February  2010, 27(1): 75-116. doi: 10.3934/dcds.2010.27.75

Multiplicity of homoclinic orbits in quasi-linear autonomous Lagrangian systems

1. 

Mathematics Section (SB/IACS/CAA), Ecole Polytechnique Fédérale - Lausanne, Station 8, CH 1015 Lausanne, Switzerland

2. 

Le Grand-Chemin 92, 1066 Epalinges, Switzerland

Received  February 2009 Revised  November 2009 Published  February 2010

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 $i=1,2$.

Citation: Boris Buffoni, Laurent Landry. Multiplicity of homoclinic orbits in quasi-linear autonomous Lagrangian systems. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 75-116. doi: 10.3934/dcds.2010.27.75
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