The existence of at least two homoclinic orbits is proved by A.
Ambrosetti and V. Coti Zelati (
Multiple homoclinic orbits for
a class of conservative systems,
Rend. Sem. Mat. Univ. Padova,
89 (1993), 177-194)
for autonomous Lagrangian systems
$\ddot{q}+V'(q)=0, ~q\in C^2(\R,\R^m),~m\geq 2 $
where $V:\R^m\rightarrow\R$ is a function of the form
$ V(q)=-\frac{|q|^2}{2}+W(q) $
with $W\in C^2(\R^m,\R)$ superquadratic, satisfying a "pinching''
hypothesis and an hypothesis on its second derivative.
The present work deals with potentials of the form $W(q,\dot{q})$
that weakly depend on $\dot{q}$. In this case an homoclinic orbit
corresponds to a classical solution to the equation
$\ddot{q}-q+W_1(q,\dot{q})-\frac{d}{dt}W_2(q,\dot{q})=0,$
where $W_i=\partial_i W$ for $i=1,2$.
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