Skip to main content
SpringerLink
Log in

Signatures of low-dimensional chaos in hourly water level measurements at coastal site of Mariupol, Ukraine

  • Original Paper
  • Published:
Stochastic Environmental Research and Risk Assessment Aims and scope Submit manuscript

Abstract

Variations of water levels in ports and estuaries are important for ship guidance and navigation and are influenced by a variety of factors. The hourly data that was collected from the coastal site at the Port of Mariupol, Ukraine during January–December 2005 were analysed with an objective to reveal features of chaotic behaviour. The concepts and methods of chaos theory (average mutual information, correlation dimension, false nearest neighbours, Lyapunov exponents) were applied. The manifestation of low-dimensional chaos was found in the time series. The possibility of nonlinear prediction was concluded.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  • Abarbanel HDI, Brown R, Sidorowich JJ, Tsimring LSh (1993) The analysis of observed chaotic data in physical systems. Rev Mod Phys 65:1331–1392

    Article  Google Scholar 

  • Bates PD, Dawson RJ, Hall JW, Horritt MS, Nicholls RJ, Wicks J, Hassan MAAM (2005) Simplified two-dimensional numerical modelling of coastal flooding and example applications. Coast Eng 52:793–810

    Article  Google Scholar 

  • Benavente J, Del Río L, Gracia FJ, Martínez-del-Pozo JA (2006) Coastal flooding hazard related to storms and coastal evolution in Valdelagrana spit (Cadiz Bay Natural Park, SW Spain). Cont Shelf Res 26:1061–1076

    Article  Google Scholar 

  • Chang H-K, Lin L-C (2006) Multi-point tidal prediction using artificial neural network with tide-generating forces. Coast Eng 53:857–864

    Article  Google Scholar 

  • Fraser AM, Swinney HL (1986) Independent coordinates for strange attractors from mutual information. Phys Rev A 33:1134–1140

    Article  Google Scholar 

  • Frison T, Abarbanel H, Earle M, Schultz J, Sheerer W (1999) Chaos and predictability in ocean water level measurements. J Geophys Res Ocean 104:7935–7951

    Article  Google Scholar 

  • Gallager RG (1968) Information theory and reliable communication. Wiley, New York

    Google Scholar 

  • Grassberger P, Procaccia I (1983) Measuring the strangeness of strange attractors. Physica D 9:189–208

    Article  Google Scholar 

  • Havstad JW, Ehlers CL (1989) Attractor dimension of nonstationary dynamical systems from small data sets. Phys Rev A 39:845–853

    Article  Google Scholar 

  • Holzfuss J, Mayer-Kress G (1986) An approach to error-estimation in the application of dimension algorithms. In: Mayer-Kress G (ed) Dimensions and entropies in chaotic systems. Springer, Berlin, pp 114–122

    Google Scholar 

  • Islam MN, Sivakumar B (2002) Characterization and prediction of runoff dynamics: a nonlinear dynamical view. Adv Water Resour 25:179–190

    Article  Google Scholar 

  • Kaplan JL, Yorke JA (1979) Chaotic behavior of multidimensional difference equations. In: Peitgen HO, Walter HO (eds) Functional differential equations and approximations of fixed points. (Lecture notes in mathematics no 730). Springer, Berlin, pp 204–227

    Chapter  Google Scholar 

  • Kennel MB, Brown R, Abarbanel HDI (1992) Determining embedding dimension for phase–space reconstruction using a geometrical construction. Phys Rev A 45:3403–3411

    Article  Google Scholar 

  • Lorenz EN (1963) Deterministic nonperiodic flow. J Atmos Sci 20:130–141

    Article  Google Scholar 

  • Mackey M, Glass L (1977) Oscillation and chaos in physiological control systems. Science 197:287–289

    Article  CAS  Google Scholar 

  • Madsen H, Jakobsen F (2004) Cyclone induced storm surge and flood forecasting in the northern Bay of Bengal. Coast Eng 51:277–296

    Article  Google Scholar 

  • Mañé R (1981) On the dimensions of the compact invariant sets of certain non-linear maps. In: Rand DA, Young LS (eds) Dynamical systems and turbulence, Warwick 1980. (Lecture notes in mathematics No 898). Springer, Berlin, pp 230–242

    Chapter  Google Scholar 

  • Nerenberg MAH, Essex C (1990) Correlation dimension and systematic geometric effects. Phys Rev A 42:7065–7074

    Article  Google Scholar 

  • Osborne AR, Provenzale A (1989) Finite correlation dimension for stochastic systems with power-law spectra. Physica D 35:357–381

    Article  Google Scholar 

  • Packard NH, Crutchfield JP, Farmer JD, Shaw RS (1980) Geometry from a time series. Phys Rev Lett 45:712–716

    Article  Google Scholar 

  • Paluš M (1995) Testing for nonlinearity using redundancies: quantitative and qualitative aspects. Physica D 80:186–205

    Article  Google Scholar 

  • Różyński G, Reeve D (2005) Multi-resolution analysis of nearshore hydrodynamics using discrete wavelet transforms. Coast Eng 52:771–792

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Schreiber T (1999) Interdisciplinary application of nonlinear time series methods. Phys Rep 308:1–64

    Article  Google Scholar 

  • Schuster HG (1989) Deterministic chaos: an introduction. Verlagsgesellschaft, Weinheim

    Google Scholar 

  • Sivakumar B (2000) Chaos theory in hydrology: important issues and interpretations. J Hydrol 227:1–20

    Article  Google Scholar 

  • Sivakumar B (2004) Chaos theory in geophysics: past, present and future. Chaos Solitons Fractals 19:441–462

    Article  Google Scholar 

  • Smith LA (1988) Intrinsic limits on dimension calculations. Phys Lett A 133:283–288

    Article  Google Scholar 

  • Sobey RJ (2006) Normal mode decomposition for identification of storm tide and tsunami hazard. Coast Eng 53:289–301

    Article  Google Scholar 

  • Srinivas K, Kesava Das V, Dinesh Kumar PK (2005) Statistical modelling of monthly man sea level at coastal tide gauge stations along the Indian subcontinent. Ind J Mar Sci 34:212–224

    Google Scholar 

  • Surkov FA, Krukier LA, Muratova GV (1990) Numerical modelling of the Sea of Azov’s dynamics resulting from narrowing of the mouth of Taganrog Bay. Phys Oceanogr 1:551–559

    Article  Google Scholar 

  • Takens F (1981) Detecting strange attractors in turbulence. In: Rand DA, Young LS (eds) Dynamical systems and turbulence, Warwick 1980. (Lecture notes in mathematics No 898). Springer, Berlin, pp 366–381

    Chapter  Google Scholar 

  • Theiler J, Eubank S, Longtin A, Galdrikian B, Farmer JD (1992) Testing for nonlinearity in time series: the method of surrogate data. Physica D 58:77–94

    Article  Google Scholar 

  • Tilburg CE, Garvine RW (2004) A simple model for coastal sea level prediction. Weather Forecast 19:511–519

    Article  Google Scholar 

  • Tsonis AA, Elsner JB (1988) The weather attractor over very short timescale. Nature 333:545–547

    Article  Google Scholar 

  • Tsonis AA, Elsner JB, Georgakakos KP (1993) Estimating the dimension of weather and climate attractors: important issues about the procedure and interpretation. J Atmos Sci 50:2549–2555

    Article  Google Scholar 

  • Wei HL, Billings SA (2006) An efficient cardinal B-spline method for high tide forecasts at the Venice Lagoon. Nonlinear Process Geophys 13:577–584

    Google Scholar 

  • Zaldivar JM, Gitiérrez E, Galván IM, Strozzi F, Tomasin A (2000) Forecasting high waters at Venice Lagoon using chaotic time series analysis and nonlinear neural networks. J Hydroinform 2:61–84

    Google Scholar 

Download references

Acknowledgments

The authors would like to thank the two anonymous reviewers for their valuable suggestions, which resulted in a more technically sound and complete presentation of the work.

Author information

Authors and Affiliations

  1. Hydrometeorological Institute, Odessa State Environmental University, PO Box 116, Odessa, 65009, Ukraine

    V. Khokhlov & N. Loboda

  2. Institute of Applied Mathematics, Odessa State Environmental University, PO Box 116, Odessa, 65009, Ukraine

    A. Glushkov & N. Serbov

  3. Hydrometeorological Observatory, Lunin av. 5, Mariupol, 87510, Ukraine

    K. Zhurbenko

Authors
  1. V. Khokhlov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Glushkov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. N. Loboda
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. N. Serbov
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. K. Zhurbenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. Khokhlov.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Khokhlov, V., Glushkov, A., Loboda, N. et al. Signatures of low-dimensional chaos in hourly water level measurements at coastal site of Mariupol, Ukraine. Stoch Environ Res Risk Assess 22, 777–787 (2008). https://doi.org/10.1007/s00477-007-0186-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00477-007-0186-2

Keywords

Search

Navigation