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An Adaptive Sub-Cells Interpolation Method to Enhance Computational Efficiency for Global Attractors of Nonlinear Dynamical Systems

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Advances in Nonlinear Dynamics

Part of the book series: NODYCON Conference Proceedings Series ((NCPS))

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Abstract

In this chapter, a previous adaptive interpolation sampling method for enhancing the subdivision technique is further improved in both efficiency and adaptability. After revealing the geometrical contribution of each order derivative terms in the Taylor expansion of an interpolation lattice on the final shape of the image of a cell, it is concluded that the third-order interpolation with more interpolation nodes cannot further enhance the passing ratio of error criterion, by which integration samplings will be replaced by interpolation ones. So the performance of the second-order interpolation with the given error criterion is adopted to measure the complexity of the dynamic behavior within a cell. For a cell that does not meet the error criterion, more objective criteria are then set up in order to set a proper number of interpolated nodes in it that are integer times for the second-order interpolation. In this way, a cell is actually divided into smaller sub-cells with smaller interpolation area and thus with higher accuracy of interpolation. So the overall computational cost is also reduced. A three-dimensional system is taken as an example to illustrate the effectiveness of the proposed method.

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References

  1. C.S. Hsu, A theory of cell-to-cell mapping dynamical-systems. J. Appl. Mech. Trans. ASME 47(4), 931–939 (1980)

    Article  MathSciNet  Google Scholar 

  2. C.S. Hsu, R.S. Guttalu, An unravelling algorithm for global analysis of dynamical-systems—an application of cell-to-cell mappings. J. Appl. Mech. Trans. ASME 47(4), 940–948 (1980)

    Article  MathSciNet  Google Scholar 

  3. J. Jiang, J.X. Xu, A method of point mapping under cell reference for global analysis of nonlinear dynamical-systems. Phys. Lett. A 188(2), 137–145 (1994)

    Article  MathSciNet  Google Scholar 

  4. J. Jiang, J.X. Xu, An iterative method of point mapping under cell reference for the global analysis of non-linear dynamical systems. J. Sound Vib. 194(4), 605–621 (1996)

    Article  MathSciNet  Google Scholar 

  5. J. Jiang, J.X. Xu, An iterative method of point mapping under cell reference for the global analysis: theory and a multiscale reference technique. Nonlinear Dyn. 15(2), 103–114 (1998)

    Article  Google Scholar 

  6. C.S. Hsu, A probabilistic theory of non-linear dynamical-systems based on the cell state-space concept. J. Appl. Mech. Trans. ASME 49(4), 895–902 (1982)

    Article  Google Scholar 

  7. C.S. Hsu, Global analysis by cell mapping. Int. J. Bifurcation Chaos 2(4), 727–771 (1992)

    Article  MathSciNet  Google Scholar 

  8. L. Hong, J.X. Xu, Crises and chaotic transients studied by the generalized cell mapping digraph method. Phys. Lett. A 262(4–5), 361–375 (1999)

    Article  MathSciNet  Google Scholar 

  9. L. Hong, J.X. Xu, Discontinuous bifurcations of chaotic attractors in forced oscillators by generalized cell mapping digraph (GCMD) method. Int. J. Bifurcation Chaos 11(3), 723–736 (2001)

    Article  MathSciNet  Google Scholar 

  10. L. Hong, J.X. Xu, Chaotic saddles in Wada basin boundaries and their bifurcations by the generalized cell-mapping digraph (GCMD) method. Nonlinear Dyn. 32(4), 371–385 (2003)

    Article  MathSciNet  Google Scholar 

  11. L. Hong, J.X. Xu, Double crises in two-parameter forced oscillators by generalized cell mapping digraph method. Chaos Solitons Fractals 15(5), 871–882 (2003)

    Article  Google Scholar 

  12. L. Hong, Y.W. Zhang, J. Jiang, A hyperchaotic crisis. Int. J. Bifurcation Chaos 20(4), 1193–1200 (2010)

    Article  Google Scholar 

  13. L. Hong, J.Q. Sun, Bifurcations of forced oscillators with fuzzy uncertainties by the generalized cell mapping method. Chaos Solitons Fractals 27(4), 895–904 (2006)

    Article  MathSciNet  Google Scholar 

  14. L. Hong, J.Q. Sun, Bifurcations of a forced Duffing oscillator in the presence of fuzzy noise by the generalized cell mapping method. Int. J. Bifurcation Chaos 16(10), 3043–3051 (2006)

    Article  Google Scholar 

  15. L. Hong, J. Jiang,, J.Q. Sun, Fuzzy responses and bifurcations of a forced Duffing oscillator with a triple-well potential. Int. J. Bifurcation Chaos 25(1) (2015)

    Google Scholar 

  16. M. Dellnitz, A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors. Numer. Math. 75(3), 293–317 (1997)

    Article  MathSciNet  Google Scholar 

  17. R. Guder, M. Dellnitz, E. Kreuzer, An adaptive method for the approximation of the generalized cell mapping. Chaos Solitons Fractals 8(4), 525–534 (1997)

    Article  MathSciNet  Google Scholar 

  18. B.H. Tongue, On obtaining global nonlinear-system characteristics through interpolated cell mapping. Physica D 28(3), 401–408 (1987)

    Article  MathSciNet  Google Scholar 

  19. B.H. Tongue, K. Gu, Interpolated cell mapping of dynamical-systems. J. Appl. Mech. Trans. ASME 55(2), 461–466 (1988)

    Article  MathSciNet  Google Scholar 

  20. B.H. Tongue, K.Q. Gu, A theoretical basis for interpolated cell mapping. SIAM J. Appl. Math. 48(5), 1206–1214 (1988)

    Article  MathSciNet  Google Scholar 

  21. W.K. Lee, C.S. Hsu, A global analysis of an harmonically excited spring-pendulum system with internal resonance. J. Sound Vib. 171(3), 335–359 (1994)

    Article  MathSciNet  Google Scholar 

  22. X.M. Liu, J. Jiang, L. Hong, D.F. Tang, Studying the global bifurcation involving Wada boundary metamorphosis by a method of generalized cell mapping with sampling-adaptive interpolation. Int. J. Bifurcation Chaos 28(2), 1830003 (2018)

    Google Scholar 

  23. X.M. Liu, J. Jiang, L. Hong, D.F. Tang, Wada boundary bifurcations induced by boundary saddle-saddle collision. Phys. Lett. A 383(2-3), 170–175 (2019)

    Article  MathSciNet  Google Scholar 

  24. X. Wang, J. Jiang, L. Hong, Enhancing subdivision technique with an adaptive interpolation sampling method for global attractors of nonlinear dynamical systems. Int. J. Dyn. Control 8(4), 1147–1160 (2020)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11972274, 11772243, 11672218).

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Authors and Affiliations

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

    Xi Wang, Jun Jiang & Ling Hong

Authors
  1. Xi Wang
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  2. Jun Jiang
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  3. Ling Hong
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Corresponding author

Correspondence to Ling Hong .

Editor information

Editors and Affiliations

  1. Department of Structural and Geotechnical Engineering, Sapienza University of Rome, Rome, Italy

    Walter Lacarbonara

  2. Department of Mechanical Engineering, University of Maryland, College Park, MD, USA

    Balakumar Balachandran

  3. Department of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, USA

    Michael J. Leamy

  4. Department of Physics, Lanzhou University of Technology, Lanzhou, Gansu, China

    Jun Ma

  5. ISEP-Institute of Engineering, Polytechnic of Porto, Porto, Portugal

    J. A. Tenreiro Machado

  6. Department of Applied Mechanics, Budapest University of Technology and Economics, Budapest, Hungary

    Gabor Stepan

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Wang, X., Jiang, J., Hong, L. (2022). An Adaptive Sub-Cells Interpolation Method to Enhance Computational Efficiency for Global Attractors of Nonlinear Dynamical Systems. In: Lacarbonara, W., Balachandran, B., Leamy, M.J., Ma, J., Tenreiro Machado, J.A., Stepan, G. (eds) Advances in Nonlinear Dynamics. NODYCON Conference Proceedings Series. Springer, Cham. https://doi.org/10.1007/978-3-030-81162-4_58

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  • DOI: https://doi.org/10.1007/978-3-030-81162-4_58

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-81161-7

  • Online ISBN: 978-3-030-81162-4

  • eBook Packages: EngineeringEngineering (R0)

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