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Nonlinear dynamics of the blood flow studied by Lyapunov exponents

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Abstract

In order to gain an insight into the dynamics of the cardiovascular system throughout which the blood circulates, the signals measured from peripheral blood flow in humans were analyzed by calculating the Lyapunov exponents. Over a wide range of algorithm parameters, paired values of both the global and the local Lyapunov exponents were obtained, and at least one exponent equaled zero within the calculation error. This may be an indication of the deterministic nature and finite number of degrees of freedom of the cardiovascular system governing the blood-flow dynamics on a time scale of minutes. A difference was observed in the Lyapunov dimension of controls and athletes.

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References

  • Abarbanel, H. D. I., R. Brown and M. B. Kennel (1991). Variation of Lyapunov exponent on a strange attractor. J. Nonlinear Sci. 1, 175–199.

    Article  MathSciNet  MATH  Google Scholar 

  • Abarbanel, H. D. I., R. Brown, J. J. Sidorowich and L. Sh. Tsimring (1993). The analysis of observed chaotic data in physical systems. Rev. Mod. Phys. 65, 1331–1392.

    Article  MathSciNet  Google Scholar 

  • Akselrod, S., D. Gordon, F. A. Ubel, D. C. Shannon, A. C. Barger and R. J. Choen (1981). Power spectrum analysis of heart rate fluctuation: A quantitative probe of beat-to-beat cardiovascular control. Science 213, 220–222.

    Google Scholar 

  • Bračič, M. and A. Stefanovska (1997). Local and global Lyapunov exponents of blood flow. Open Sys. & Information Dyn. 4, 435–456.

    Article  MATH  Google Scholar 

  • Bračič, M. and A. Stefanovska. Wavelet analysis in studying the dynamics of blood circulation. Open Sys. & Information Dyn. (in press)

  • Brown, R., P. Bryant and H. D. I. Abarbanel (1991). Computing the Lyapunov spectrum of a dynamical system from an observed time series. Phys. Rev. A43, 2787–2806.

    MathSciNet  Google Scholar 

  • Di Renzo, M., G. Mancia, G. Parati, A. Pedotti and A. Zanchetti (Eds) (1995). Computer Analysis of Cardiovascular Signals. Amsterdam, IOS Press.

    Google Scholar 

  • Eckmann, J.-P. and D. Ruelle (1985). Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617–656.

    Article  MathSciNet  Google Scholar 

  • Eckmann, J.-P. and D. Ruelle (1992). Fundamental limitations for estimating dimensions and Lyapunov exponents in dynamical systems. Physica D56, 185–187.

    MathSciNet  Google Scholar 

  • Fell, J. and P. Beckmann (1994). Resonance-like phenomena in Lyapunov calculations from data reconstructed by time-delay method. Phys. Rev. Lett. A190, 172–176.

    MathSciNet  Google Scholar 

  • Fell, J., J. Roschke and P. Beckmann (1993). Deterministic chaos and the first positive Lyapunov exponent: a nonlinear analysis of the humam electroencephalogram during sleep. Biol. Cyber. 69, 139–146.

    Article  MATH  Google Scholar 

  • Glass, L. and D. Kaplan (1993) Time series analysis of complex dynamics in physiology and medicine. Medical Progress through Technology 19, 115–128.

    Google Scholar 

  • Goldhirsch, I., P. L. Sulem and S. A. Orszag (1987). Stability and Lyapunov stability of dynamical systems: a differential approach and a numerical method. Physica D27, 311–337.

    MathSciNet  Google Scholar 

  • Griffith, T. M. (1996). Temporal chaos in the microcirculation. Cardiovascular Research 31, 342–358.

    Article  Google Scholar 

  • Hoffmann, U., A. Yanar, U. K. Franzeck, J. M. Edwards and A. Bollinger (1990). The frequency histogram—a new method for the evaluation of laser Doppler flux motion. Microvascular Research 40, 293–301.

    Article  Google Scholar 

  • Holzfuss, J. and U. Parlitz (1990) Lyapunov exponents from time series. Lecture Notes in Mathematics 1486, 263–270.

    MathSciNet  Google Scholar 

  • Hyndman, B. W., R. I. Kitney and B. McA Sayers (1971). Spontaneous rhythms in physiological control systems. Nature 233, 339–341.

    Article  Google Scholar 

  • Kaplan, J. L. and J. A. Yorke (1978). Numerical solution of a generalized eigenvalue problem for even mappings. Lecture Notes in Mathematics 730, 228–237.

    MathSciNet  Google Scholar 

  • Kvernmo, H. D. and B. Østerud (1997). The effect of physical conditioning suggests adaptation in procagulant and fibrinolytic potential. Thromb. Res. 87, 559–569.

    Article  Google Scholar 

  • Lyapunov, A. M. (1892). General problem of stability of motion. Comm. Soc. Math. Kharkov 2, 265–272.

    Google Scholar 

  • Mane, R. (1981). On the dimension of the compact invariant sets of certain nonlinear maps. Lecture Notes in Mathematics 898, 230–242.

    MATH  MathSciNet  Google Scholar 

  • Nilsson, G. E., T. Tenland and P. A. Öberg (1980). Evaluation of a laser Doppler flowmeter for measurement of tissue blood flow. IEEE Trans. Biomed. Eng. 27, 597–604.

    Google Scholar 

  • Oseledec, V. I. (1968). A multiplicative ergodic theorem. Trans. Moscow Math. Soc. 19, 197–231.

    MATH  MathSciNet  Google Scholar 

  • Packard, N., J. Crutchfiels, D. Farmer and R. Shaw (1980). Geometry from time series. Phys. Rev. Lett. 45, 712–716.

    Article  Google Scholar 

  • Parlitz, U. (1992) Identification of true and spurious exponents from time series. Int. J. Bif. Chaos 2, 155–165.

    Article  MATH  MathSciNet  Google Scholar 

  • Rosenstein, M. T., J. J. Collins and C. J. DeLuca (1993). A practical method for calculating the largest Lyapunov exponent from small data sets. Physica, D65, 117–134.

    MathSciNet  Google Scholar 

  • Rosenstein, M. T., J. J. Collins and C. J. DeLuca (1994). Reconstruction expansion as a geometry-based framework for choosing proper delay times. Physica, D73, 82–98.

    MathSciNet  Google Scholar 

  • Sano M. and Y. Sawada (1985). Measurement of the Lyapunov spectrum from a chaotic time series. Phys. Rev. Lett. 55, 1082–1085.

    Article  MathSciNet  Google Scholar 

  • Sato, S., M. Sano and Y. Sawada (1987). Practical methods of measuring the generalised dimension and the largest Lyapunov exponent in high dimensional chaotic systems. Prog. Theor. Phys. 77, 1–5.

    Article  MathSciNet  Google Scholar 

  • Stefanovska, A. (1992). Dissertation, University of Ljubljana, Ljubljana.

    Google Scholar 

  • Stefanovska, A., P. Krošelj, M. Bračič and S. Strle (1997). Reconstructing cardiovascular dynamics. IFAC Meeting on Modelling and Control in Biomedical Systems, Warwick, UK, 23–26 March 1997.

  • Stern, M. D. (1975). In vivo observation of microcirculation by coherent light scattering. Nature 254, 56–58.

    Article  Google Scholar 

  • Takens, F. (1981). Detecting strange attractors in turbulence. Lecture Notes in Mathematics 898, 366–381.

    Article  MATH  MathSciNet  Google Scholar 

  • Teichert, H. (1995). Asymptotic analysis of the Lyapunov exponents of certain linear Hamiltonian systems with small noise. Math. Nachr. 172, 291–311.

    MATH  MathSciNet  Google Scholar 

  • Wolf, A., J. B. Swift, H. Swinney and J. A. Vastano (1985). Determining Lyapunov exponents from time series. Physica D16, 285–317.

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Faculty of Electrical Engineering, Laboratory of Nonlinear Dynamics and Synergetics, University of Ljubljana, Ljubljana, Slovenia

    Maja Bračič & Aneta Stefanovska

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  1. Maja Bračič
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  2. Aneta Stefanovska
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Correspondence to Maja Bračič.

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Bračič, M., Stefanovska, A. Nonlinear dynamics of the blood flow studied by Lyapunov exponents. Bull. Math. Biol. 60, 417–433 (1998). https://doi.org/10.1006/bulm.1997.0007

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  • DOI: https://doi.org/10.1006/bulm.1997.0007

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