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Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

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Detecting Causality in Uni-directionally Coupled Chaotic Oscillators with Small Frequency Mismatch

Journal of Applied Nonlinear Dynamics 9(1) (2020) 31--36 | DOI:10.5890/JAND.2020.03.003

Kazimieras Pukenas

Department of Applied Biology and Rehabilitation, Lithuanian Sports University, Sporto 6, LT-44221, Kaunas, Lithuania

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In the present work, we present a new method for assessing causality in uni-directionally coupled chaotic oscillators with small frequency mismatch. This method is based on the correlation between changes in the phase dynamics of the slave oscillator and the dynamics of the phase difference between the oscillators. Application of the proposed approach to master-slave R¨ossler systems showed that the new algorithm is well-suited for assessing the presence and direction of coupling, especially in the case of weak coupling.


  1. [1]  Papana, A., Kyrtsou, C., Kugiumtzis, D., and Diks, C. (2013), Simulation study of direct causality measures in multivariate time series, Entropy, 15, 2635-2661.
  2. [2]  Granger, C.W.J. (1969), Investigating causal relations by econometric models and cross-spectral methods, Econometrica, 37, 424-438.
  3. [3]  Krakovska, A. and Hanzely, F. (2016), Testing for causality in reconstructed state spaces by an optimized mixed prediction method, Physical Review E, 94, 052203.
  4. [4]  Chen, Y., Rangarajan, G., Feng, J., and Ding, M. (2004), Analyzing multiple nonlinear time series with extended Granger causality, Physical Letters A, 324, 26-35.
  5. [5]  Sugihara, G., May, R., Ye, H., Hsieh, C.H., Deyle, E., Fogarty, M., and Munch, S. (2012), Detecting causality in complex ecosystems, Science, 338, 496-500.
  6. [6]  Lusch, B., Maia, P.D., and Kutz, J.N. (2016), Inferring connectivity in networked dynamical systems: Challenges using Granger causality, Physical Review E, 94, 032220.
  7. [7]  Schreiber, T. (2000), Measuring information transfer, Physical Review Letters, 85, 461-464.
  8. [8]  Paluš, M. and Vejmelka, M. (2007), Directionality of coupling from bivariate time series: How to avoid false causalities and missed connections, Physical Review E, 75, 056211.
  9. [9]  Rosenblum, M.G. and Pikovsky, A.S. (2001), Detecting direction of coupling in interacting oscillators, Physical Review E, 64, 045202(R).
  10. [10]  Smirnov, D.A. and Bezruchko, B.P. (2003), Estimation of interaction strength and direction from short and noisy time series, Physical Review E, 68, 046209.
  11. [11]  Arnhold, J., Grassberger, P., Lehnertz, K., and Elger, C.E. (1999), A robust method for detecting interdependences: application to intracranially recorded EEG, Physica D, 134(4), 419-430.
  12. [12]  Quiroga, R.Q., Kraskov, A., Kreuz, T., and Grassberger, P. (2002), Performance of different synchronization measures in real data: a case study on electroencephalographic signals, Physical Review E, 65, 041903.
  13. [13]  Andrzejak, R.G., Kraskov, A., Stögbauer, H., Mormann, F., and Kreuz, T. (2003), Bivariate surrogate techniques: Necessity, strengths, and caveats, Physical Review E, 68, 066202.
  14. [14]  Smirnov, D.A. and Andrzejak, R.G. (2005), Detection of weak directional coupling: Phase-dynamics approach versus state-space approach, Physical Review E, 71, 036207.
  15. [15]  Chicharro, D. and Andrzejak, R.G. (2009), Reliable detection of directional couplings using rank statistics, Physical Review E, 80, 026217.
  16. [16]  Pyragas, K. (2007), Conditional Lyapunov exponents from time series, Physical Review E, 56, 5183.
  17. [17]  Zou, Y., Pazó, D., Romano, M.C., Thiel, M., and Kurths, J. (2007), Distinguishing quasiperiodic dynamics from chaos in short-time series, Physical Review E, 76, 016210.
  18. [18]  Rosenblum, M.G., Pikovsky, A.S., and Kurths, J. (1996), Phase Synchronization of Chaotic Oscillators, Physical Review Letters, 76(11), 1804.
  19. [19]  Boccaletti, S., Kurths, J., Osipov, G., Valladares, D.L., and Zhou, C.S. (2002), The synchronization of chaotic systems, Physics Reports, 366, 1-101.
  20. [20]  Krakovská, A., Jakubík, J., Budáčová, H., and Holecyová, M. (2016), Causality studied in reconstructed state space, Examples of Uni-directionally Connected Chaotic Systems,