Journal of Systems Science and Complexity

, Volume 27, Issue 3, pp 494–506 | Cite as

Measuring and locating zones of chaos and irregularity

  • David Matthew GarnerEmail author
  • Bingo Wing-Kuen Ling


The new measures computed here are the spectral detrended fluctuation analysis (sDFA) and spectral multi-taper method (sMTM). sDFA applies the standard detrended fluctuation analysis (DFA) algorithm to power spectra. sMTM exploits the minute increases in the broadband response, typical of chaotic spectra approaching optimal values. The authors chose the Brusselator, Lorenz, and Duffing as the proposed models to measure and locate chaos and severe irregularity. Their series of chaotic parametric responses in short time-series is advantageous. Where cycles have only a limited number of slow oscillations such as for systems biology and medicine. It is difficult to create, locate, or monitor chaos. From 50 linearly increasing starting points applied to the chaos target function (CTF); the mean percentage increases in Kolmogorov-Sinai entropy (KS-Entropy) for the proposed chosen models; and p-values when the models were compared statistically by Kruskal-Wallis and ANOVA1 test with distributions assumed normal are Duffing (CTF: 31%: p < 0.03); Lorenz (CTF: 2%: p < 0.03), and Brusselator (CTF: 8%: p < 0.01). Principal component analysis (PCA) is applied to assess the significance of the objective functions for tuning the chaotic response. From PCA the conclusion is that CTF is the most beneficial objective function overall delivering the highest increases in mean KS-Entropy.


Chaos entropy optimization signal processing systems biology 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Goldberger A L and West B J, Chaos and order in the human body, MD Comput. 1992, 9(1): 25–34.Google Scholar
  2. [2]
    Goldberger A L, Rigney D R, and West B J, Chaos and fractals in human physiology, Sci. Am. 1990, 262(2): 42–49.CrossRefGoogle Scholar
  3. [3]
    Amaral L A, Diaz-Guilera A, Moreira A A, Goldberger A L, and Lipsitz L A, Emergence of complex dynamics in a simple model of signaling networks, Proc. Natl. Acad. Sci. USA, 2004, 101(44): 15551–15555.CrossRefzbMATHMathSciNetGoogle Scholar
  4. [4]
    Costa M, Goldberger A L, and Peng C K, Multiscale entropy analysis of biological signals, Phys. Rev. E. Stat. Nonlin. Soft. Matter Phys., 2005, 71(2): 021906.CrossRefMathSciNetGoogle Scholar
  5. [5]
    Peng C K, Havlin S, Stanley H E, and Goldberger A L, Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series, Chaos, 1995, 5(1): 82–87.CrossRefGoogle Scholar
  6. [6]
    Gao Z K and Jin N D, Scaling analysis of phase fluctuations in experimental three-phase flows, Physica A: Statistical Mechanics and Its Applications 2011, 390: 3541–3550.CrossRefGoogle Scholar
  7. [7]
    Gao ZK and Jin ND, Nonlinear characterization of oil gas water three-phase flow in complex networks, Chemical Engineering Science, 2011, 66: 2660–2671.CrossRefGoogle Scholar
  8. [8]
    Zhang J, Zhang K, Feng J, and Small M, Rhythmic dynamics and synchronization via dimensionality reduction: Application to human gait, PLoS. Comput. Biol., 2010, 6: 1001033.CrossRefMathSciNetGoogle Scholar
  9. [9]
    Javorka M, Trunkvalterova Z, Tonhajzerova I, Javorkova J, Javorka K, and Baumert M, Shortterm heart rate complexity is reduced in patients with type 1 diabetes mellitus, Clin. Neurophysiol, 2008, 119(5): 1071–1081.CrossRefGoogle Scholar
  10. [10]
    Seitsonen E R, Korhonen I K, van Gils M J, Huiku M, Lotjonen J M, Korttila K T, and Yli-Hankala A M, EEG spectral entropy, heart rate, photoplethysmography, and motor responses to skin incision during sevoflurane anaesthesia, Acta Anaesthesiol. Scand., 2005, 49(3): 284–292.CrossRefGoogle Scholar
  11. [11]
    Choi S R, Lim Y H, Lee S C, Lee J H, and Chung C J, Spectral entropy monitoring allowed lower sevoflurane concentration and faster recovery in children, Acta Anaesthesiol. Scand., 2010, 54(7): 859–862.CrossRefGoogle Scholar
  12. [12]
    Vakkuri A, Yli-Hankala A, Talja P, Mustola S, Tolvanen-Laakso H, Sampson T, and Viertio-Oja H, Time-frequency balanced spectral entropy as a measure of anesthetic drug effect in central nervous system during sevoflurane, propofol, and thiopental anesthesia, Acta Anaesthesiol. Scand., 2004, 48(2): 145–153.CrossRefGoogle Scholar
  13. [13]
    Johnson R W and Shore J E, Which is the better entropy expression for speech processing: −S logS or log S?, IEEE Trans. Acoust., 1984.Google Scholar
  14. [14]
    Osipov V V and Ponizovskaya E V, Stochastic resonance in the brusselator model, Phys. Rev. E. Stat. Phys. Plasmas. Fluids Relat Interdiscip. Topics, 2000, 61(4): 4603–4605.Google Scholar
  15. [15]
    Pena B and Perez-Garcia C, Stability of turing patterns in the Brusselator model, Phys. Rev. E. Stat. Nonlin. Soft. Matter Phys., 2001, 64(5): 056213.CrossRefMathSciNetGoogle Scholar
  16. [16]
    Kubicek M, Ryzler V, and Marek M, Spatial structures in a reaction-diffusion system-detailed analysis of the Brusselator, Biophys. Chem., 1978, 8(3): 235–246.CrossRefGoogle Scholar
  17. [17]
    Chakravarti S, Marek M, and Ray W H, Reaction-diffusion system with Brusselator kinetics: Control of a quasiperiodic route to chaos, Phys. Rev. E. Stat. Phys. Plasmas. Fluids Relat Interdiscip. Topics, 1995, 52(3): 2407–2423.Google Scholar
  18. [18]
    Frland J and Alfsen K H, Lyapunov-exponent spectra for the Lorenz model, Phys. Rev. A, 1984, 29(5): 2928–2931.CrossRefGoogle Scholar
  19. [19]
    Lorenz E N, Deterministic nonperiodic flow, J. Atmos. Sci., 1963, 20(2): 130–141.CrossRefGoogle Scholar
  20. [20]
    Bonatto C, Gallas J A, and Ueda Y, Chaotic phase similarities and recurrences in a dampeddriven Duffing oscillator, Phys. Rev. E. Stat. Nonlin. Soft. Matter Phys., 2008, 77(2): 026217.CrossRefGoogle Scholar
  21. [21]
    Skinner J E, Pratt C M, and Vybiral T, A reduction in the correlation dimension of heartbeat intervals precedes imminent ventricular fibrillation in human subjects, Am. Heart J., 1993, 125(3): 731–743.CrossRefGoogle Scholar
  22. [22]
    Kauffman S A and Johnsen A, Coevolution to the edge of chaos: Coupled fitness landscapes, poised states, and coevolutionary avalanches, J. Theor. Biol., 1991, 149(4): 467–505.CrossRefGoogle Scholar
  23. [23]
    Teusink B and Smid E J, Modelling strategies for the industrial exploitation of lactic acid bacteria, Nat. Rev. Microbiol., 2006, 4(1): 46–56.CrossRefGoogle Scholar
  24. [24]
    Anier A, Lipping T, Melto S, and Hovilehto S, Higuchi fractal dimension and spectral entropy as measures of depth of sedation in intensive care unit, Conf. Proc. IEEE Eng Med. Biol. Soc., 2004, 1: 526–529.Google Scholar
  25. [25]
    Weil G, Passot S, Servin F, and Billard V, Does spectral entropy reflect the response to intubation or incision during propofol-remifentanil anesthesia?, Anesth. Analg., 2008, 106(1): 152–159.CrossRefGoogle Scholar
  26. [26]
    Shannon C E, A mathematical theory of communication, The Bell System Technical Journal, 1948, 27: 379–423.CrossRefzbMATHMathSciNetGoogle Scholar
  27. [27]
    Alkan A and Kiymik M K, Comparison of AR and Welch methods in epileptic seizure detection, J. Med. Sys., 2006, 6(30): 413–419.CrossRefGoogle Scholar
  28. [28]
    Yule G U, On a method of investigating periodicities in disturbed series, with special reference to wolfer’s sunspot numbers, Phil. Trans. Roy. Soc., 1927, 226-A: 269–298.Google Scholar
  29. [29]
    Kantz H and Schreiber T, Nonlinear Time Series Analysis, Cambridge University Press, Cambridge, 2000.Google Scholar
  30. [30]
    Zyczkowski K, Renyi extrapolation of Shannon entropy, Open Systems and Information Dynamics 2003, 3(10): 297–310.CrossRefMathSciNetGoogle Scholar
  31. [31]
    Zhang J, Luo X D, Nakamura T, Sun J F, and Small M, Detecting temporal and spatial correlations in pseudoperiodic time series, Physical Review E, 2007, 75(1).Google Scholar
  32. [32]
    Zhang J, Luo X, and Small M, Detecting chaos in pseudoperiodic time series without embedding, Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 2006, 73(1): 016216.CrossRefGoogle Scholar
  33. [33]
    Zhang J and Small M, Complex network from pseudoperiodic time series: Topology versus dynamics, Phys. Rev. Lett., 2006, 96(23): 238701.CrossRefGoogle Scholar
  34. [34]
    Percival D B and Walden A T, Spectral Analysis for Physical Applications: Multitaper and Conventional Univariate Techniques, Cambridge University Press, Cambridge, 1993.CrossRefzbMATHGoogle Scholar
  35. [35]
    Ghil M, The SSA-MTM toolkit: Applications to analysis and prediction of time series, Applications of Soft Computing, 1997, 3165: 216–230.CrossRefGoogle Scholar
  36. [36]
    Slepian S, Prolate spheroidal wave functions, Fourier analysis, and uncertainty, the discrete case, Bell Syst. Tech. J., 1978, 57: 1371–1430.CrossRefzbMATHGoogle Scholar
  37. [37]
    Blackman R B and Tukey J W, The Measurement of Power Spectra: From the Point of View of Communication Engineering, Dover, New York, 1958.Google Scholar
  38. [38]
    Dirac P, New Notation for Quantum Mechanics, Proceedings of the Cambridge Philosophical Society, 1939, 35: 416.CrossRefMathSciNetGoogle Scholar
  39. [39]
    Nelder J A and Mead R, A simplex method for function minimization, Computer Journal, 1965, 7: 308–313.CrossRefzbMATHGoogle Scholar
  40. [40]
    Leatherbarrow R J and Fersht A R, Protein engineering, Protein Eng., 1986, 1(1): 7–16.CrossRefGoogle Scholar
  41. [41]
    Bjorkman M and Holmstrom K, Global optimization of costly nonconvex functions using radial basis functions, Optimization and Engineering, 2000, 1: 373–397.CrossRefMathSciNetGoogle Scholar
  42. [42]
    Wolf A, Swift J B, Swinney H L, and Vastano J A, Determining Lyapunov exponents from a time-series, Physica D, 1985, 3(16): 285–317.CrossRefMathSciNetGoogle Scholar
  43. [43]
    Kruskal W H and Wallis W A, Use of ranks in one-criterion variance analysis, Journal of the American Statistical Association, 1952, 260(47): 583–621.CrossRefGoogle Scholar
  44. [44]
    Jolliffe I T, Principal Component Analysis, Series: Springer Series in Statistics, Springer, New York, 2002.Google Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Biological and Medical SciencesOxford Brookes UniversityOxfordUK
  2. 2.University of LincolnLincolnUK

Personalised recommendations