# American Institute of Mathematical Sciences

October  2019, 24(10): 5569-5596. doi: 10.3934/dcdsb.2019072

## Interplay of time-delay and velocity alignment in the Cucker-Smale model on a general digraph

 1 Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China 2 Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea 3 School of Mathematics, Korea Institute for Advanced Study, Seoul 02455, Republic of Korea

* Corresponding author: Jiu-Gang Dong

Received  August 2018 Revised  November 2018 Published  April 2019

Fund Project: The work of S.-Y. Ha was supported by the Samsung Science and Technology Foundation under Project Number SSTF-BA1401-03. The work of J.-G. Dong was supported in part by NSFC grant 11671109.

We study dynamic interplay between time-delay and velocity alignment in the ensemble of Cucker-Smale (C-S) particles(or agents) on time-varying networks which are modeled by digraphs containing spanning trees. Time-delayed dynamical systems often appear in mathematical models from biology and control theory, and they have been extensively investigated in literature. In this paper, we provide sufficient frameworks for the mono-cluster flocking to the continuous and discrete C-S models, which are formulated in terms of system parameters and initial data. In our proposed frameworks, we show that the continuous and discrete C-S models exhibit exponential flocking estimates. For the explicit C-S communication weights which decay algebraically, our results exhibit threshold phenomena depending on the decay rate and depth of digraph. We also provide several numerical examples and compare them with our analytical results.

Citation: Jiu-Gang Dong, Seung-Yeal Ha, Doheon Kim. Interplay of time-delay and velocity alignment in the Cucker-Smale model on a general digraph. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5569-5596. doi: 10.3934/dcdsb.2019072
##### References:

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##### References:
Digraph connection topology $\mathcal C$
The convergence trajectories of the first component velocities satisfying the condition (6.1). Left: Digraph $\mathcal C$ and right: all-to-all graph
The convergence trajectories of the first component velocities not satisfying the condition (6.1). Left: Digraph ${\mathcal C}$ and right: all-to-all graph
The convergence trajectories of the first component velocities satisfying the condition in Corollary 3.1
The trajectories of the first component velocities not satisfying the condition in Corollary 3.1
 $\boldsymbol x_1(t)$ $(1, 0)$ $\boldsymbol v_1(t)$ $\frac{e^{-10}}{672 \sqrt{2}}(1, -2)$ $\boldsymbol x_2(t)$ $(0, 1)$ $\boldsymbol v_2(t)$ $\frac{e^{-10}}{672 \sqrt{2}}(3, -4)$ $\boldsymbol x_3(t)$ $(-1, 0)$ $\boldsymbol v_3(t)$ $\frac{e^{-10}}{672 \sqrt{2}}(5, 6)$ $\boldsymbol x_4(t)$ $(0, -1)$ $\boldsymbol v_4(t)$ $\frac{e^{-10}}{672 \sqrt{2}}(-7, -8)$
 $\boldsymbol x_1(t)$ $(1, 0)$ $\boldsymbol v_1(t)$ $\frac{e^{-10}}{672 \sqrt{2}}(1, -2)$ $\boldsymbol x_2(t)$ $(0, 1)$ $\boldsymbol v_2(t)$ $\frac{e^{-10}}{672 \sqrt{2}}(3, -4)$ $\boldsymbol x_3(t)$ $(-1, 0)$ $\boldsymbol v_3(t)$ $\frac{e^{-10}}{672 \sqrt{2}}(5, 6)$ $\boldsymbol x_4(t)$ $(0, -1)$ $\boldsymbol v_4(t)$ $\frac{e^{-10}}{672 \sqrt{2}}(-7, -8)$
 $\boldsymbol x_1(t)$ $(1, 0)$ $\boldsymbol v_1(t)$ $(1, -2)$ $\boldsymbol x_2(t)$ $(0, 1)$ $\boldsymbol v_2(t)$ $(3, -4)$ $\boldsymbol x_3(t)$ $(-1, 0)$ $\boldsymbol v_3(t)$ $(5, 6)$ $\boldsymbol x_4(t)$ $(0, -1)$ $\boldsymbol v_4(t)$ $(-7, -8)$
 $\boldsymbol x_1(t)$ $(1, 0)$ $\boldsymbol v_1(t)$ $(1, -2)$ $\boldsymbol x_2(t)$ $(0, 1)$ $\boldsymbol v_2(t)$ $(3, -4)$ $\boldsymbol x_3(t)$ $(-1, 0)$ $\boldsymbol v_3(t)$ $(5, 6)$ $\boldsymbol x_4(t)$ $(0, -1)$ $\boldsymbol v_4(t)$ $(-7, -8)$
 $\boldsymbol x_1(t)$ $(1, 0)$ $\boldsymbol v_1(t)$ $\frac{e^{-10}}{7056 \sqrt{2}}(1, -2)$ $\boldsymbol x_2(t)$ $(0, 1)$ $\boldsymbol v_2(t)$ $\frac{e^{-10}}{7056 \sqrt{2}}(3, -4)$ $\boldsymbol x_3(t)$ $(-1, 0)$ $\boldsymbol v_3(t)$ $\frac{e^{-10}}{7056 \sqrt{2}}(5, 6)$ $\boldsymbol x_4(t)$ $(0, -1)$ $\boldsymbol v_4(t)$ $\frac{e^{-10}}{7056 \sqrt{2}}(-7, -8)$
 $\boldsymbol x_1(t)$ $(1, 0)$ $\boldsymbol v_1(t)$ $\frac{e^{-10}}{7056 \sqrt{2}}(1, -2)$ $\boldsymbol x_2(t)$ $(0, 1)$ $\boldsymbol v_2(t)$ $\frac{e^{-10}}{7056 \sqrt{2}}(3, -4)$ $\boldsymbol x_3(t)$ $(-1, 0)$ $\boldsymbol v_3(t)$ $\frac{e^{-10}}{7056 \sqrt{2}}(5, 6)$ $\boldsymbol x_4(t)$ $(0, -1)$ $\boldsymbol v_4(t)$ $\frac{e^{-10}}{7056 \sqrt{2}}(-7, -8)$
 $\boldsymbol x_1(t)$ $(1, 0)$ $\boldsymbol v_1(t)$ $(1, -2)$ $\boldsymbol x_2(t)$ $(0, 1)$ $\boldsymbol v_2(t)$ $(3, -4)$ $\boldsymbol x_3(t)$ $(-1, 0)$ $\boldsymbol v_3(t)$ $(5, 6)$ $\boldsymbol x_4(t)$ $(0, -1)$ $\boldsymbol v_4(t)$ $(-7, -8)$
 $\boldsymbol x_1(t)$ $(1, 0)$ $\boldsymbol v_1(t)$ $(1, -2)$ $\boldsymbol x_2(t)$ $(0, 1)$ $\boldsymbol v_2(t)$ $(3, -4)$ $\boldsymbol x_3(t)$ $(-1, 0)$ $\boldsymbol v_3(t)$ $(5, 6)$ $\boldsymbol x_4(t)$ $(0, -1)$ $\boldsymbol v_4(t)$ $(-7, -8)$
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