Ames W.F., Roger C. Nonlinear equations in the applied sciences. Volume 185
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140
Shui-Nee
Chow
and
Masahiro Yamashita
Here p1,
p2,
p3 and p4
are
parameters, and
e
is assumed to be sufficiently small. We
consider first the unperturbed system
(c
=
0).
As
easily seen, this unperturbed system
is
a
Hamiltonian system. Let
il
=
23
and
i2
=
24,
and let
I
=
(51, z2,13,24). Then
the unperturbed system
can
be written in the form
(11.10)
x
=
XH(5)
where the Hamiltonian function
H(s)
is given by
(11.11)
Furthermore, system
(11.10)
has one more first integral
which results from the conservation of the angular momentum.
Since
{F,
H)(z)
=
dF(Z)XH(Z)
r
53
1
(11.13)
J
1''
52
-245:
+
5;)
=
154
-
53
-
52511
z1
-251(5;
+
5;)
=O
and since
dF(s)
and
dH(s)
are
linearly independent for any
I
E
lR4\{O}
unperturbed
system,
(11.10)
is
a
completely integrable system in
R4\{O}.
So
we shall utilize this
special structure
(see
Proposition 9.3(ii) and Theorem
9.4)
to derive the Melnikov
vector even though the perturbed system is not
a
Hamiltonian system.
Next we notice that the unperturbed system
(11.10)
has
a
homoclinic orbit
r(t,
0)
to the origin.
Geometry of
the
Melnikov
Vector
141
where
p(t)
=
sech
t.
In terms of the complete integrability, we know that the stable and the unstable
manifolds of system
(ll.lO),
both
of
which have dimension two, must coincide along
~(t,
0)
and in fact, by using
a
symmetry property of
XH,
this ‘homoclinic manifold’
can be expressed
as
r(t,v)
=(p(t)cosv,p(t)sinv,p(t)cosv,j(t)sinv),
(11.15)
v
E
[0,2*),
t
E
R.
That
is,
system
(11.10)
has
a
family
of
homoclinic orbits parametrized by
v.
Thus
system
(11.10)
is
an example to
case
(ii) in Section
8.
Now we go back to the original perturbed system
(8.8).
Let
As
we mentioned in case(ii) in Section
8,
the first approximation
of
the Melnikov
vector can be used to detect
a
point of transversal
or
tangential intersection of the
perturbed stable and unstable manifolds
of
system
(11.16).
Fhrthermore, by virture
of Proposition 9.3(ii), bounded solutions of the adjoint system of the linearized system
of system
(11.10)
along
an
orbit
7(t,
v)
are given by
dH(r(t,
v))
=
(-p(t)
cos
v
+
2p(ty
cos
v,
-p(t)
sin
v
(1
1.17)
+
2p(q3
sin
v,
p(t)
cos
v,
li(t)
sin
v)
and
(11.18)
dF(
-y(
t,
v))
=
(j(t)
sin
v,
-j(
t)
cos
v,
-p(
t)
sin
v,
p(
t)
cos
v).
142
Shui-Nee Chow
and
Masahiro Yamashita
It is easily shown that
dH(-y(t,
v))
and
dF(-y(t,
u))
are linearly independent.
We can now compute the
first
approximation of the Melnikov vector
fi(cr,
v)
=
(Lfl(a,
v),
fiz(Q,
v))
as
follows.
J-m
(11.19)
(v3~osv+p~sinv)p(t)~fi(t)cosw(t
-a)}&
6
1
+wZ
+-
2
RW
=
-3pz
-
RW
sec
h(
-)(-p3
sin
v
+
p4
coswcr)
2
m
h2(%
v)
=
Lm
dF(-Y(t,
v))!dt
-
a,
7(t,
.),P)dt
=
2p1
sin
2v
+
c(
-p3
sin
v
+
p4
cos
v)
coswcr.
Let
c
=
R
sec
h(
y).
Then the Melnikov vector becomes
1
w(p3
cos
v
+
p4
sin
v)
sin
wcr
2pl
sin
2v
+
c(
-pa
sin
v
+
p4
cos
v)
cos
wa.
'
(11.20)
To
find points of intersection, consider, for example, the case
v
=
0.
In this case the
Melnikov vector becomes
(11.21)
Solving
Lf(cr,
0;
p)
=
0,
we have the following bifurcation set
S
in the parameter space
(PI
7
pZ,
p3,
p4).
(11.22) S=AUBUC,
Geometry
of
the Melnikov Vector
143
Next we examine the transversality and the tangency
of
intersection in the case
w
=
0.
The derivatives
of
M
are
given by
1
a
-w2p3 cos
WQ
(
1
1.23)
and
(11.24)
1
lp
-
[
4p1
-
cp3
COSWQ
-wp4
sin
wa
a
--n;l(a,O
)-
aw
Since, in this example, the stable and the unstable manifolds
of
the unperturbed
system coincide and constitute
a
two dimensional manifold in
R4,
rank[&;n;l(cr,O;
p)
&M(a,O;p)]
=
2 implies
a
transversal intersection
(see
Proposition 8.3).
(i) Let
p
E
A.
Since
intersection is always transversal.
(ii) Let
p
E
B.
In this case we have
(ii-1) If
pi
#
~C~~COSWW",
then the rank
of
the above matrix is 2 and
so
we have a
transversal intersection.
(ii-2)
If
pi
=
fcp3
COSWQ,
then
&$(a,
0,
p)
=
0.
By computing the rank of
(8.16),
we
have
a
tangential intersection.
(iii) Let
p
E
C.
In this case
144
Shui-Nee Chow
and
Masahiro Yamashita
By computing the rank
of
(8.16),
we have
a
tangential intersection
f
#
0,143
#
0.
Example
3.
The aim of this example is to give an example in which condition (ii) of Theorem
5.5
is
not satisfied but the stable and unstable manifolds intersect transversally. To
this end
we
modify the system in Example
2
slightly and consider the following system.
51
=x3
(
1
1.26)
W"
=
(p(a)
cos
v,
p(a)
sin
v,
p(a)
cos
v,
Ka)
sin
v,
Y
1,
(11.27)
W'
=
(p(a)
cos
v,
p(a)
sin
v,
p(a)
cos
v,
p(a)
sin
v,
0)
where
p(t)
=
sech
t
and
a,
y
E
R,
v
E
[0,2r].
Note that dim
W"
=
3
and dim
W'
=
2,
and the 'homoclinic manifold' is
W"
n
W'.
From
(11.17)
and
(11,18),
it is clear that the adjoint system of the linearized system of
the unperturbed system has two linearly independent bounded solutions on
R
which
are given by
(11.28)
+*(t)
=(-p(t)cosv
+
2~(t)~cosv,
-p(t)sinv
+
2p(
t)3
sin
v,
i(t)
cos
v,
p(
t)
sin
v,
0)
Geometry
of
the
Melnikov Vector
145
and
(11.29)
&(t)
=
(p(t)
sin
v,
-dt)
cos
v,
-p(t)
sin
v,
p(t)
cos
v,
0).
Hence the Nelnikov vector for system
(11.25)
is precisely the same
m
(11.20).
Consider
the
case
v
=
0.
From now on we assume that
(1
1.30)
where
C
is
defined in
(11.22).
$fi(a, 0;
p)
#
0.
Thus condition (ii) in Theorem
5.5
is not satisfied. However,
we shall show that there exists the transversal intersection of the stable and unstable
manifolds of system
(11.25).
First recall, by using the notation in
(5.16)
and
(5.17),
that we have the following situation.
p
E
C
and
pi
#
0
Then we know that
&&f(a,O;p)
=
0
and
(11.31)
(11.32)
Therefore if
&m:
&fi(a,O
:
p)
=
0
which means that
&mi
=
&mi.
#
0,
we have the transversal intersection even though
Note that the linearized
146
Shui-Nee
Chow
and
Masahiro
Yamashita
system of the unperturbed system of
(11.25)
has an unbounded solution
(O,O,
O,O,
ce')
on
[a,
ce)
where
c
is
a
nonzero constant. Let
(11.33)
dj(t)
=
(o,o,
O,O,
ce-').
Then
(11.34)
=
/:
ce-'
cos
w(t
-
a)dt
So
we have
(11.35)
Thus, in this example, there exists the transversal intersection but the condition by
the Melnikov vector cannot be used to show it. In other words, the Melnikov vector
cannot give complete information about the transversal intersection. This limitation of
the Melnikov vector in higher dimensional
cases
comes from the fact that the Melnikov
vector is the projection
of
the real distance between the stable and unstable manifolds
to the space
of
the completely unbounded solutions, i.e., the complement subspace
of
R(P(a))
+
R(1
-
Q(a)).
Therefore the Melnikov vector drops the information
about the projection of the real distance to other subspaces. See the decomposition
in Section
5
of
TAp)IRn.
Finally, we examine an example for which the Melnikov function cannot be applied
to
detect the intersection of the stable and unstable manifolds. Before doing this, we
recall system
(11.1)
in Example
1.
The first approximation
of
the Melnikov function of this system was
~2(a)
=
-c
sin
a
Geometry
of
the
Melnikov
Vector
147
where
c
#
0
is
a
constant. The reason that
&?(a)
can be used to detect the intersection
of the stable and unstable manifolds of this system is that the distance
d
between the
stable and unstable manifolds is expressed
as
(11.36)
d
=
€(&?(a)
+
O(
lei))
I
That is,
n;r(a)
constitutes the leading term.
Now we consider the following rapidly forced system.
(
1
1.37)
3
t
2
€1
x
-I
+
-2
=
c
cos(-)
where
e
<<
1
and
el
<<
1.
In this case the Melnikov function takes the form
(11.38)
7r
7r
a
c
€
e
&(a,c)
=
--cosech(-) sin(-).
Hence
&?(a,
c)
cannot be the leading term of the expansion
of
d
in terms of
c.
See also
Holmes, Marsden and Scheule
[ll].
This is
a
serious limitation of the perturbation
method used in the theory of the Melnikov function we developed before and in fact
it relates to one of the fundamental problems in dynamics since the time
of
Poincark.
Resolution of this difficulty has to wait for future study.
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[l]
Arnold, V.I.,
Mathematical Methods
of
Classical Mechanics,
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Acad.
Sci. Novi Lyncaei,
1(1935), 85-216.
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Arnold, V.I.,
Instability
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Birkhoff, G.D.,
Nouvelles recherche3 sur les systemes dynamiques,
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Chow, S.N., Hale,
J.K.,
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Coppel, W.A.,
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The ezistence of homoclinic orbits and the method of Melnikov for
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Preprint
.
Nonlinear Waves
Andrea Donato
Department
of
Mathematics
University
of
Messina, Italy
Abstract
A
quasilinear hyperbolic system which is invariant under scaling
is transformed into
a
system whose stationary solutions in canoni-
cal variables are the similarity solutions. The propagation of weak
discontinuities in this special non-constant state is considered. The
transformed system may have
a
constant state solution that becomes
a
non-constant state solution in the original variables. The possibil-
ity for
a
similarity line to be
a
discontinuity line is also investigated.
1
Introduction
A
precise definition of
wave,
which embraces all the physical phe-
nomena commonly referred
to
by this name, is unattainable.
From
an intuitive point of view, any kind of disturbance which propagates
in
a
medium with
a
finite speed
as
time increases may be considered
as
a
wave. The signal associated with the perturbation may distort,
change its magnitude and velocity but it is still clearly identifiable.
A
diversity
of
wave propagation problems are described by partial
differential equations of hyperbolic type. However, wave phenomena
in the above mentioned general sense can
also
be described by non-
linear parabolic equations which arise out
of
asymptotic analysis of
more complicated systems.
In
many physical situations involving wave interactions, the ap-
propriate descripting partial differential equations are nonlinear. In
the case
of
hyperbolic systems, the nonlinearity commonly leads to
the solution developing discontinuities (shocks) even for smooth and
small initial data.
Nonlinear Equations
in
the Applied Sciences
149
Copyright
0
1992
by
Academic Press, Inc.
All
rights
of
reproduction
in
any
form
reserved.
ISBN
0-12-056752-0