Cost function based on hidden Markov models for parameter estimation of chaotic systems

  • Yasser Shekofteh
  • Sajad Jafari
  • Karthikeyan Rajagopal
Methodologies and Application


In this note, we deal with parameter estimation methods of chaotic systems. The parameter estimation of the chaotic systems has some significant issues due to their butterfly effects. It can be formulated as an optimization problem and needs a suitable cost function. In this paper, we propose a new cost function based on a hidden Markov model which is a statistical tool for modeling of time series data. It can model dynamical characteristics of the chaotic systems. Moreover, the use of dynamical features of their strange attractors is investigated to achieve a better cost function in the procedure of parameter estimation. Our experimental results indicate the success of the proposed cost function in the one-dimensional parameter estimation of a new four-dimensional chaotic system and Lorenz system as a well-known three-dimensional chaotic system.


Parameter estimation System identification Chaotic systems State space Cost function Hidden Markov model 



This work was supported by the research grant from Shahid Beheshti University G.C. (Grant Number SAAD-600-1076). Sajad Jafari was supported by Iran National Science Foundation (No. 96000815).

Author Contributions

Yasser Shekofteh designed the study and contributed to the experiment and algorithm design. Yasser Shekofteh and Sajad Jafari wrote the paper. Sajad Jafari and Karthikeyan Rajagopal performed the chaotic analysis of the paper.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Computer Science and EngineeringShahid Beheshti UniversityVelenjak, TehranIran
  2. 2.Department of Biomedical EngineeringAmirkabir University of TechnologyTehranIran
  3. 3.Center for Nonlinear Dynamics, College of EngineeringDefence UniversityBishoftuEthiopia

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