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New invariant measures to track slow parameter drifts in fast dynamical systems

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Abstract

Estimates of quantitative characteristics of nonlinear dynamics, e.g., correlation dimension or Lyapunov exponents, require long time series and are sensitive to noise. Other measures (e.g., phase space warping or sensitivity vector fields) are relatively difficult to implement and computationally intensive. In this paper, we propose a new class of features based on Birkhoff ergodic theorem, which are fast and easy to calculate. They are robust to noise and do not require large data or computational resources. Application of these metrics in conjunction with the smooth orthogonal decomposition to identify/track slowly changing parameters in nonlinear dynamical systems is demonstrated using both synthetic and experimental data.

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Notes

  1. The same derivation can also be repeated for the characteristic position metric to show that \(\mathbf {P}(\mathbf y)\) is also a direct function of \(\varvec{\phi }\).

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Authors and Affiliations

  1. Department of Mechanical, Industrial and Systems Engineering, University of Rhode Island, Kingston, RI, 02881, USA

    Son Hai Nguyen & David Chelidze

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  1. Son Hai Nguyen
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  2. David Chelidze
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Correspondence to David Chelidze.

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Nguyen, S.H., Chelidze, D. New invariant measures to track slow parameter drifts in fast dynamical systems. Nonlinear Dyn 79, 1207–1216 (2015). https://doi.org/10.1007/s11071-014-1737-y

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  • DOI: https://doi.org/10.1007/s11071-014-1737-y

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