  We study analytically the behavior of the largest Lyapunov exponent
$\lambda_1$ for a one-dimensional chain of coupled nonlinear oscillators, by
combining the transfer integral method and a Riemannian geometry approach. We
apply the results to a simple model, proposed for the DNA denaturation, which
emphasizes a first order-like or second order phase transition depending on the
ratio of two length scales: this is an excellent model to characterize
$\lambda_1$ as a dynamical indicator close to a phase transition.
