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Least squares parameter estimation in chaotic differential equations

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Abstract

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.

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Kallrath, J., Schlöder, J.P. & Bock, H.G. Least squares parameter estimation in chaotic differential equations. Celestial Mech Dyn Astr 56, 353–371 (1993). https://doi.org/10.1007/BF00699746

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Key words

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