Skip to main content
SpringerLink
Log in

Transients in the synchronization of asymmetrically coupled oscillator arrays

  • Regular Article
  • Session B: Papers I
  • Published:
The European Physical Journal Special Topics Aims and scope Submit manuscript

Abstract

We consider the transient behavior of a large linear array of coupled linear damped harmonic oscillators following perturbation of a single element. Our work is motivated by modeling the behavior of flocks of autonomous vehicles. We first state a number of conjectures that allow us to derive an explicit characterization of the transients, within a certain parameter regime Ω. As corollaries we show that minimizing the transients requires considering non-symmetric coupling, and that within Ω the computed linear growth in N of the transients is independent of (reasonable) boundary conditions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. P. Barooah, P.G. Mehta, J.P. Hespanha, IEEE Trans. Autom. Contr. 54, 2100 (2009)

    Article  MathSciNet  Google Scholar 

  2. C.E. Cantos, J.J.P. Veerman, D.K. Hammond, Eur. Phys. J. Special Topics 225, 1115 (2016)

    Google Scholar 

  3. R. Courant, D. Hilbert, Methods of Mathematical Physics, Vol. 2 (John Wiley and Sons, 1962)

  4. R.E. Chandler, R. Herman, E.W. Montroll, Op. Res. 6, 165 (1958)

    Article  MathSciNet  Google Scholar 

  5. K.-C. Chu, Transp. Sci. 8, 361 (1974)

    Article  Google Scholar 

  6. R. Herman, E.W. Montroll, R.B. Potts, R.W. Rothery, Op. Res. 7, 86 (1959)

    Article  MathSciNet  Google Scholar 

  7. I. Herman, D. Martinec, Z. Hurák, M. Sebek, PDdE-based Analysis of Vehicular Platoons with Spatio-Temporal Decoupling, Proc. 4th Workshop on Distributed Estimation and Control in Networked systems (Koblenz, Germany, 2013), p. 144

  8. D. Martinec, I. Herman, Z. Hurák, M. Sebek, Wave-absorber Vehicular Platoon Controller [ArXiv:1311.2095]

  9. R.H. Middleton, J.H. Braslavsky, Trans. Aut. Contr. 55, 1519 (2010)

    Article  MathSciNet  Google Scholar 

  10. D.A. Paley, A.K. Bahrani, American Control Conf., 4628 (2010)

  11. A. Pant, P. Seiler, K. Hedrick, IEEE Trans. Aut. Contr. 47, 403 (2002)

    Article  MathSciNet  Google Scholar 

  12. L.E. Peppard, IEEE Trans. Aut. Contr., 579 (1974)

  13. C. Robinson, Dynamical Systems, Stability, Symbolic Dynamics, and Chaos, 2nd ed. (CRC Press, 1999)

  14. W. Sullivan, Boundary Conditions and a One Lane Linear model of Traffic Flow (Portland State University, Master’s Thesis, 2010)

  15. D. Swaroop, J.K. Hedrick, C.C. Chien, P. Ioannou, Vehicle Syst. Dyn. 23, 597 (1994)

    Article  Google Scholar 

  16. D. Swaroop, J.K. Hedrick, Trans. Aut. Contr. 41, 349 (1996)

    Article  MathSciNet  Google Scholar 

  17. D. Swaroop, J.K. Hedrick, J. Dyn. Sys. Measurement Contr. 41, 462 (1999)

    Article  Google Scholar 

  18. F.M. Tangerman, J.J.P. Veerman, B.D. Stošić, Trans. Autom. Contr. 57, 2844 (2012)

    Article  Google Scholar 

  19. L.N. Trefethen, A.E. Trefethen, S.C. Reddy, T.A. Driscoll, Science 261, 578 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  20. L.N. Trefethen, SIAM REV. 39, 383 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  21. J.J.P. Veerman, J.S. Caughman, G. Lafferriere, A. Williams, J. Stat. Phys. 121, 901 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  22. J.J. P. Veerman, Stability of Large Flocks: an Example [arXiv:1002.0768] (2009)

  23. J.J.P. Veerman, F.M Tangerman, Impulse Stability of Large Linear Flocks: an Example [arXiv:1002.0782] (2009)

  24. J.J.P. Veerman, Symmetry and Stability of Homogeneous Flocks (a Position Paper), Proc. 1st Int’l Conf. on Pervasive and Embedded Computing and Communication Systems (Algarve, 2010)

  25. J.J.P. Veerman, B.D. Stosic, F.M. Tangerman, J. Stat. Phys. 137, 189 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  26. S.K. Yadlapalli, S. Darbha, K.R. Rajagopal, IEEE Trans. Autom. Contr. 51, 1315 (2006)

    Article  MathSciNet  Google Scholar 

  27. N.W. Ashcroft, N.D. Merman, Solid State Physics (Harcourt College Publishers, 1976)

Download references

Author information

Authors and Affiliations

  1. Maseeh Department of Mathematics & Statistics, Portland State University, Portland, USA

    C.E. Cantos

  2. Applied Mathematics Department, Oregon Institute of Technology, Wilsonville, USA

    D.K. Hammond

  3. Maseeh Department of Mathematics & Statistics, Portland State University, Portland, USA, CQCN, Dept of Physics, University of Crete, Heraklion, Greece

    J.J.P. Veerman

Authors
  1. C.E. Cantos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D.K. Hammond
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J.J.P. Veerman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to C.E. Cantos, D.K. Hammond or J.J.P. Veerman.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cantos, C., Hammond, D. & Veerman, J. Transients in the synchronization of asymmetrically coupled oscillator arrays. Eur. Phys. J. Spec. Top. 225, 1199–1209 (2016). https://doi.org/10.1140/epjst/e2016-02658-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjst/e2016-02658-y

Search

Navigation