Nonlinear Dynamics

, Volume 70, Issue 1, pp 381–391

Parameter estimation for chaotic systems using a geometric approach: theory and experiment

Original Paper

DOI: 10.1007/s11071-012-0461-8

Cite this article as:
Olson, C.C., Nichols, J.M. & Virgin, L.N. Nonlinear Dyn (2012) 70: 381. doi:10.1007/s11071-012-0461-8


A method for estimating model parameters based on chaotic system response data is described. This estimation problem is made challenging by sensitive dependence to initial conditions. The standard maximum likelihood estimation method is practically infeasible due to the non-smooth nature of the likelihood function. We bypass the problem by introducing an alternative, smoother function that admits a better-defined maximum and show that the parameters that maximize this new function are asymptotically equivalent to maximum likelihood estimates. We use simulations to explore the influence of noise and available data on model Duffing and Lorenz oscillators. We then apply the approach to experimental data from a chaotic Duffing system. Our method does not require estimation of initial conditions and parameter estimates may be obtained even when system dynamics have been estimated from a delay embedding.


Parameter estimation Chaotic dynamics Maximum likelihood Differential evolution Evolutionary algorithms 

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Sotera Defense Solutions, Inc.CroftonUSA
  2. 2.Naval Research LaboratoryWashington, DCUSA
  3. 3.Department of Mechanical EngineeringDuke UniversityDurhamUSA

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