Abstract
In this chapter, we consider formation of oscillatory patterns in networks of identical nonlinear systems with time-delays. First of all, by applying the harmonic balance method, we derive the corresponding harmonic balance equations for networks of identical nonlinear systems with delay couplings. Then, solving the equations by reducing to the stability problem of linear retarded systems, we estimate the oscillation profile such as the frequency, amplitudes and phases of coupled systems. Based on this analysis method, we also develop a design method of networks for nonlinear systems that can achieve prescribed oscillation profiles. The effectiveness of the proposed methods is shown by numerical examples.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Chen, J., Gu, G., Nett, C.N.: A new method for computing delay margins for stability of linear delay systems. Systems & Control Letters 26, 107–117 (1995)
Engelborghs, K., Luzyaninna, T., Roose, D.: Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL. ACM Trans. Math. Softw. 28(1), 1–21 (2002)
Golubitsky, M., Stewart, I.: The Symmetry Perspective - From Equilibrium to Chaos in Phase Space and Physical Space. Birkhäuser (2004)
Iwasaki, T.: Multivariable harmonic balance for central pattern generators. Automatica 44(12), 3061–3069 (2008)
Neefs, P., Steur, E., Nijmeijer, H.: Network complexity and synchronous behavior an experimental approach. Int. J. of Neural Systems 20(3), 233–247 (2010)
Newman, M., Barabási, A.L., Watts, D.J.: The Structure and Dynamics of Networks. Princeton University Press (2006)
Niculescu, S.I., Fu, P., Chen, J.: Stability switches and reversals of linear systems with commensurate delays: A matrix pencil characterization. In: Proceedings of the 16th IFAC World Congress, Prague, Czech Republic, IFAC-PapersOnline, World Congress, vol. 16, Part I, pp. 637–642 (2005)
Oguchi, T., Nijmeijer, H., Yamamoto, T.: Synchronization in networks of chaotic systems with time-delay coupling. Chaos 18(3), 037108 (2008)
Oguchi, T., Yamamoto, T., Nijmeijer, H.: Synchronization of bidirectionally coupled nonlinear systems with time-varying delay. In: Loiseau, J.J., Michiels, W., Niculescu, S.-I., Sipahi, R. (eds.) Topics in Time Delay Systems. LNCIS, vol. 388, pp. 391–401. Springer, Heidelberg (2009)
Pogromsky, A., Glad, T., Nijmeijer, H.: On diffusion driven oscillations in coupled dynamical systems. International Journal of Bifurcation and Chaos 9(4), 629–644 (1999)
Pogromsky, A., Santoboni, G., Nijmeijer, H.: Partial synchronization: from symmetry towards stability. Physica D 172(2), 65–87 (2002)
Steur, E., Oguchi, T., van Leeuwen, C., Nijmeijer, H.: Partial synchronization in diffusively time-delay coupled oscillator networks. Chaos 22(4), 43144 (2012)
Uchida, E., Oguchi, T.: Pattern Formation in Networks of Nonlinear Systems with Delay Couplings. In: Proceedings of the 18th IFAC World Congress 2011, pp. 5118–5123 (2011)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Oguchi, T., Uchida, E. (2014). Analysis and Design of Pattern Formation in Networks of Nonlinear Systems with Delayed Couplings. In: VyhlÃdal, T., Lafay, JF., Sipahi, R. (eds) Delay Systems. Advances in Delays and Dynamics, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-319-01695-5_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-01695-5_11
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-01694-8
Online ISBN: 978-3-319-01695-5
eBook Packages: EngineeringEngineering (R0)