Optimal synchronization of oscillatory chemical reactions with complex pulse, square, and smooth waveforms signals maximizes Tsallis entropy

Hisa-Aki Tanaka, Isao Nishikawa, Jürgen Kurths, Yifei Chen, and István Z. Kiss
Europhysics Letters, 2015


Synchronization, coupled oscillators, Nonlinear dynamics and chaos, Neural networks


We show that the mathematical structure of Tsallis entropy underlies an important and ubiquitous problem in nonlinear science related to an efficient synchronization of weakly forced nonlinear oscillators. The maximization of the locking range of oscillators with the use of phase models is analyzed with general constraints that encompass forcing waveform power, magnitude, or area. The optimization problem is then recasted as a general form of Tsallis entropy maximization. The solution of these optimization problems is shown to be a direct consequence from Hölder’s inequality. The resulting new maximization principle is confirmed in numerical simulations and experiments with chemical oscillations with nickel electrodissolution. While weakly nonlinear oscillators have generic optimal waveforms (sinusoidal, 50% duty cycle square wave, and equally paced bipolar pulses for power-, area-, and magnitude-constraints, respectively), strongly nonlinear oscillators require more complex waveforms such as smooth, square, and pulse ones.

Download PDF

Figures at a glance


  1. HARADA T., TANAKA H.-A., HANKINS M. J. and KISS I. Z., Phys. Rev. Lett., 105 (2010) 088301.
  2. ZLOTNIK A., CHEN Y., KISS I. Z., TANAKA H.-A. and LI J. S., Phys. Rev. Lett., 111 (2013) 024102.
  3. BAGHERI N., STELLING J. and DOYLE F. J., PLoS Comput. Biol., 4 (2008) e1000104.
  4. AIHARA K., MATSUMOTO G. and IKEGAYA Y., J. Theor. Biol., 109 (1984) 249.
  5. JACKSON J. C., WINDMILL J. F. C., POOK V. G. and ROBERT D., Proc. Natl. Acad. Sci. U.S.A., 106 (2009) 10177.
  6. FUKUDA H., MURASE H. and TOKUDA I. T., Sci. Rep., 3 (2013) 1533.
  7. LUTHER S. et.al., Nature, 475 (2011) 235.
  8. TAKANO K., MOTOYOSHI M. and FUJISHIMA M., Proc. IEEE Asian SolidState Circuits Conference. Korea., (2007) p.336.
  9. WILSON D. andMOEHLIS J., SIAM. J. Appl. Dyn. Sys., 13 (2014) 276.
  10. TANAKA H.-A., Physica D: Nonlinear Phenomena, 288 (2014) 1.
  11. TANAKA H.-A., J. Phys. A: Mathematical and Theoretical, 47 (2014) 402002.
  12. HARDY G., LITTLEWOOD J. E. and PÓLYA G., Inequalities, Second Edition (Cambridge Mathematical Library) 1988.
  13. TSALLIS C., Introduction to Nonextensive Statistical Mechanics (Springer, New York) 2009.
  14. PRATO D. and TSALLIS C., Phys. Rev. E, 60 (1999) 2398.
  15. KURAMOTO Y., Chemical Oscillations, Waves and Turbulence (Springer, Berlin) 1984.
  16. WINFREE A. T., The Geometry of Biological Time 2nd 418 edn (Springer, New York) 2001.
  17. PIKOVSKY A. S., ROSENBLUM M. G. and KURTHS J., Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge University Press, Cambridge) 2001.
  18. NABI A. and MOEHLIS J., J. Math. Biol., 64 (2012) 981.
  19. PYRAGAS K., PYRAGAS, V., KISS, I. Z. and HUDSON, J. L., Phys. Rev. Lett., 89 (2002) 244103.
  20. NAGASHIMA T., WEI X., TANAKA H.-A. and SEKIYA H., IEEE Trans. Circuits Syst., 61 (2014) 2904.
  21. KUREBAYASHI W., SHIRASAKA S., and NAKAO H., Phys. Rev. Lett., 111 (2013) 214101.