Advertisement

On the Limit of Large Couplings and Weighted Averaged Dynamics

  • Wim WiegerinckEmail author
  • Willem Burgers
  • Frank Selten
Chapter
Part of the Understanding Complex Systems book series (UCS)

Abstract

We consider a network of deterministic nonlinear oscillators with nonidentical parameters. Interactions between the different oscillators are linear, but the coupling coefficients for each interaction may differ. We consider the case where coupling coefficients are sufficiently large, so that the different oscillators will have their state variables strongly tied together and variables of the different oscillators will rapidly become (almost) synchronized. We will argue that the dynamics of the network is approximated by the dynamics of weighted averages of the vector fields of the different oscillators. Our focus of application will be on so-called supermodeling, a recently proposed model combination approach in which different existing models are dynamically coupled together aiming to improved performance. With large coupling theory, we are able to analyze and better understand earlier reported supermodeling results. Furthermore, we explore the behavior in partially coupled networks, in particular supermodeling with incomplete models, each modeling a different aspect of the truth. Results are illustrated numerically for the Lorenz 63 model.

Keywords

Ground Truth Pressure Field Nonlinear Oscillator Effective Parameter Large Coupling 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

This work has been supported by FP7 FET Open Grant # 266722 (SUMO project).

References

  1. 1.
    van den Berge, L.A., Selten, F.M., Wiegerinck, W., Duane, G.S.: A multi-model ensemble method that combines imperfect models through learning. Earth Syst. Dyn. 2(1), 161–177 (2011)CrossRefADSGoogle Scholar
  2. 2.
    Bishop, C.: Pattern Recognition and Machine Learning. Springer, Berlin (2006)zbMATHGoogle Scholar
  3. 3.
    Duane, G., Tribbia, J., Kirtman, B.: Consensus on long-range prediction by adaptive synchronization of models. In: Arabelos, D.N., Tscherning, C.C. (eds.) EGU General Assembly Conference Abstracts. EGU General Assembly Conference Abstracts, vol. 11, p. 13324 (2009)Google Scholar
  4. 4.
    Duane, G., Tribbia, J., Weiss, J., et al.: Synchronicity in predictive modelling: a new view of data assimilation. Nonlinear Process. Geophys. 13(6), 601–612 (2006)CrossRefADSGoogle Scholar
  5. 5.
    Kocarev, L., Shang, A., Chua, L.: Transition in dynamical regimes by driving: A unified method of control and synchronization of chaos. Int. J. Bifurcation Chaos 3(3), 479–483 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Lorenz, E.: Deterministic nonperiodic flow. Atmos. J. Sci. 20, 130–141 (1963)CrossRefADSGoogle Scholar
  7. 7.
    Lorenz, E.: Irregularity: A fundamental property of the atmosphere. Tellus A 36(2), 98–110 (1984)CrossRefADSGoogle Scholar
  8. 8.
    Olfati-Saber, R., Fax, J., Murray, R.: Consensus and cooperation in networked multi-agent systems. Proc. IEEE 95(1), 215–233 (2007)CrossRefGoogle Scholar
  9. 9.
    Pecora, L., Carroll, T.: Synchronization in chaotic systems. Phys. Rev. Lett. 64(8), 821–824 (1990)MathSciNetCrossRefADSGoogle Scholar
  10. 10.
    Pikovsky, A., Rosenblum, M., Kurths, J.: Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge Nonlinear Science Series, vol. 12. Cambridge University Press, Cambridge (2003)Google Scholar
  11. 11.
    Rössler, O.: An equation for continuous chaos. Phys. Lett. A 57(5), 397–398 (1976)CrossRefADSGoogle Scholar
  12. 12.
    Sun, J., Bollt, E., Nishikawa, T.: Master stability functions for coupled nearly identical dynamical systems. Europhys. Lett. 85, 60,011 (2009)Google Scholar
  13. 13.
    Tebaldi, C., Knutti, R.: The use of the multi-model ensemble in probabilistic climate projections. Phil. Trans. Roy. Soc. A: Math. Phys. Eng. Sci. 365(1857), 2053 (2007)MathSciNetCrossRefADSGoogle Scholar
  14. 14.
    Wiegerinck, W., Selten, F.: Supermodeling: Combining imperfect models through learning. In: NIPS Workshop on Machine Learning for Sustainability (MLSUST) (2011). URL http://people.csail.mit.edu/kolter/mlsust11/lib/exe/fetch.php?media=wiegerinck-mlsust.pdf
  15. 15.
    Yang, S., Baker, D., Li, H., Cordes, K., Huff, M., Nagpal, G., Okereke, E., Villafañe, J., Kalnay, E., Duane, G.: Data assimilation as synchronization of truth and model: Experiments with the three-variable lorenz system. J. Atmos. Sci. 63(9), 2340–2354 (2006)CrossRefADSGoogle Scholar
  16. 16.
    Yu, W., Chen, G., Cao, M., Kurths, J.: Second-order consensus for multiagent systems with directed topologies and nonlinear dynamics. IEEE Trans. Syst. Man Cybernet. B: Cybernet. 40(3), 881–891 (2010)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Donders Institute for Brain, Cognition and BehaviourRadboud University NijmegenNijmegenThe Netherlands
  2. 2.Royal Netherlands Meteorological InstituteDe BiltThe Netherlands

Personalised recommendations