Biological Cybernetics

, Volume 92, Issue 1, pp 38–53 | Cite as

On partial contraction analysis for coupled nonlinear oscillators

Article
First Online:
Received:
Accepted:

Abstract

We describe a simple yet general method to analyze networks of coupled identical nonlinear oscillators and study applications to fast synchronization, locomotion, and schooling. Specifically, we use nonlinear contraction theory to derive exact and global (rather than linearized) results on synchronization, antisynchronization, and oscillator death. The method can be applied to coupled networks of various structures and arbitrary size. For oscillators with positive definite diffusion coupling, it can be shown that synchronization always occurs globally for strong enough coupling strengths, and an explicit upper bound on the corresponding threshold can be computed through eigenvalue analysis. The discussion also extends to the case when network structure varies abruptly and asynchronously, as in “flocks” of oscillators or dynamic elements.

Keywords

Network Structure Coupling Strength Diffusion Coupling Nonlinear Oscillator Study Application 
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.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aronson DG, Ermentrout GB, Kopell N (1990) Amplitude response of coupled oscillators. Physica D 41:403–449Google Scholar
  2. 2.
    Bar-Eli K (1985) On the stability of coupled chemical oscillators. Physica D 14:242–252Google Scholar
  3. 3.
    Barahona M, Pecora LM (2002) Synchronization in small-world systems. Phys Rev Lett 89(5):054101Google Scholar
  4. 4.
    Bay John S, Hemami H (1987) Modeling of a neural pattern generator with coupled nonlinear oscillators. IEEE Trans Biomed Eng 34(4):297–306Google Scholar
  5. 5.
    Bressloff PC, Coombes S, Souza B de (1997) Dynamics of a ring of pulse-coupled oscillators: group theoretic approach. Phys Rev Lett 79(15):2791–2794Google Scholar
  6. 6.
    Brody CD, Hopfield JJ (2003) Simple networks for spike-timing-based computation, with application to olfactory processing. Neuron 37:843–852Google Scholar
  7. 7.
    Bruckstein AM, Mallows CL, Wagner IA (1997) Cooperative cleaners: a study in ant-robotics. Am Math Monthly 104(4):323–343Google Scholar
  8. 8.
    Chakraborty T, Rand RH (1988) The transition from phase locking to drift in a system of two weakly coupled Van der Pol oscillators. Int J Nonlin Mech 23:369–376Google Scholar
  9. 9.
    Chua LO (1998) CNN: a paradigm for complexity. World Scientific, Hong KongGoogle Scholar
  10. 10.
    Collins JJ, Stewart IN (1993a) Coupled nonlinear oscillators and the symmetries of animal gaits. J Nonlin Sci 3:349–392Google Scholar
  11. 11.
    Collins JJ, Stewart IN (1993b) Hexapodal gaits and coupled nonlinear oscillator models. Biol Cybern 68:287–298Google Scholar
  12. 12.
    Combescot C, Slotine JJE (2000) A study of coupled oscillators. MIT Nonlinear Systems Laboratory, Report MIT-NSL-000801Google Scholar
  13. 13.
    Coombes S (2001) Phase-locking in networks of pulse-coupled McKean relaxation oscillators. Physica D 160(3–4):173–188Google Scholar
  14. 14.
    Cutts C, Speakman J (1994) Energy savings in formation flight of pink-footed geese. J Exp Biol 189:251–261Google Scholar
  15. 15.
    Dragoi V, Grosu I (1998) Synchronization of locally coupled neural oscillators. Neural Process Lett 7(2):199–210Google Scholar
  16. 16.
    Fiedler M (1973) Algebraic connectivity of graphs. Czechoslovak Math J 23(98):298–305Google Scholar
  17. 17.
    FitzHugh RA (1961) Impulses and physiological states in theoretical models of nerve membrane. Biophys J 1:445–466Google Scholar
  18. 18.
    Godsil C, Royle G (2001) Algebraic graph theory. Springer, Berlin Heidelberg New YorkGoogle Scholar
  19. 19.
    Golubitsky M, Stewart I, Buono PL, Collins JJ (1998) A modular network for legged locomotion. Physica D 115:56–72Google Scholar
  20. 20.
    Golubitsky M, Stewart I, Buono PL, Collins JJ (1999) The role of symmetry in locomotor central pattern generators and animal gaits. Nature 401: 693–695Google Scholar
  21. 21.
    Golubitsky M, Stewart I (2002) Patterns of oscillation in coupled cell systems. Geometry dynamics and mechanics: 60th birthday volume for J.E. Marsden. Springer, Berlin Heidelberg New York, pp 243–286Google Scholar
  22. 22.
    Hahnloser RH, Seung HS, Slotine JJE (2003) Permitted and forbidden sets in symmetric threshold-linear networks. Neural Comput 15:621–638Google Scholar
  23. 23.
    Hirsch M, Smale S (1974) Differential equations, dynamical systems, and linear algebra. Academic, New YorkGoogle Scholar
  24. 24.
    Hopfield JJ, Brody CD (2001) What is a moment? Transient synchrony as a collective mechanism for spatiotemporal integration. Proc Natl Acad Sci USA 98:1282–1287Google Scholar
  25. 25.
    Horn RA, Johnson CR (1985) Matrix analysis. Cambridge University Press, Cambridge, UKGoogle Scholar
  26. 26.
    Horn RA, Johnson CR (1989) Topics in matrix analysis. Cambridge University Press, Cambridge, UKGoogle Scholar
  27. 27.
    Jadbabaie A, Lin J, Morse AS (2003) Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans Automat Control 48(6):988–1001Google Scholar
  28. 28.
    Ketterle W (2002) When atoms behave as waves: Bose-Eintein condensation and the atom laser. Rev Mod Phys 74:1131–1151Google Scholar
  29. 29.
    Kopell N, Ermentrout GB (1986) Symmetry and phase-locking in chains of weakly coupled oscillators. Commun Pure Appl Math 39:623–660Google Scholar
  30. 30.
    Kopell N, Somers DC (1995) Anti-phase solutions in relaxation oscillators coupled through excitatory interactions. J Math Biol 33:261–280Google Scholar
  31. 31.
    Kopell N (2000) We got rhythm: dynamical systems of the nervous system. (published version of the 1998 Gibbs lecture of the AMS) 47:6–16Google Scholar
  32. 32.
    Krishnaprasad PS, Tsakiris D (2001) Oscillations, SE(2)-snakes and motion control: a study of the roller racer. Dynam Stabil Syst 16(4):347–397Google Scholar
  33. 33.
    Kuramoto Y (1984) Chemical oscillations, wave, and turbulence. Springer, Berlin Heidelberg New YorkGoogle Scholar
  34. 34.
    Langbort C, D’Andrea R (2004) Distributed control of spatially reversible interconnected systems with boundary conditions. SIAM J Control Optimizat (in press)Google Scholar
  35. 35.
    Latham PE, Deneve S, Pouget A (2003) Optimal computation with attractor networks. J Physiol Paris 97:683–694Google Scholar
  36. 36.
    Leonard NE, Fiorelli E (2001) Virtual leaders, artificial potentials and coordinated control of groups. In: 40th IEEE conference on decision and controlGoogle Scholar
  37. 37.
    Lin Z, Broucke M, Francis B (2004) Local control strategies for groups of mobile autonomous agents. IEEE Trans Automat Control 49(4):622–629Google Scholar
  38. 38.
    Loewenstein Y, Sompolinsky H (2002) Oscillations by symmetry breaking in homogeneous networks with electrical coupling. Phys Rev E 65:1–11Google Scholar
  39. 39.
    Lohmiller W (1999) Contraction analysis of nonlinear systems. PhD thesis, Department of Mechanical Engineering, MIT, Cambridge, MAGoogle Scholar
  40. 40.
    Lohmiller W, Slotine JJE (1998) On contraction analysis for nonlinear systems. Automatica 34(6):683–696Google Scholar
  41. 41.
    Lohmiller W, Slotine JJE (2000) Control system design for mechanical systems using contraction theory. IEEE Trans Automat Control 45(5):984–989Google Scholar
  42. 42.
    Lohmiller W, Slotine JJE (2000) Nonlinear process control using contraction theory. AIChE J 46(3):588–596Google Scholar
  43. 43.
    Manor Y, Nadim F, Ritt J, Epstein S, Marder, E, Kopell N (1999) Network oscillations generated by balancing asymmetric inhibition in passive neurons. J Neurosci 19:2765–2779Google Scholar
  44. 44.
    May RM, Gupta S, McLean AR (2001) Infectious disease dynamics: what characterizes a successful invader? Phil Trans R Soc Lond B 356:901–910Google Scholar
  45. 45.
    Micchelli CA (1986) Interpolation of scattered data. Construct Approximat 2:11–22Google Scholar
  46. 46.
    Mirollo RE, Strogatz SH (1990) Synchronization of pulse-coupled biological oscillators. SIAM J Appl Math 50:1645–1662Google Scholar
  47. 47.
    Mohar B (1991) Eigenvalues, diameter, and mean distance in graphs. Graphs Combinator 7:53–64Google Scholar
  48. 48.
    Murray JD (1993) Mathematical biology. Springer, Berlin Heidelberg New YorkGoogle Scholar
  49. 49.
    Nagumo J, Arimoto S, Yoshizawa S (1962) An active pulse transmission line simulating nerve axon. Proc Inst Radio Eng 50:2061–2070Google Scholar
  50. 50.
    Neuenschwander S, Castelo-Branco M, Baron J, Singer W (2002) Feed-forward synchronization: propagation of temporal patterns along the retinothalamocortical pathway. Philos Trans Biol Sci 357(1428):1869–1876Google Scholar
  51. 51.
    Olfati-Saber R, Murray RM (2003) Agreement problems in networks with directed graphs and switching topology. In: IEEE conference on decision and controlGoogle Scholar
  52. 52.
    Pecora LM, Carroll TL (1990) Synchronization in chaotic systems. Phys Rev Lett 64:821–824Google Scholar
  53. 53.
    Pecora LM (1998) Synchronization conditions and desynchronizing patterns in coupled limit-cycle and chaotic systems. Phys Rev E 64(8):821–824Google Scholar
  54. 54.
    Pecora LM, Carroll TL (1998) Master stability functions for synchronized coupled systems. Phys Rev Lett 80(10):2109–2112Google Scholar
  55. 55.
    Pikovsky A, Rosenblum M, Kurths J (2003) Synchronization: a universal concept in nonlinear sciences. Cambridge University Press, Cambridge, UKGoogle Scholar
  56. 56.
    Pogromsky A, Santoboni G, Nijmeijer H (2002) Partial synchronization: from symmetry towards stability. Physica D 172:65–87Google Scholar
  57. 57.
    Rand RH, Holmes PJ (1980) Bifurcation of periodic motions in two weakly coupled van der pol oscillators. Int J Nonlin Mech 15:387–399Google Scholar
  58. 58.
    Ravasz ZNE, Vicsek T, Brechet T, Barabási AL (2000) Physics of the rhythmic applause. Phys Rev E 61(6):6987–6992Google Scholar
  59. 59.
    Reddy DVR, Sen A, Johnston GL (1998) Time delay induced death in coupled limit cycle oscillators. Phys Rev Lett 80(23):5109–5112Google Scholar
  60. 60.
    Reynolds C (1987) Flocks, birds, and schools: a distributed behavioral model. Comput Graph 21:25–34Google Scholar
  61. 61.
    Rock I, Palmer S (1990) The legacy of gestalt psychology. Sci Am 263:84–90Google Scholar
  62. 62.
    Seiler P, Pant A, Hedrick JK (2003) A systems interpretation for observations of bird V-formations. J Theor Biol 221:279–287Google Scholar
  63. 63.
    Seung HS (1998) Continuous attractors and oculomotor control. Neural Netw 11:1253–1258Google Scholar
  64. 64.
    Singer W (1999) Neuronal synchrony: a versatile code for the definition of relations. Neuron 24:49–65Google Scholar
  65. 65.
    Slotine JJE, Li W (1991) Applied nonlinear control. Prentice-Hall, Englewood Cliffs, NJGoogle Scholar
  66. 66.
    Slotine JJE, Lohmiller W (2001) Modularity, evolution, and the binding problem: a view from stability theory. Neural Netw 14(2):137–145Google Scholar
  67. 67.
    Slotine JJE (2003) Modular stability tools for distributed computation and control. Int J Adapt Control Signal Process 17(6):397–416Google Scholar
  68. 68.
    Slotine JJE, Wang W (2004) A study of synchronization and group cooperation using partial contraction theory. In: Morse S, Leonard N, Kumar V (eds) Proceedings of Block Island workshop on cooperative control. Lecture notes in control and information science, vol 309. Springer, Berlin Heidelberg New YorkGoogle Scholar
  69. 69.
    Smale S (1976) A mathematical model of two cells via Turing’s equation. In: The Hopf bifurcation and its applications. Springer, Berlin Heidelberg New York, pp 354–367Google Scholar
  70. 70.
    Somers DC, Kopell N (1993) Rapid synchronization through fast threshold modulation. Biol Cybern 68:393–407Google Scholar
  71. 71.
    Somers DC, Kopell N (1995) Waves and synchrony in networks of oscillators of relaxation and non-relaxation type. Physica D 89(1–2):169–183Google Scholar
  72. 72.
    Storti DW, Rand RH (1982) Dynamics of two strongly coupled Van der Pol oscillators. Int J Nonlin Mech 17:143–152Google Scholar
  73. 73.
    Strogatz SH, Stewart I (1993) Coupled oscillators and biological synchronization. Sci Am 269(6):102–109Google Scholar
  74. 74.
    Strogatz SH (1994) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. Addison-Wesley, Reading, MAGoogle Scholar
  75. 75.
    Strogatz SH (2000) From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Phys D Nonlin Phenom 143:1–4Google Scholar
  76. 76.
    Strogatz SH (2001) Exploring complex networks. Nature 410:268–276Google Scholar
  77. 77.
    Strogatz S (2003) Sync: the emerging science of spontaneous order. Hyperion, New YorkGoogle Scholar
  78. 78.
    Tanner H, Jadbabaie A, Pappas GJ (2003a) Stable flocking of mobile agents, Part I: Fixed topology; Part II: Dynamic topology. In: IEEE conference on decision and control, MauiGoogle Scholar
  79. 79.
    Tanner H, Jadbabaie A, Pappas GJ (2003b) Coordination of multiple autonomous vehicles. In: IEEE Mediterranean conference on control and automation, Rhodes, GreeceGoogle Scholar
  80. 80.
    Turing A (1952) The chemical basis of morphogenesis. Philos Trans R Soc B 237(641):37–72Google Scholar
  81. 81.
    Vicsek T, Czirok A, Jacob EB, Cohen I, Schochet O (1995) Novel type of phase transitions in a system of self-driven particles. Phys Rev Lett 75:1226–1229Google Scholar
  82. 82.
    Wang W, Slotine JJE (2004) A study of continuous attractors and singularly perturbed systems using contraction theory. MIT Nonlinear Systems Laboratory Report 010804, Cambridge, MAGoogle Scholar
  83. 83.
    Watts DJ, Strogatz SH (1998) Collective dynamics of ‘Small-World’ networks. Nature 393:440–442Google Scholar
  84. 84.
    Watts DJ (1999) Small Worlds. Princeton University Press, Princeton, NJGoogle Scholar
  85. 85.
    Winfree AT (1967) Biological rhythms and the behavior of populations of coupled oscillators. J Theor Biol 16:15–42Google Scholar
  86. 86.
    Winfree AT (2000) The geometry of biological time. Springer, Berlin Heidelberg New YorkGoogle Scholar
  87. 87.
    Wolf JA, Schroeder LF, Finkel LH (2001) Computational modeling of medium spiny projection neurons in nucleus accumbens: toward the cellular mechanisms of afferent stream integration. Proc IEEE 89(7):1083–1092Google Scholar
  88. 88.
    Yen SC, Menschik ED, Finkel LH (1999) Perceptual grouping in striate cortical networks mediated by synchronization and desynchronization. Neurocomputing 26(7):609– 616Google Scholar

Copyright information

© Springer-Verlag 2004

Authors and Affiliations

  1. 1.Nonlinear Systems LaboratoryMassachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations