Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

Least squares parameter estimation in chaotic differential equations

  • 109 Accesses

  • 22 Citations


A recent least squares algorithm, which is designed to adapt implicit models to given sets of data, especially models given by differential equations or dynamical systems, is reviewed and used to fit the Hénon-Heiles differential equations to chaotic data sets.

This numerical approach for estimating parameters in differential equation models, called theboundary value problem approach, is based on discretizing the differential equations like a boundary value problem,e.g. by a multiple shooting or collocation method, and solving the resulting constrained least squares problem with a structure exploiting generalized Gauss-Newton-Method (Bock, 1981).

Dynamical systems like the Hénon-Heiles system which can have initial values and parameters that lead to positive Lyapunov exponents or phase space filling Poincaré maps give rise to chaotic time series. Various scenarios representing ideal and noisy data generated from the Hénon-Heiles system in the chaotic region are analyzedw.r.t. initial conditions, parameters and Lyapunov exponents. The original initial conditions and parameters are recovered with a given accuracy. The Lyapunov spectrum is then computed directly from the identified differential equations and compared to the spectrum of the “true” dynamics.

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


  1. Abramowitz, M. and Stegun, I.A.: 1970,Handbook of Mathematical Functions, Dover Publication, 9th printing.

  2. Baake, E., Baake, M., Bock, H.G. and Briggs, K.M.: 1992, ‘Fitting Ordinary Differential Equations to Chaotic Data’,Phys.Rev. A45.

  3. Bennetin, G., Galgani, L. and Strelcyn, J.M.: 1976, ‘Kolmogorov entropy and numerical experiments’,Phys.Rev. A14, 2338–2345.

  4. Bennetin, G., Galgani, L., Giorgilli, A. and Strelcyn, J.M.: 1980, ‘Lyapunov Characteristic Exponents for Smooth Dynamical Systems — A Method for Computing all of them’,Meccania (March), 9–20, 21–30.

  5. Bock, H.G.: 1981. in: Ebert, K.H., Deuflhard, P. & Jäger, W. (Eds.)Modelling of Chemical Reaction Systems, Springer Series in Chemical Physics, Springer, Heidelberg.

  6. Bock, H.G.: 1987. ‘Randwertproblemmethoden zur Parameteridentifizierung in Systemen nichtlinearer Differentialgleichungen’,Bonner Mathematische Schriften 183, Bonn.

  7. Bock, H.G. and Schlöder, J.P.: 1986, ‘Recent progress in the development of algorithm and software for large-scale parameter estimation problems in chemical reaction systems’, in: Kotobh, P.(Ed.)Automatic Control in Petrol, Petrochemical and Desalination Industries, IFAC Congress, Pergamon, Oxford.

  8. Bock, H.G., Eich, E. and Schlöder, J.P.: 1988, ‘Numerical Solution of Constrained Least Squares Boundary Value Problems in Differential Algebraic Equations’, in: Strehmel (Ed.)Numerical Treatment of Differential Equations, BG Teubner, Leipzig.

  9. Bulirsch, R. and Stoer, J.: 1966, ‘Numerical Treatment of Ordinary Differential Equations by Extrapolations Methods’,Numerische Mathematik 8, 1–13.

  10. Eckmann, J.-P. and Ruelle, D.: 1985, ‘Ergodic Theory of chaos and strange attractors’,Rev.Mod.Phys. 57, 617–657.

  11. Eichhorn, H.: 1992, ‘Generalized Least Squares Adjustment, a Timely but Much Neglected Tool’,Celestial Mechanics and Dynamical Astronomy, this issue.

  12. Deuflhard, P.: 1974, ‘A modified Newton Methode for the Solution of Ill-conditioned Systems of Nonlinear Equations with Applications to Multiple Shooting’,Numerische Mathematik 22, 289–311.

  13. Froeschlé, C.: 1984, ‘The Lyapunov Characteristic Exponents — Applications to Celestial Mechanics’,Celest. Mech. 34, 95–115.

  14. Froeschlé, C. and Gonczi, R.: 1988, ‘On the Stochasticity of Halley like Comets’,Celest. Mech. 43, 325–330.

  15. Gauss, C.F.: 1809, ‘Theoria Motus Corporum Coelestium in Sectionibus Conicus Solem Ambientium’, Perthes, F. and Besser, J. H., Hamburg, [reprinted in Carl Friedrich Gauss Werke, Vol. VII, Königliche Gesellschaft der Wissenschaften zu Göttingen, Göttingen 1871.]

  16. Gonczi, R. and Froeschlé, C.: 1981, ‘The Lyapunov Characteristic Exponents as Indicators of Stochasticity in the Restricted Three-Body problem’,Celest. Mech. 25, 271–280.

  17. Hénon, M. and Heiles, C.: 1964, ‘The Applicability of the Third Integral of Motion: Some Numerical Experiments’,Astron.J. 69, 73–79.

  18. Holzfuss, J. and Parlitz, U.: 1991, ‘Lyapunov exponents from time series’, in: Arnold, L., Cranel, H. and Eckmann, J.-P.,Lyapunov Exponents, Proceedings, Oberwolfach 1990, 263–271.

  19. Schlöder, J.P.: 1988, ‘Numerische Methoden zur Behandlung hochdimensionaler Aufgaben der Parameteridentifizierung’,Bonner Mathematische Schriften,187, Bonn.

  20. Wolf, A., Swift, J.B., Swinney, H.L. and Vastano, J.A.: 1985, ‘Determining Lyapunov Exponents from a Time Series’,Physica 16D, 285–317.

  21. Zeng, X., Eykholt, R. and Pielke, R.A.: 1991, ‘Estimating the Lyapunov-Exponents from Short Time Series of Low Precision’,Physical Review Letters,66, 3229–3232.

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kallrath, J., Schlöder, J.P. & Bock, H.G. Least squares parameter estimation in chaotic differential equations. Celestial Mech Dyn Astr 56, 353–371 (1993).

Download citation

Key words

  • least squares techniques
  • numerical parameter estimation
  • boundary value problem approach
  • dynamical systems
  • Hénon-Heiles system
  • Lyapunov spectrum