Nonlinear Dynamics

, Volume 94, Issue 4, pp 3077–3100 | Cite as

Non-delayed synchronization of non-autonomous dynamical systems on Riemannian manifolds and its applications

  • Simone FioriEmail author
Original Paper


The present paper aims at tackling the non-delayed synchronization of two first-order, non-autonomous dynamical systems whose state spaces are (curved) Riemannian manifolds. The present research endeavor borrows notions from system theory, differential geometry, control theory and numerical calculus to design a general synchronization theory and a set of numerical methods to implement the devised synchronization theory on a computing platform. The features of these synchronization algorithms are illustrated by means of five sets of numerical experiments including the synchronization of the attitude of a fleet of flying bodies and the secure transmission of a message by the modulation of a system-generated carrier.


Control theory Differential geometry Non-autonomous dynamical system Numerical calculus Riemannian manifold System synchronization 



I wish to gratefully thank the anonymous reviewers, who contributed significantly to enrich the quality of the present paper by a number of interesting observation and suggestions.

Compliance with ethical standards

Conflicts of interest

The authors declare that they have no conflict of interest.

Supplementary material

Supplementary material 1 (mp4 3260 KB)


  1. 1.
    Abraham, R., Marsden, J.E., Ratiu, T.S.: Manifolds, Tensor Analysis, and Applications. Springer, Berlin (1988)CrossRefGoogle Scholar
  2. 2.
    Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, Berlin (1989)CrossRefGoogle Scholar
  3. 3.
    Blasius, B., Huppert, A., Stone, L.: Complex dynamics and phase synchronization in spatially extended ecological systems. Nature 399(6734), 354–359 (1999)CrossRefGoogle Scholar
  4. 4.
    Blekhman, I.I., Landa, P.S., Rosenblum, M.G.: Synchronization and chaotization in interacting dynamical systems. Appl. Mech. Rev. 48(11), 733–752 (1995)CrossRefGoogle Scholar
  5. 5.
    Bloch, A.M.: Nonholonomic Mechanics and Control. Springer, Berlin (2000)Google Scholar
  6. 6.
    Bullo, F., Lewis, A.D.: Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Mechanical Control Systems. Springer, Berlin (2005)CrossRefGoogle Scholar
  7. 7.
    Castrejón-Pita, A.A., Read, P.L.: Synchronization in a pair of thermally coupled rotating baroclinic annuli: understanding atmospheric teleconnections in the laboratory. Phys. Rev. Lett. 104, 204501 (2010)CrossRefGoogle Scholar
  8. 8.
    Chen, S., Zhao, L., Zhang, W., Shi, P.: Consensus on compact Riemannian manifolds. Inf. Sci. 268, 220–230 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Courant, R., Friedrichs, K., Lewy, H.: On the partial difference equations of mathematical physics. IBM J. 11, 215–234 (1967)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ding, K., Han, Q.-L.: Master-slave synchronization of nonautonomous chaotic systems and its application to rotating pendulums. Int. J. Bifurc. Chaos 22, 1250147 (21 pages) (2012)zbMATHGoogle Scholar
  11. 11.
    Ermentrout, G.B., Rinzel, J.: Beyond a pacemaker’s entrainment limit-phase walk-through. Am. J. Physiol. Regul. Integr. Comp. Physiol. 246(1), R102–R106 (1984)CrossRefGoogle Scholar
  12. 12.
    Ermentrout, B., Wechselberger, M.: Canards, clusters, and synchronization in a weakly coupled interneuron model. SIAM J. Appl. Dyn. Syst. 8, 253–278 (2009)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Fiori, S.: Nonlinear damped oscillators on Riemannian manifolds: fundamentals. J. Syst. Sci. Complex. 29(1), 22–40 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ge, Z.-M., Lin, T.-N.: Chaos, chaos control and synchronization of a gyrostat system. J. Sound Vib. 251(3), 519–542 (2002)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  16. 16.
    Hammouch, Z., Mekkaoui, T.: Chaos synchronization of a fractional nonautonomous system. Nonauton. Dyn. Syst. 1, 61–71 (2014)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Haxholdt, C., Kampmann, C., Mosekilde, E., Sterman, J.D.: Entrainment in a disaggregated economic long wave model. Syst. Dyn. Rev. 11, 177–198 (1995)CrossRefGoogle Scholar
  18. 18.
    Kloeden, P.E.: Synchronization of nonautonomous dynamical systems. Electron. J. Differ. Equ. 1, 1–10 (2003)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Lambert, J.D., Lambert, D.: Numerical Methods for Ordinary Differential Systems: The Initial Value Problem, Chapter 5. Wiley, New York (1991)zbMATHGoogle Scholar
  20. 20.
    Li, J.-S., Dasanayake, I., Ruths, J.: Control and synchronization of neuron ensembles. IEEE Trans. Autom. Control 58(8), 1919–1930 (2013)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Luo, A.C.J.: A theory for synchronization of dynamical systems. Commun. Nonlinear Sci. Numer. Simul. 14(5), 1901–1951 (2009)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Magdy, M.A., Ng, T.S.: Regulation and control effort in self-tuning controllers. IEE Proc. D Control Theory Appl. 133(6), 289–292 (1986)CrossRefGoogle Scholar
  23. 23.
    Masroor, S., Peng, C., Ali, Z.A.: Event triggered multi-agent consensus of DC motors to regulate speed by LQR scheme. Math. Comput. Appl. 22(14), 12 (2017)MathSciNetGoogle Scholar
  24. 24.
    Pikovsky, A., Rosenblum, M., Kurths, J.: Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge (2001)CrossRefGoogle Scholar
  25. 25.
    Sastry, S.: Nonlinear Systems: Analysis, Stability and Control. Springer, Berlin (1999)CrossRefGoogle Scholar
  26. 26.
    Schäfer, C., Rosenblum, M.G., Kurths, J., Abel, H.H.: Heartbeat synchronised with ventilation. Nature 392(6673), 239–240 (1998)CrossRefGoogle Scholar
  27. 27.
    Seuret, A., Dimarogonas, D.V., Johansson, K.H.: Consensus under communication delays. In: Proceedings of the 47th IEEE Conference on Decision and Control (Cancun, Mexico, December 2008), pp. 4922–4927 (2009)Google Scholar
  28. 28.
    Shooshtari, B.K., Forouzanfar, A.M., Molaei, M.R.: Identical synchronization of a non-autonomous unified chaotic system with continuous periodic switch. Springerplus 5(1), 1667 (2016)CrossRefGoogle Scholar
  29. 29.
    Stankovski, T.: Tackling the inverse problem for non-autonomous systems: application to the life sciences. In: Springer Theses. Springer (2014) ISBN: 9783319007526Google Scholar
  30. 30.
    Tyagi, A., Davis, J.W.: A recursive filter for linear systems on Riemannian manifolds. In: Proceedings of the 2008 IEEE conference on computer vision and pattern recognition (Anchorage, AK, June 2008) (2008)Google Scholar
  31. 31.
    Zeestraten, M.J.A., Havoutis, I., Silvério, J., Calinon, S., Caldwell, D.G.: An approach for imitation learning on Riemannian manifolds. IEEE Robot. Autom. Lett. 2(3), 1240–1247 (2017)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Dipartimento di Ingegneria dell’InformazioneUniversità Politecnica delle MarcheAnconaItaly

Personalised recommendations