Dynamical phase transition in the simplest molecular chain model

We consider the dynamics of the simplest chain of a large number $N$ of particles. In the double scaling limit, we find the partition of the parameter space into two domains: for one domain, the supremum over the time interval $(0,\infty)$ of the relative extension of the chain tends to $1$ as $N\to\infty$, and for the other domain, to infinity.