[ 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

Synchronisation Results for an Interconnected Network of Nonlinear Systems with Diffusive Nonlinear Coupling using Contraction

Journal of Applied Nonlinear Dynamics 12(3) (2023) 591-608 | DOI:10.5890/JAND.2023.09.012

Yashasvi Chauhan, B. B. Sharma

Department of Electrical Engineering, National institute of technology, Hamirpur, 177005, H.P, India

Download Full Text PDF

 

Abstract

This paper presents a contraction theory based methodology for synchronisation of non-linearly coupled dynamical systems interconnected to constitute a complex network. Here, a systematic control procedure is presented to achieve synchronisation of complex network of proposed strict-feedback like class of nonlinear systems. The non-linear diffusive coupling function between different systems of the network is assumed to be in the form of bidirectional links. The proposed methodology can be applicable to any arbitrarily structure of linear/non-linear, bidirectional or unidirectional N-coupled systems in a network. Rigorous analytical results have been derived for coupled systems interacting through specific nonlinear coupling function which are interconnected in different networked topologies including Ring, Global, Star, Arbitrary etc. The analytical conditions for synchronisation are expressed in terms of bounds on coupling strength which are derived using partial contraction concepts blended with graph theory results. An example of complex networks is simulated to verify the theoretical results.

Acknowledgments

The DST-SERB, the Government of India, and the National Institute of Technology Hamirpur are all appreciated by the authors for supporting to carry out the research.

References

  1. [1]  Chen, G., Wang, X., and Li, X. (2015), Fundamentals of Complex Networks: Models, Structures and Dynamics, Wiley-Blackwell.
  2. [2]  Fujisaka, H. and Yamada, T. (1983), Stability theory of synchronised motion in coupled-oscillator systems, Progress of Theoretical Physics, 69, 32-47.
  3. [3]  Pecora, L. and Carroll, T. (1990), Synchronization in chaotic system, Physical Review Letters, 64, 821-824.
  4. [4]  Park, J.H. (2006), Synchronization of genesio chaotic system via backstepping approach, Chaos, Solitons $\&$ Fractals, 27, 1369-1375.
  5. [5]  Wu, C.W. (2007), Synchronization in complex networks of nonlinear dynamical systems, World scientific.
  6. [6]  Guo, R. (2017), Projective synchronization of a class of chaotic systems by dynamic feedback control method, Nonlinear Dynamics, 90, 53–64.
  7. [7]  Lahav, N., Sendi{\~n}a-Nadal, I., Hens, C., Ksherim, B., Barzel, B., Cohen, R., and Boccaletti, S. (2022), Topological synchronization of chaotic systems, Scientific reports, 12, 1-10.
  8. [8]  Joshi, S.K. (2021), Synchronization of chaotic dynamical systems, International Journal of Dynamics and Control, 9, 1285-1302.
  9. [9]  Wu, X., Guan, Z.H., and Li, T. (2007), Chaos synchronization between unified chaotic system and genesio system, International Symposium on Neural Networks, Springer.
  10. [10]  Lü, J., Zhou, T., and Zhang, S. (2002), Chaos synchronization between linearly coupled chaotic systems, Chaos, Solitons and Fractals, 14, 529-541.
  11. [11]  Zhang, C., Deng, F., Zhang, W., Hou, T., and Yang, Z. (2019), Anti-synchronization and synchronization of coupled chaotic system with ring connection and stochastic perturbations, IEEE Access, 07, 76902-76909.
  12. [12]  Lü, J., Yu, X., and Chen, G. (2004), Chaos synchronization of general complex dynamical networks, Physica A: Statistical Mechanics and its Applications, 334, 281-302.
  13. [13]  Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., and Hwang, D.U. (2006), Complex networks: Structure and dynamics, Physics Reports, 424, 175-308.
  14. [14]  Jian, X., Yehong, Y., and Jushu, L. (2013), Synchronisation of complex networks with derivative coupling via adaptive control, International Journal of Systems Science, 44, 2183-2189.
  15. [15]  Garg, V. and Sharma, B.B. (2018), Complete amplitude scaling and projective synchronisation of a class of nonlinear dynamical systems, International Journal of Systems, Control and Communications, 9, 85-105.
  16. [16]  Liao, X. and Chen, G. (2003), On feedback-controlled synchronization of chaotic systems, International Journal of Systems Science, 34, 453-461.
  17. [17]  Tao, X.R., Tang, L., and He, P. (2018), Synchronization of unified chaotic system via output feedback control scheme, Journal of Applied Nonlinear Dynamics, 7, 383-392.
  18. [18]  Strogatz, S.H. (2001), Exploring complex networks, Nature, 410, 268-276.
  19. [19]  Angeli, D. (2002), A lyapunov approach to incremental stability properties, IEEE Transactions on Automatic Control, 47, 410-421.
  20. [20]  Handa, H. and Sharma, B.B. (2016), Synchronization of a set of coupled chaotic fitzhugh–nagumo and hindmarsh–rose neurons with external electrical stimulation, Nonlinear Dynamics, 85, 1517–1532.
  21. [21]  Jovic, B. (2011), Chaotic Synchronization, Conditional Lyapunov Exponents and Lyapunov's Direct Method, Springer Berlin Heidelberg.
  22. [22]  Zhou, T., Chen, G., Lu, Q., and Xiong, X. (2006), On estimates of lyapunov exponents of synchronized coupled systems, Chaos: An Interdisciplinary Journal of Nonlinear Science, 16, 033123.
  23. [23]  Singh, A.K., Yadav, V.K., and Das, S. (2019), Nonlinear control technique for dual combination synchronization of complex chaotic systems, Journal of Applied Nonlinear Dynamics, 8, 261-277.
  24. [24]  Lohmiller, W. and Slotine, J.J.E. (2000), Nonlinear process control using contraction theory, Aiche Journal, 46, 588-596.
  25. [25]  Jouffroy, J. and Slotine, J.E. (2004), Methodological remarks on contraction theory, 43rd IEEE Conference on Decision and Control (CDC).
  26. [26]  Sharma, B.B. and Kar, I.N. (2011), Observer-based synchronization scheme for a class of chaotic systems using contraction theory, Nonlinear Dynamics, 63, 429–445.
  27. [27]  Wang, W. and Slotine, J.-J.E. (2005), On partial contraction analysis for coupled nonlinear oscillators, Biological Cybernetics, 92, 38-53.
  28. [28]  Sharma, B.B. and Kar, I.N. (2008), Design of asymptotically convergent frequency estimator using contraction theory, IEEE Transactions on Automatic Control, 53, 1932-1937.
  29. [29]  Li, K., Sun, W., and Fu, X. (2009) Synchronisation of complex networks via partial contraction principle, International Journal of Systems, Control and Communications, 1, 414-425.
  30. [30]  Sharma, B. and Kar, I. (2010), Contraction theory-based recursive design of stabilising controller for a class of non-linear systems, IET Control Theory and Applications, 4, 1005-1018.
  31. [31]  Wu, C.W. and Chua, L.O. (1995), Application of graph theory to the synchronization in an array of coupled nonlinear oscillators, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 42, 494-497.
  32. [32]  Anand, P. and Sharma, B.B. (2020), Simplified synchronizability scheme for a class of nonlinear systems connected in chain configuration using contraction, Chaos, Solitons and Fractals, 141, 110331.
  33. [33]  Lohmiller, W. and Slotine, J.J.E. (1998), On contraction analysis for non-linear systems, Automatica, 34, 683-696.
  34. [34]  Slotine, J.J.E. and Wang, W. (2005), A Study of Synchronization and Group Cooperation Using Partial Contraction Theory, Springer Berlin Heidelberg.

Copyright © 2011-2024 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates