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Response Analysis of a Forced Duffing Oscillator with Fuzzy Uncertainty

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EVOLVE - A Bridge between Probability, Set Oriented Numerics, and Evolutionary Computation V

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 288))

Abstract

The transient and steady-state membership distribution functions (MDFs) of fuzzy response of a forced Duffing oscillator with fuzzy uncertainty are studied by means of the Fuzzy Generalized Cell Mapping (FGCM) method. The FGCM method is first introduced. A rigorous mathematical foundation of the FGCM is established with a discrete representation of the fuzzy master equation for the possibility transition of continuous fuzzy processes. The FGCM offers a very effective approach for solutions to the fuzzy master equation based on the min-max operator of fuzzy logic. Fuzzy response is characterized by its topology in the state space and its possibility measure of MDFs. The response topology is obtained based on the qualitative analysis of the FGCM involving the Boolean operation of 0 and 1. The evolutionary process of transient and steady-state MDFs is determined by the quantitative analysis of the FGCM with the min-max calculations. It is found that the evolutionary orientation of MDFs is in accordance with invariant manifolds leading to invariant sets. In the evolutionary process of a steady-state fuzzy response with an increase of the intensity of fuzzy noise, a merging bifurcation is observed in a sudden change of the MDFs from two sharp peaks of most possibility to one peak band around unstable manifolds.

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Author information

Authors and Affiliations

  1. State Key Lab for Strength and Vibration, Xi’an Jiaotong University, Xi’an, 710049, China

    Ling Hong & Jun Jiang

  2. School of Engineering, University of California at Merced, Merced, CA, 95344, USA

    Jian-Qiao Sun

Authors
  1. Ling Hong
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  2. Jun Jiang
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  3. Jian-Qiao Sun
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Corresponding author

Correspondence to Ling Hong .

Editor information

Editors and Affiliations

  1. Interdisciplinary Centre for Security, Reliability and Trust, University of Luxembourg, Luxembourg, Luxembourg

    Alexandru-Adrian Tantar

  2. Interdisciplinary Centre for Security, Reliability and Trust, University of Luxembourg, Luxembourg, Luxembourg

    Emilia Tantar

  3. School of Engineering, University of California, Merced, California, USA

    Jian-Qiao Sun

  4. College of Mechanical Engineering, Beijing University of Technology, Beijing, China

    Wei Zhang

  5. Department of Mechanics, Tianjin University, Tianjin, China

    Qian Ding

  6. Depto. Computación, CINVESTAV-IPN, Mexico City, Mexico

    Oliver Schütze

  7. Leiden Institute of Advanced Computer Science, Leiden University, Leiden, The Netherlands

    Michael Emmerich

  8. Université de Bordeaux, Bordeaux, France

    Pierrick Legrand

  9. School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales, Australia

    Pierre Del Moral

  10. Computer Science Department, CINVESTAV-IPN, Mexico City, Mexico

    Carlos A. Coello Coello

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© 2014 Springer International Publishing Switzerland

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Hong, L., Jiang, J., Sun, JQ. (2014). Response Analysis of a Forced Duffing Oscillator with Fuzzy Uncertainty. In: Tantar, AA., et al. EVOLVE - A Bridge between Probability, Set Oriented Numerics, and Evolutionary Computation V. Advances in Intelligent Systems and Computing, vol 288. Springer, Cham. https://doi.org/10.1007/978-3-319-07494-8_1

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  • DOI: https://doi.org/10.1007/978-3-319-07494-8_1

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-07493-1

  • Online ISBN: 978-3-319-07494-8

  • eBook Packages: EngineeringEngineering (R0)

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