We consider the Newtonian system
−
q
¨
+
B
(
t
)
q
=
W
q
(
q
,
t
)
-\ddot q+B(t)q = W_q(q,t)
with
B
B
,
W
W
periodic in
t
t
,
B
B
positive definite, and show that for each isolated homoclinic solution
q
0
q_0
having a nontrivial critical group (in the sense of Morse theory), multibump solutions (with
2
≤
k
≤
∞
2\le k\le \infty
bumps) can be constructed by gluing translates of
q
0
q_0
. Further we show that the collection of multibumps is semiconjugate to the Bernoulli shift. Next we consider the Schrödinger equation
−
Δ
u
+
V
(
x
)
u
=
g
(
x
,
u
)
-\Delta u+V(x)u = g(x,u)
in
R
N
\mathbb {R}^N
, where
V
V
,
g
g
are periodic in
x
1
,
…
,
x
N
x_1,\ldots ,x_N
,
σ
(
−
Δ
+
V
)
⊂
(
0
,
∞
)
\sigma (-\Delta +V)\subset (0,\infty )
, and we show that similar results hold in this case as well. In particular, if
g
(
x
,
u
)
=
|
u
|
2
∗
−
2
u
g(x,u)=|u|^{2^*-2}u
,
N
≥
4
N\ge 4
and
V
V
changes sign, then there exists a solution minimizing the associated functional on the Nehari manifold. This solution gives rise to multibumps if it is isolated.