Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Random Parametric Resonance in Time-Dependent Networks of Adaptive Frequency Oscillators

Discontinuity, Nonlinearity, and Complexity 3(3) (2014) 347--365 | DOI:10.5890/DNC.2014.09.009

Julio Rodriguez

tecData AG, Bahnhofstrasse 108/114, 9240 Uzwil, Switzerland

Download Full Text PDF



We consider a network of interacting phase oscillators endowed with adaptive mechanisms, leading the collective motion to a consensual dynamical state. Specifically, for a given network topology (i.e. an adjacency matrix) governing the mutual interactions, the adaptive mechanisms enable all oscillators to ultimately adopt a consensual frequency. Once reached, the consensual frequency subsists even if interactions between the oscillators are switched off. For the class of models we consider, the consensual frequency is independent of the network topology. Even though this independence might suggest that extension totime-dependent networks is straightforward, this is not true here. For time-dependent networks and spectra of the underlying Laplacian matrices, one may observe the emergence of more complex dynamics. Due to their high degree of complexity, these dynamics generally offer little hope for analytical tractability. In this paper, we focus on connected time-dependent networks with circulant adjacency matrices. The simple spectral structures and commutativity properties enjoyed by circulant matrices enable an analytical stability analysis of the consensus state. Ultimately, we are able to reduce the stability analysis to a dissipative harmonic oscillator with parametric pumping.


The author warmly thanks Dr. Martin ANDEREGG for the interesting discussions on commuting matrices. This paper was supported by the DFG-IRTG 1132 (Deutsche Forschungsgemeinschaft - International Research Training Group) under the project entitled “Internationales Graduiertenkolleg - Stochastics and Real World Models”.


  1. [1]  Acebrón, J., Spigler, R.(1998), Adaptive frequency model for phase-frequency synchronization in large populations of globally coupled nonlinear oscillators, Physical Review Letters, 81, 2229-2232.
  2. [2]  Arnold, V. (1976), Les Méthodes Mathématiques de la Mécanique Classique, Editions Mir, Moscou.
  3. [3]  Belykh, V. N., Belykh, I. V., and Hasler, M. (2004), Connection graph stability method for synchronized coupled chaotic systems, Physica D, 195, 159-187.
  4. [4]  Bhatia, N. P., Szegö, G. P. (1970), Stability Theory of Dynamical Systems, Springer , Berlin & Heidelberg & New York.
  5. [5]  Boccaletti, S., Hwang, D.-U., Chavez, M., Amann, A., Kurths, J., and Pecora, L. M. (2006), Synchronization in dynamical networks: Evolution along commutative graphs, Physical Review E, 74, 016102.
  6. [6]  Davis, P.J. (1979), Circulant Matrices, Wiley, New York.
  7. [7]  Duc, L. H., Ilchmann, A., Siegmund, S., and Taraba, P. (2006), On stability of linear time-varying second-order differential equations, Quarterly of Applied Mathematics, 64, 137-151.
  8. [8]  Ermentrout, B. (1991), An adaptive model for synchrony in the firefly pteroptyx malaccae, Journal of Mathematical Biology, 29, 571-585.
  9. [9]  Grimshaw, R. (1990), Nonlinear Ordinary Differential Equations, Blackwell Scientific Publications, Oxford.
  10. [10]  Lindenberg,K., Seshadri, V., andWest, B.J. (1980), Brownianmotion of harmonic systems with fluctuating parameters. ii. relation between moment instabilities and parametric resonance, Physical Review A, 22, 2171-2179.
  11. [11]  Lu, W., Atay, F. M., and Jost, J. (2007), Synchronization of discrete-time dynamical networks with time-varying couplings, SIAM Journal on Mathematical Analysis, 39, 1231-1259.
  12. [12]  Righetti, L., Buchli, J., and Ijspeert, A. (2006), Dynamic Hebbian Learning in Adaptive Frequency Oscillators, Physica D, 216, 269-281.
  13. [13]  Rodriguez, J. (2011), Networks of Self-Adaptive Dynamical Systems, PhD thesis, Ecole Polythechnique F∩ed∩erale de Lausanne.
  14. [14]  Rodriguez, J. (2013), Noise and Delays in Adaptive Interacting Oscillatory Systems, PhD thesis, Universitä Bielefeld.
  15. [15]  Rodriguez, J. and Hongler, M.-O. (2009), Networks of Mixed Canonical-Dissipative Systems and Dynamic Hebbian Learning, International Journal of Computational Intelligence Systems, 2, 140-146.
  16. [16]  Rodriguez, J. and Hongler, M.-O. (2012), Networks of Self-Adaptive Dynamical Systems, IMA Journal of Applied Mathematics. doi: 10.1093/imamat/hxs057.
  17. [17]  Rodriguez, J., Hongler, M.-O., Blanchard, Ph. (2013), Time Delayed Interactions in Networks of Self-Adapting Hopf Oscillators, ISRN Mathematical Analysis, 2013, doi:10.1155/2013/816353.
  18. [18]  Stilwell, D. J., Bollt, E. M., and Roberson, D. G. (2006), Sufficient Conditions for Fast Switching Synchronization in Time-Varying Network Topologies, SIAM Journal on Applied Dynamical Systems, 5, 140-156.
  19. [19]  Tanaka, H.-A., Lichtenberg, A. J., and Oishi, S. (1997), Self-synchronization of coupled oscillators with hysteretic responses, Physica D, 100, 279-300.
  20. [20]  Taylor, D., Ott, E., and Restrepo, J.G. (2010), Spontanous synchronization of coupled oscillator systems with frequency adaptation, Physical Review E, 81, 046214.
  21. [21]  Wang, L. and Wang, Q. (2011), Synchronization in complex networks with switching topology. Physics Letters A, 375, 3070-3074.