Skip to main content
SpringerLink
Log in

Oscillators as Systems and Synchrony as a Design Principle

  • Chapter
Current Trends in Nonlinear Systems and Control

Part of the book series: Systems and Control: Foundations & Applications ((SCFA))

Summary

The chapter presents an expository survey of ongoing research by the author on a system theory for oscillators. Oscillators are regarded as open systems that can be interconnected to robustly stabilize ensemble phenomena characterized by a certain level of synchrony. The first part of the chapter provides examples of design (stabilization) problems in which synchrony plays an important role. The second part of the chapter shows that dissipativity theory provides an interconnection theory for oscillators.

This chapter presents research results of the Belgian Program on Interuniversity Attraction Poles, initiated by the Belgian Federal Science Policy Office. The scientific responsibility rests with its author.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Angeli D (2002) A Lyapunov approach to incremental stability properties. IEEE Trans. on Automatic Control 47:410–422

    Article  MathSciNet  Google Scholar 

  2. Brown E, Moehlis J, Holmes P (2004) On phase reduction and response dynamics of neural oscillator populations. Neural Computation 16(4):673–715

    Article  MATH  Google Scholar 

  3. Gérard M, Sepulchre R (2004) Stabilization through weak and occasional interactions: a billiard benchmark. In: 6th IFAC Symposium on Nonlinear Control Systems, Stuggart, Germany

    Google Scholar 

  4. Goldbeter A (1996) Biochemical oscillations and cellular rhythms. Cambridge University Press, Cambridge, UK

    MATH  Google Scholar 

  5. Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiology 117:500–544

    Google Scholar 

  6. Holmes PJ (1982) The dynamics of repeated impacts with a sinusoidally vibrating table. Journal of Sound and Vibration 84(2):173–189

    Article  MATH  MathSciNet  Google Scholar 

  7. Holmes PJ, Full RJ, Koditschek D, Guckenheimer J (2006) Dynamics of legged locomotion: Models, analyses and challenges. SIAM Review, to appear

    Google Scholar 

  8. Hopfield JJ (1982) Neural networks and physical systems with emergent collective computational abilities. Proceedings of the National Academy of Sciences USA 79:2554–2558

    Article  MathSciNet  Google Scholar 

  9. Justh EW, Krishnaprasad PS (2002) A simple control law for UAV formation flying Technical report Institute for Systems Research, University of Maryland

    Google Scholar 

  10. Justh EW, Krishnaprasad PS (2003) Steering laws and continuum models for planar formations. In: IEEE 42nd Conf. on Decision and Control, Maui, HI

    Google Scholar 

  11. Kumar V, Leonard N, Morse A (eds) (2004) Cooperative control, vol. 309 of Lecture Notes in Control and Information Sciences. Springer-Verlag, London

    MATH  Google Scholar 

  12. Kuramoto Y (1984) Chemical oscillations, waves, and turbulence. Springer-Verlag, London

    MATH  Google Scholar 

  13. Lehtihet BN, Miller HE (1986) Numerical study of a billiard in a gravitational field. Physica 21D:93–104

    MathSciNet  Google Scholar 

  14. Lohmiller W, Slotine JJE (1998) On contraction analysis for nonlinear systems. Automatica 34(6):683–696

    Article  MATH  MathSciNet  Google Scholar 

  15. Ogren P, Fiorelli E, Leonard NE (2004) Cooperative control of mobile sensor networks: Adaptive gradient climbing in a distributed environment. IEEE Trans. on Automatic Control 49(8):1292–1302

    Article  MathSciNet  Google Scholar 

  16. Ortega R, Loria A, Nicklasson PJ, Sira-Ramirez H (1998) Passivity-based Control of Euler-Lagrange Systems. Springer-Verlag, London

    Google Scholar 

  17. Paley D, Leonard N, Sepulchre R (2004) Collective motion: bistability and trajectory tracking. In: IEEE 43rd Conf. on Decision and Control, Atlantis, Bahamas

    Google Scholar 

  18. Pavlov A, Pogromsky AY, van de Wouw N, Nijmeijer H (2004) Convergent dynamics, a tribute to Boris Pavlovich Demidovich. Systems & Control Letters 52:257–261

    Article  MathSciNet  Google Scholar 

  19. Pogromsky A (1998) Passivity based design of synchronizing systems. Int. J. Bifurcation and Chaos 8:295–319

    Article  MATH  MathSciNet  Google Scholar 

  20. Ronsse R, Lefevre P, Sepulchre R (2004a) Open-loop stabilization of 2D impact juggling. In: 6th IFAC Symposium on Nonlinear Control Systems, Stuggart, Germany

    Google Scholar 

  21. Ronsse R, Lefevre P, Sepulchre R (2004b) Sensorless stabilization of bounce juggling Technical report, Department of Electrical Engineering and Computer Science, University of Liège

    Google Scholar 

  22. Sastry S (1999) Nonlinear systems. Springer-Verlag, London

    MATH  Google Scholar 

  23. Sepulchre R, Gérard M (2003) Stabilization of periodic orbits in a wedge billiard. 1568–1573. In: IEEE 42nd Conf. on Decision and Control Maui, HI

    Google Scholar 

  24. Sepulchre R, Jankovic M, Kokotovic P (1997) Constructive nonlinear control. Springer-Verlag, London, UK

    MATH  Google Scholar 

  25. Sepulchre R, Paley D, Leonard N (2004a) Collective motion and oscillator synchronization. 189–205. In: Cooperative control, Kumar V, Leonard N, Morse A (eds), Springer-Verlag, London

    Google Scholar 

  26. Sepulchre R, Paley D, Leonard N (2004b) Collective stabilization of n steered particles in the plane, in preparation

    Google Scholar 

  27. Sepulchre R, Stan GB (2004) Feedback mechanisms for global oscillations in Lure systems. Systems & Control Letters 54(8):809–818

    Article  MathSciNet  Google Scholar 

  28. Slotine JJ, Wang W (2004) A study of synchronization and group cooperation using partial contraction theory. 207–228. In: Cooperative control, Kumar V, Leonard N, Morse A (eds) Springer-Verlag, London

    Google Scholar 

  29. Stan GB, Sepulchre R (2004) Dissipativity theory and global analysis of limit cycles, Technical report, Department of Electrical Engineering and Computer Science, University of Liège

    Google Scholar 

  30. Strogatz S (2003) Sync: the emerging science of spontaneous order. Hyperion

    Google Scholar 

  31. Strogatz SH (2000) From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D 143:1–20

    Article  MATH  MathSciNet  Google Scholar 

  32. van der Schaft AJ (2000) L 2-gain and passivity techniques in nonlinear control. Springer-Verlag, London

    MATH  Google Scholar 

  33. Willems JC (1972) Dissipative dynamical systems. Arch. Rational Mechanics and Analysis 45:321–393

    Article  MATH  MathSciNet  Google Scholar 

  34. Wilson H (1999) Spikes, decisions, and actions. Oxford University Press, Oxford, UK

    MATH  Google Scholar 

  35. Winfree A (2000) The geometry of biological time, second edition. Springer-Verlag, London

    MATH  Google Scholar 

  36. Wojtkowski MP (1998) Hamiltonian systems with linear potential and elastic constraints. Communications in Mathematical Physics 194:47–60

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Electrical Engineering and Computer Science Institute, Montefiore B28, B-4000, Liège, Belgium

    Rodolphe Sepulchre

Authors
  1. Rodolphe Sepulchre
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dipartimento di Informatica Sistemi e Produzione, Università di Roma “Tor Vergata”, Via del Politecnico 1, I-00133, Roma, Italia

    Laura Menini  & Luca Zaccarian  & 

  2. Department of Electrical and Computer Engineering, University of New Mexico, EECE Building MSC01 1100, Albuquerque, NM, 87131-0001, USA

    Chaouki T. Abdallah

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Birkhäuser Boston

About this chapter

Cite this chapter

Sepulchre, R. (2006). Oscillators as Systems and Synchrony as a Design Principle. In: Menini, L., Zaccarian, L., Abdallah, C.T. (eds) Current Trends in Nonlinear Systems and Control. Systems and Control: Foundations & Applications. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4470-9_7

Download citation

Publish with us

Policies and ethics

Search

Navigation