[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Journal of Applied Nonlinear Dynamics
ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Mathematical Models of Nonlinear Uniform Consensus II

Journal of Applied Nonlinear Dynamics 7(1) (2018) 95--104 | DOI:10.5890/JAND.2018.03.008

Mansoor Saburov, Khikmat Saburov

Faculty of Science, International Islamic University Malaysia, 25200 Kuantan, Pahang, Malaysia

College of Engineering and Technology, American University of the Middle East, 54200, Egaila, Kuwait

Mathematical Modeling Laboratory, MIMOS Berhad, 57000 Kuala Lumpur, Malaysia

Download Full Text PDF

 

Abstract

This paper is a continuation of our previous studies on nonlinear consensus. We have considered a nonlinear protocol for a structured time-invariant and synchronous multi-agent system. We present an opinion sharing dynamics of the multi-agent system as a trajectory of a polynomial stochastic operator associated with a stochastic multidimensional hyper-matrix. We provide a criterion for a uniform consensus in the multi-agent system. Particularly, the uniform consensus is achieved in the multi-agent system if all entries of the stochastic multidimensional hyper-matrix are positive. Some numerical results are also presented to support our theoretical results.

Acknowledgments

This work has been done under the MOHE grant FRGS14-141-0382. The Author (M.S.) is grateful to the Junior Associate scheme of the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.

References

  1. [1]  DeGroot, M.H. (1974), Reaching a consensus, Journal of the American Statistical Association, 69, 118-121.
  2. [2]  Vicsek, T., Czirok, A., Ben-Jacob, E., Cohen, I., and Shochet, O. (1995), Novel type of phase transition in a system of self-driven particles, Physical Review Letters, 75, 1226-1229.
  3. [3]  Olfati-Saber, R. and Murray, R.M. (2004), Consensus problems in networks of agents with switching topology and time-delays, IEEE Transactions on Automatic Control, 49, 1520-1533.
  4. [4]  Liu, X., Chen, T., and Lu, W. (2009), Consensus problem in directed networks of multi-agents via nonlinear protocols, Physics Letters A, 373, 3122-3127.
  5. [5]  Wen, G., Duan, Z., Yu, W., and Chen, G. (2013), Consensus of multi-agent systems with nonlinear dynamics and sampled-data information: a delayed-input approach, International Journal of Robust and Nonlinear Control, 23, 602-619.
  6. [6]  Yu, W., Chen, G., and Cao, M. (2011), Consensus in directed networks of agents with nonlinear dynamics, IEEE Transactions on Automatic Control, 56, 1436-1441.
  7. [7]  Saburov, M. and Saburov, Kh. (2014), Mathematical models of nonlinear uniform consensus, ScienceAsia, 40(4), (2014), 306-312.
  8. [8]  Saburov, M. and Saburov, Kh. (2014), Reaching a nonlinear consensus: polynomial stochastic operators, International Journal of Control, Automation and Systems, 12(6), 1276-1282.
  9. [9]  Saburov, M. and Saburov, Kh. (2016), Reaching a nonlinear consensus: a discrete nonlinear time-varying case, International Journal of Systems Science, 47(10), 2449-2457.
  10. [10]  Berger, R.L. (1981), A necessary and sufficient condition for reaching a consensus using DeGroot's method, Journal of the American Statistical Association, 76, 415-418.
  11. [11]  Chatterjee, S. and Seneta, E. (1977), Towards consensus: some convergence theorems on repeated averaging, Journal of Applied Probability, 14, 89-97.
  12. [12]  Lyubich, Y.I. (1992), Mathematical Structures in Population Genetics, Springer-Verlag: Berlin Heidelberg.
  13. [13]  Ganikhodjaev, N., Saburov, M., and Jamilov, U. (2013), Mendelian and non-Mendelian quadratic operators, Applied Mathematics & Information Sciences, 7, 1721-1729.
  14. [14]  Ganikhodjaev, N., Saburov, M., and Nawi, A. M. (2014), Mutation and chaos in nonlinear models of heredity, The Scientific World Journal, 2014, 1-11.
  15. [15]  Ganikhodzhaev, R., Mukhamedov, F., and Rozikov, U. (2011), Quadratic stochastic operators and processes: results and open problems, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 14, 279-335.
  16. [16]  Mukhamedov, F. and Saburov, M. (2010), On homotopy of volterrian quadratic stochastic operator, Applied Mathematics & Information Sciences, 4, 47-62.
  17. [17]  Mukhamedov, F. and Saburov, M. (2014), On dynamics of Lotka-Volterra type operators, Bulletin of the Malaysian Mathematical Sciences Society, 37, 59-64.
  18. [18]  Mukhamedov, F., Saburov, M., and Qaralleh, I. (2013), On ξ(s)−quadratic stochastic operators on twodimensional simplex and their behavior, Abstract and Applied Analysis, 2013, 1-12.
  19. [19]  Saburov, M. (2013), Some strange properties of quadratic stochastic volterra operators,World Applied Sciences Journal, 21, 94-97.
  20. [20]  Krause, U. (2009), Compromise, consensus, and the iteration of means, Elemente der Mathematik, 64, 1-8.
  21. [21]  Krause, U. (2009), Markov chains, Gauss soups, and compromise dynamics, Journal of Contemporary Mathematical Analysis, 44, 59-66.

Copyright © 2011-2023 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates