Detailed asymptotics for a convex Hamiltonian system with two degrees of freedom (Q1209578)

The authors investigate the simultaneous system \(u''+u(1+u^ 2+v^ 2)=0\), \(v''+v(k+u^ 2+v^ 2)=0\), where \(k>1\). The system has two conserved energies given by \[ E(u,v,u',v')={1\over 2}u^{'2}+{1\over 2}v^{'2}+{1\over 2}u^ 2+{k\over 2}v^ 2+{1\over 4}(u^ 2+v^ 2)^ 2, \] and \[ F(u,v,u',v')=-{(uv'-u'v)^ 2\over 2(k-1)}+v^{'2}+kv^ 2+{1\over 2}v^ 2(u^ 2+v^ 2). \] It is shown that every solution with \(F\neq 0\) is quasi-periodic with at most two basic frequencies. On the other hand, if \(F=0\) and if the energy \(E\) is sufficiently large, then \(v(t)\) and \(v'(t)\) always tend exponentially to 0 as \(T\to\pm\infty\), and \(u(t)\) is exponentially asymptotic to a periodic solution of \(w''+w+w^ 3=0\).