Affiliation:
1. Department of Mathematics , Zhejiang Normal University , Jinhua , Zhejiang 321004 , P. R. China
2. Department of Mathematics , University of Arizona , Tucson , AZ 85750 , USA
Abstract
Abstract
This paper discusses a high-order Melnikov method for periodically perturbed equations. We introduce a new method to compute
M
k
(
t
0
)
{M_{k}(t_{0})}
for all
k
≥
0
{k\geq 0}
, among which
M
0
(
t
0
)
{M_{0}(t_{0})}
is the traditional Melnikov function, and
M
1
(
t
0
)
,
M
2
(
t
0
)
,
…
{M_{1}(t_{0}),M_{2}(t_{0}),\ldots\,}
are its high-order correspondences. We prove that, for all
k
≥
0
{k\geq 0}
,
M
k
(
t
0
)
{M_{k}(t_{0})}
is a sum of certain multiple integrals, the integrand of which we can explicitly compute. In particular, we obtain explicit integral formulas for
M
0
(
t
0
)
{M_{0}(t_{0})}
and
M
1
(
t
0
)
{M_{1}(t_{0})}
. We also study a concrete equation for which the explicit formula of
M
1
(
t
0
)
{M_{1}(t_{0})}
is used to prove the existence of a transversal homoclinic intersection in the case of
M
0
(
t
0
)
≡
0
{M_{0}(t_{0})\equiv 0}
.
Funder
National Natural Science Foundation of China
Subject
General Mathematics,Statistical and Nonlinear Physics
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