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Delayed Cucker-Smale model with and without noise revisited. (arXiv:1810.01084v1 [math.CA])

来源于:arXiv
Delay in information processing has been incorporated in many models that describe emergence in biological systems. In particular, different versions of the Cucker-Smale model have been considered with processing delay in previous works. In this paper, we study the delayed Cucker-Smale-type system proposed by Erban, Haskovec and Sun \cite{ErHaSu}, in which all terms in the communication weight and the velocity coupling have a positive constant delay $\tau>0$. %We give an affirmative answer to an open question We show that flocking occurs for the communication weight originally proposed by Cucker and Smale, $\psi(r)=(1+r^{2})^{-\beta}$, $r \geq 0$, if $0<\beta <\frac{1}{2}$ and the delay $\tau$ is sufficiently small. In addition, we prove that the delayed system with multiplicative white noises exhibits flocking behavior if the intensity of the noise is sufficiently small. Both results rely on stability estimates for the Cucker-Smale delayed flow. 查看全文>>