Instabilities of coupled quasi-homoclinic limit cycles

It has been recently shown that the continuous coupling of non-linear oscil lators close to an Andronov homoclinic bifurcation generically leads to two types of instabilities: a self-focusing Kuramoto like instability or a fin ite wavelength period doubling instability. The purpose of this paper is to study simple examples where these instabilities have been observed. It inc ludes the coupling of two pendula, the sine-Gordon chain and the coupling o f dissipative non-linear oscillators.