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We discuss the dynamics of integrable and nonintegrable chains of coupledoscillators under continuous weak position measurements in the semiclassicallimit. We show that, in this limit, the dynamics is described by a standardstochastic Langevin equation, and a measurement-induced transition appears as anoise- and dissipation-induced chaotic-to-nonchaotic transition akin tostochastic synchronization. In the nonintegrable chain of anharmonicallycoupled oscillators, we show that the temporal growth and the ballisticlight-cone spread of a classical out-of-time correlator characterized by theLyapunov exponent and the butterfly velocity, are halted above a noise or belowan interaction strength. The Lyapunov exponent and the butterfly velocity bothact like order parameter, vanishing in the nonchaotic phase. In addition, thebutterfly velocity exhibits a critical finite-size scaling. For the integrablemodel, we consider the classical Toda chain and show that the Lyapunov exponentchanges nonmonotonically with the noise strength, vanishing at the zero noiselimit and above a critical noise, with a maximum at an intermediate noisestrength. The butterfly velocity in the Toda chain shows a singular behaviorapproaching the integrable limit of zero noise strength.

Semiclassical Limit of Measurement-Induced Transition in Many-Body Chaos in Integrable and Nonintegrable Oscillator Chains