Journal of Nonlinear Science

, Volume 21, Issue 6, pp 875–895 | Cite as

On Bifurcations in Nonlinear Consensus Networks

  • Vaibhav SrivastavaEmail author
  • Jeff Moehlis
  • Francesco Bullo


The theory of consensus dynamics is widely employed to study various linear behaviors in networked control systems. Moreover, nonlinear phenomena have been observed in animal groups, power networks and in other networked systems. These observations inspire the development in this paper of three novel approaches to define distributed nonlinear dynamical interactions. The resulting dynamical systems are akin to higher-order nonlinear consensus systems. Over connected undirected graphs, the resulting dynamical systems exhibit various interesting behaviors that we rigorously characterize.


Consensus network Networked systems Bifurcation theory 

Mathematics Subject Classification (2000)

34C23 34K18 68M12 


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  1. Arcak, M.: Passivity as a design tool for group coordination. IEEE Trans. Autom. Control 52(8), 1380–1390 (2007) MathSciNetCrossRefGoogle Scholar
  2. Ashwin, P., Burylko, O., Maistrenko, Y.: Bifurcation to heteroclinic cycles and sensitivity in three and four coupled phase oscillators. Physica D 237(4), 454–466 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  3. Bogacz, R., Brown, E., Moehlis, J., Holmes, P., Cohen, J.D.: The physics of optimal decision making: A formal analysis of performance in two-alternative forced choice tasks. Psychol. Rev. 113(4), 700–765 (2006) CrossRefGoogle Scholar
  4. Bullo, F., Cortés, J., Martínez, S.: Distributed Control of Robotic Networks. Applied Mathematics Series. Princeton University Press, Princeton (2009). Available at zbMATHGoogle Scholar
  5. Couzin, I.D., Krause, J., Franks, N.R., Levin, S.A.: Effective leadership and decision-making in animal groups on the move. Nature 433(7025), 513–516 (2005) CrossRefGoogle Scholar
  6. Dionne, B., Golubitsky, M., Stewart, I.: Coupled cells with internal symmetry. Nonlinearity 9, 559–599 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  7. Dörfler, F., Francis, B.: Geometric analysis of the formation problem for autonomous robots. IEEE Trans. Automat. Contol 55(10), 2379–2384 (2010) CrossRefGoogle Scholar
  8. Fax, J.A., Murray, R.M.: Information flow and cooperative control of vehicle formations. IEEE Trans. Autom. Control 49(9), 1465–1476 (2004) MathSciNetCrossRefGoogle Scholar
  9. Jadbabaie, A., Lin, J., Morse, A.S.: Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans. Autom. Control 48(6), 988–1001 (2003) MathSciNetCrossRefGoogle Scholar
  10. Khalil, H.K.: Nonlinear Systems, 3rd edn. Prentice Hall, New York (2002) zbMATHGoogle Scholar
  11. Kimura, M., Moehlis, J.: Novel vehicular trajectories for collective motion from coupled oscillator steering control. SIAM J. Appl. Dyn. Syst. 7(4), 1191–1212 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  12. Kwatny, H., Pasrija, A., Bahar, L.: Static bifurcations in electric power networks: Loss of steady-state stability and voltage collapse. IEEE Trans. Circuits Syst. 33(10), 981–991 (1986) zbMATHCrossRefGoogle Scholar
  13. Lin, J., Morse, A.S., Anderson, B.D.O.: The multi-agent rendezvous problem. Part 1: The synchronous case. SIAM J. Control Optim. 46(6), 2096–2119 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  14. Lorenz, J.: Continuous opinion dynamics under bounded confidence: A survey. Int. J. Mod. Phys. C 18(12), 1819–1838 (2007) zbMATHCrossRefGoogle Scholar
  15. Nabet, B., Leonard, N.E., Couzin, I.D., Levin, S.A.: Dynamics of decision making in animal group motion. J. Nonlinear Sci. 19(4), 399–435 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  16. Olfati-Saber, R., Fax, J.A., Murray, R.M.: Consensus and cooperation in networked multi-agent systems. Proc. IEEE 95(1), 215–233 (2007) CrossRefGoogle Scholar
  17. Olfati-Saber, R., Murray, R.M.: Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans. Autom. Control 49(9), 1520–1533 (2004) MathSciNetCrossRefGoogle Scholar
  18. Olien, L., Bélair, J.: Bifurcations, stability, and monotonicity properties of a delayed neural network model. Physica D 102(3-4), 349–363 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  19. Papachristodoulou, A., Jadbabaie, A.: Synchronization in oscillator networks with heterogeneous delays, switching topologies and nonlinear dynamics. In: IEEE Conf. on Decision and Control, San Diego, CA, December 2006, pp. 4307–4312 (2006) Google Scholar
  20. Poulakakis, I., Scardovi, L., Leonard, N.E.: Coupled stochastic differential equations and collective decision making in the Two-Alternative Forced-Choice task. In: American Control Conference, pp. 69–74 (2010) Google Scholar
  21. Roxin, A., Ledberg, A.: Neurobiological models of two-choice decision making can be reduced to a one-dimensional nonlinear diffusion equation. PLoS Comput. Biol. 4(3), e1000046 (2008) MathSciNetCrossRefGoogle Scholar
  22. Spanos, D.P., Olfati-Saber, R., Murray, R.M.: Approximate distributed Kalman filtering in sensor networks with quantifiable performance. In: Symposium on Information Processing of Sensor Networks, Los Angeles, CA, April 2005, pp. 133–139 (2005) CrossRefGoogle Scholar
  23. Srivastava, V., Moehlis, J., Bullo, F.: On bifurcations in nonlinear consensus networks. In: American Control Conference, Baltimore, MD, June 2010, pp. 1647–1652 (2010) Google Scholar
  24. Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Perseus Books Group, New York City (2000) Google Scholar
  25. Tanner, H.G., Jadbabaie, A., Pappas, G.J.: Flocking in fixed and switching networks. IEEE Trans. Autom. Control 52(5), 863–868 (2007) MathSciNetCrossRefGoogle Scholar
  26. Wei, J., Ruan, S.: Stability and bifurcation in a neural network model with two delays. Physica D 130(3-4), 255–272 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  27. Zou, F., Nossek, J.A.: Bifurcation and chaos in cellular neural networks. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 40(3), 166–173 (1993) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Vaibhav Srivastava
    • 1
    Email author
  • Jeff Moehlis
    • 1
  • Francesco Bullo
    • 1
  1. 1.Department of Mechanical EngineeringUniversity of California Santa BarbaraSanta BarbaraUSA

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