Dynamical phase transition in the simplest molecular chain model

We consider the dynamics of the simplest chain of large number $N$ of particles. We find, in the double scaling limit, the partition of the parameter space onto two domains, where for one domain the supremum (over all time interval $(0,\mathrm{\infty })$ of relative extension tenda to $1$ as $N\to \mathrm{\infty }$. and for the other this supremum tends to infinity as $N\to \mathrm{\infty }$.