We discuss the dynamics of integrable and non-integrable chains of coupled
oscillators under continuous weak position measurements in the semiclassical
limit. We show that, in this limit, the dynamics is described by a standard
stochastic Langevin equation, and a measurement-induced transition appears as a
noise- and dissipation-induced chaotic-to-non-chaotic transition akin to
stochastic synchronization. In the non-integrable chain of anharmonically
coupled oscillators, we show that the temporal growth and the ballistic
light-cone spread of a classical out-of-time correlator characterized by the
Lyapunov exponent and the butterfly velocity, are halted above a noise or below
an interaction strength. The Lyapunov exponent and the butterfly velocity both
act like order parameter, vanishing in the non-chaotic phase. In addition, the
butterfly velocity exhibits a critical finite size scaling. For the integrable
model we consider the classical Toda chain, and show that the Lyapunov exponent
changes non-monotonically with the noise strength, vanishing at the zero noise
limit and above a critical noise, with a maximum at an intermediate noise
strength. The butterfly velocity in the Toda chain shows a singular behaviour
approaching the integrable limit of zero noise strength.Comment: 4p + eps Main text; 13p Supplementar