
implies that
(Tw) d
1
d
2
h(x, x') > 0 for all
x
and
x'.
We refer to Mather [10,11] and Aubry and le Daeron [2] for more details.
h
q
Given
the generating functions
h
u
...,h
q
we define W in C
2
(X) as follows:
W{x)=
Σ
jjj
where
x
=
X
p
t
q
= {(
x
j)jeZ'Xj+
q
= Xj + P
f
°
r
ally}
is the space we defined in the introduction. This function
will
enable us to study the
fixed
points
of the composite map F
q
°
F
q
_
x
o...
o
F
x
. If F
}
= F for all
j
then such
fixed
points
are
of
course
q
periodic points
of
the single map F. Note that the
composite map
F
q
o F
q
_
1
o...
o
F
x
need not be
a
twist map.
Lemma
2.1 Any point
,
b) in
R
2
satisfies
if and
only
if
there
is a
crtical
point
x
in
X of
W
which
satisfies
x
ί
=a,
-d
2
h
1
{x
o
,x
1
) =
b.
Proof.
The gradient
of
W is given by
djW=
d
2
hj(Xj_
l9
x
j
) + d
1
h
j+
^Xj,
x
j+1
).
Hence
if
one defines y
/
= d
1
/ +1
(x
i7
,x_
/
.
+
1
),
then
(
x
j>yj) =
F
j(
χ
j-i>yj-i)
if and only
if
djW=0, which proves the lemma.
In
view
of this result we shall identify any orbit of
F
u
..., F
q
with its sequence of
x
coordinates.
There
are several structures on
X
which are relevant for our problem. First of
all
X is a
manifold diffeomorphic to R
q
. Furthermore
X
has
a
partial ordering
defined by
xf^y
iff
XiSyi for alii.
We also define
x<y
iff x^y and x
+ y,
x<ζy
iff
Xi<y
t
for all/.
With this ordering
X
becomes
a
lattice.

Periodic
Orbits 357
The
gradient
flow
of W
will
be denoted by
t
}
t
>
0
. It is defined by the
differential equation
x'(t)
= gmdW(x(ή)
onl.
Lemma
2.2. For all x, y in X and t>0 we have:
(a)
if x^y then
φ
t
(x)^φ
t
(y),
(b) if x<y then φ
t
(x)<φ
t
(y)
Proof.
Let V
s
denote
djW
9
then
θ
if
j<i-ί
or
ί
3iδ
2
Λ
ι
<x
j
,x
ί
) if j = i + l
id
2
hj{Xi,Xj)
if 7 = i-l.
Hence
all the off diagonal elements of the Jacobian of grad W are nonnegative,
which implies (a). The other assertion
follows
from the fact that the upper and
lower diagonal only contain strictly positive elements (see Hirsch [8]).
The
analogue of this lemma for the parabolic equation (II) mentioned in the
introduction
is of course the comparison principle. Lemma 2.2 may be restated as
follows:
//
the
graphs
of x and y (think of them as
piecewise
linear
functions
on R) do not
intersect,
then neither do the
graphs
of φ
t
(x) and
φ
t
(y).
For
parabolic equations this has been generalized by Matano [9]. His result
states roughly that, if x(s, t) and y(s, t) are solutions of the parabolic equation (II),
then
the number of intersections of the graphs of
ί-»x(s,ί)
and t-+y(s,t) is a
nonincreasing function of
5
(note that for the PDE we have called the time variable
s)
1
.
The discrete version of this principle is given in Smillie [13] in the context of
competitive and cooperative differential equations.
To
define the "number of intersections" we introduce some more notation.
For
x, y in X we say that x intersects y transversally, in symbols xcfiy, if, for any
integer z, x—^ implies that x
i
_
ί
—y
i
_
ί
and x
i + 1
—y
ί+1
have opposite signs.
If x and y intersect transversally then we define
I(x,y),
the number of
intersections of x with y, to be the largest integer k for which there are
such that
holds for j = 0,1,2,... ,k-l.
Clearly k must be even, and fe ^ q.
The
following result was proved by Smillie [13]:
Proposition
2.3.
Given
x and y in X, the set of t in R for
which
φ
t
(x) and φ
t
(y) do not
intersect
transversally
is
discrete
and
I(φ
t
(x),
φ
t
(y)) is a
nonincreasing
function
of t
1
Actually Matano doesn't prove this. Our statement is an easy consequence of his results however

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References
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TL;DR: In this paper, Newtonian mechanics: experimental facts investigation of the equations of motion, variational principles Lagrangian mechanics on manifolds oscillations rigid bodies, differential forms symplectic manifolds canonical formalism introduction to pertubation theory.
Abstract: Part 1 Newtonian mechanics: experimental facts investigation of the equations of motion. Part 2 Lagrangian mechanics: variational principles Lagrangian mechanics on manifolds oscillations rigid bodies. Part 3 Hamiltonian mechanics: differential forms symplectic manifolds canonical formalism introduction to pertubation theory.

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TL;DR: The prerequisite for the study of this book is a knowledge of matrices and the essentials of functions of a complex variable as discussed by the authors, which is a useful text in the application of differential equations as well as for the pure mathematician.
Abstract: The prerequisite for the study of this book is a knowledge of matrices and the essentials of functions of a complex variable. It has been developed from courses given by the authors and probably contains more material than will ordinarily be covered in a one-year course. It is hoped that the book will be a useful text in the application of differential equations as well as for the pure mathematician.

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