Estimating Lyapunov Exponents from Approximate Return Maps

  • J. N. Glover
Part of the NATO ASI Series book series (NSSB, volume 208)


Phase portrait reconstructions from experimental time series are widely used for the calculation of dimensions, Lyapunov exponents and entropy, all of which can be used to characterise a dynamical system and investigate the effect of varying external parameters (Schuster 1984). Often it is desirable to reduce the dimensionality of the system by the method of Poincaré sections (Hirsch and Smale 1974). The reduction in the amount of data available for calculating quantities such as correlation dimension is compensated for by the improvement in quantitative and qualitative deductions that can be made (Brandstater and Swinney 1987). An alternative approach, sometimes possible in low dimensions, is to consider an approximate one-dimensional return map of the flow, for example by plotting successive minima of some component (Sparrow 1982, Chapman 1989). If the trajectories end up near an attractor with Hausdorff dimension slightly greater than two, then the resulting return map is close to one dimensional and the comprehensive theory of one-dimensional maps can be used to characterise bifurcations in the system as a parameter is varied. In some cases this approach can be made rigorous and when it works it is simpler than using the reconstructed flow. In this report the two approaches are related: the first produces a ‘Poincaré section of a Reconstruction’ while the second produces a ‘Reconstruction of a Poincaré section’. The one-dimensional approximate return map can then be used to estimate the largest Lyapunov exponent of a Poincaré map of the flow. Only the case in ℝ3 is considered and when the measured signal is the first component of the state vector x(t).


Lyapunov Exponent Hausdorff Dimension Strange Attractor Large Lyapunov Exponent Successive Minimum 


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  1. Brandstater A., Swinney H.L., 1987 Strange Attractors in Weakly Turbulent Cuoette-Taylor Flow,Phys. Rev.A, 2207 (35).CrossRefGoogle Scholar
  2. Chapman P.B., Glover J.N., Mees A.I. 1989 The Dynamics of the Rikitake Equations from the Stiff` Limit submitted for publication.Google Scholar
  3. Hirsch M.W., Smale S.,1974 Differential Equations, Dynamical Systems, and Linear Algebra,Academic Press, New York.Google Scholar
  4. Schuster H.G.,1984 Deterministic Chaos: An Introduction,VCH, New York, 117–139.Google Scholar
  5. Sparrow C.,1982 The Lorenz Equations: Bifurcations,Chaos and Strange Attractors,Apppl. Math. Sci. 41. Springer-Verlag, New York.Google Scholar
  6. Takens F.,1980 Detecting Strange Attractor in Turbulence.,Dynamical Systems and Turbulence. Lecture Notes in Mathematics. Vol. 898, Rand D.A. and Young L.S., eds, 365–381. Springer-Verlag, New York.Google Scholar

Copyright information

© Plenum Press, New York 1989

Authors and Affiliations

  • J. N. Glover
    • 1
  1. 1.Department of MathematicsUniversity of Western AustraliaNedlandsAustralia

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