Stochastic transition in a classical nonlinear dynamical system: A Lennard–Jones chain

In this paper we present and discuss the results of various computer experiments performed on a Lennard–Jones chain for a number of particles $N$ ranging from three to one thousand. These experiments indicate that this system exhibits a transition from near integrable to stochastic behavior, as one goes from low specific energies to higher ones. More precisely, it is possible to characterize two values of the energy per particle, $E_{c_1}$ and $E_{c_2}$ , such that, for energies lower than $E_{c_1}$, the overwhelming majority of initial conditions lead to ordered motion and, for energies higher than $E_{c_2}$, the overwhelming majority of initial conditions lead to stochastic motion. The most interesting conclusion of these computations is that the above mentioned critical values seem to be roughly independent of the number of degrees of freedom, if this number is sufficiently large (greater than ten). On the contrary, when $N$ is small (from three to ten), $E_{c_1}$ and $E_{c_2}$ are strongly dependent on both the number of degrees of freedom and the initial conditions.