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We study the asymptotic behaviour of a finite network of oscillators(harmonic or anharmonic) coupled to a number of deterministic Lagrangianthermostats of finite energy. In particular, we consider a chain of oscillatorsinteracting with two thermostats situated at the boundary of the chain. Underappropriate assumptions we prove that the vector $(p,q)$ of moments andcoordinates of the oscillators in the network satisfies $(p,q)(t)\to (0,q_c)$when $t\to\infty$, where $q_c$ is a critical point of some effective potential,so that the oscillators just stop. Moreover, we argue that the energy transportin the system stops as well without reaching the thermal equilibrium. Thisresult is in contrast to the situation when the energies of the thermostats areinfinite, studied for a similar system in [14] and subsequent works, where theconvergence to a non-trivial limiting regime was established. The proof isbased on a method developed in [22], where it was observed that the thermostatsproduce some effective dissipation despite the Lagrangian nature of the system.

Asymptotic behaviour of a network of oscillators coupled to thermostats of finite energy