Emergent behaviors of the Justh-Krishnaprasad model with uncertain communications

Abstract

We study stochastic alignment of the Justh-Krishnaprasad(J-K) model under a multiplicative noise. Justh and Krishnaprasad proposed a planar Newtonian particle model for an ensemble of flocking agents moving with a unit speed, where the dynamics of the heading angles (of velocities) is governed by a generalized Kuramoto-type model with a communication weight. In this paper, we are interested in the random perturbation of the communication weight, which is represented by a multiplicative noise. Unlike the additive white noise perturbation, the multiplicative one is known not to violate collective behaviors. We present sufficient conditions leading to the nematic alignment of velocities in terms of the system parameters and initial data. Our analysis begins with the two-body system, where it turns out to be stable under the effect of noise, showing that the nematic alignment occurs when the communication is sufficiently strong with respect to the noise strength. For the general many-body system with a corresponding condition, we showed the accumulation of heading angles modulo π/2 and the stochastic stability of nematic alignment under the assumption of the constant communication weight, which suggests a strong evidence for the nematic alignment. This analysis is done by transporting the system into a similar form to the stochastic Kuramoto model, where we refined the order parameter analysis in order to extend local stochastic stability results to the whole circle of heading angles. We present several numerical simulations and compare them with analytical results.