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Piecewise Isometries: Applications in Engineering

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Abstract

This paper defines and describes a class of two-dimensional discontinuous mappings known as piecewise isometries (PWIs), briefly reviews two known instances in which such mappings arise in an electronic engineering context, and gives preliminary results in a third case. In relation to the last example, the concept of a ‘PWI with contraction’ – a piecewise similarity – is introduced.

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References

  1. Chua, L.O., Lin, T. ‘Chaos in digital filters’, IEEE Trans. Circuits Syst. 35(6) (1988)

  2. Kocarev, L., Wu, C.W. and Chua, L.O., ‘Complex behaviour in Digital filters with overflow nonlinearity: analytical results’, IEEE Trans. CAS-II 43 (1996) 234–246.

    Google Scholar 

  3. M.J. Ogorzałek (1992) ArticleTitle‘Complex behaviour in digital filters’ Int. J. Bifurcat. Chaos 2 IssueID1 11–29 Occurrence Handle10.1142/S0218127492000033

    Article  Google Scholar 

  4. P. Ashwin Xin-Chu Fu (2002) ArticleTitle‘On the geometry of orientation preserving planar piecewise isometries J. Nonlinear Sci. 12 207–240 Occurrence Handle01837429 Occurrence Handle1905204 Occurrence Handle10.1007/s00332-002-0477-1 Occurrence Handle2002JNS....12..207A

    Article  MATH  MathSciNet  ADS  Google Scholar 

  5. Feely, O. and Fitzgerald, D., ‘Non-ideal and chaotic behaviour in bandpass sigma-delta modulators in: Proceedings of NDES 1996, Sevilla, Spain, 1996, pp. 399–404.

  6. Feely, O., ‘Nonlinear dynamics of bandpass sigma–delta modulation’, in: Proceedings of NDES, Dublin, 1995, pp. 33–36.

  7. Ashwin, P., Deane, J.H.B. and Fu, X.-C., ‘Dynamics of a bandpass sigma-delta modulator as a piecewise isometry,’ Proceedings, ISCAS 2001 (Proceedings of the IEEE International Symposium on Circuits and Systems, 2001), Sydney, Australia, 2001, pp. III-811–III-814.

  8. J.H.B. Deane (2002) ArticleTitle‘Global attraction in the sigma-delta piecewise isometry’ Dynamical Systems 17 IssueID4 377–388 Occurrence Handle1022.37026 Occurrence Handle1975120 Occurrence Handle10.1080/1468936021000043922

    Article  MATH  MathSciNet  Google Scholar 

  9. Deane, J.H.B. and Goetz, A., ‘Piecewise isometries and the buck converter’ (to be submitted).

  10. A. Goetz (1998) ArticleTitle‘Perturbations of 8-attractors and births of satellite systems’ Int. J. Bifurcat. Chaos 8 IssueID10 1937–1956 Occurrence Handle0956.37029 Occurrence Handle1670619 Occurrence Handle10.1142/S0218127498001613

    Article  MATH  MathSciNet  Google Scholar 

  11. Goetz, A., ‘Piecewise isometries – an emerging area of dynamical systems’, Trends Math: Fractals in Graz 2001 (2001) 135–144.

  12. M. Boshernitzan A. Goetz (2003) ArticleTitle‘A dichotomy for a two parameter piecewise rotation’ Ergodic Theory Dyn. Syst. 23 1–12 Occurrence Handle1992662 Occurrence Handle10.1017/S0143385702001396

    Article  MathSciNet  Google Scholar 

  13. A. Goetz (2000) ArticleTitle‘Dynamics of piecewise isometries’ Illinois J. Math. 44 IssueID3 465–478 Occurrence Handle0964.37009 Occurrence Handle1772421

    MATH  MathSciNet  Google Scholar 

  14. Goetz, A., ‘A self-similar example of a piecewise isometric attractor’, in: Dynamical systems (Luminy-Marseille), World Scientific Publishing, River Edge NJ, (1998) pp. 248–258.

  15. P. Ashwin Xin-Chu. Fu (2002) ArticleTitle‘On the geometry of orientation-preserving planar piecewise isometries’ J. Nonlinear Sci. 12 207–240 Occurrence Handle01837429 Occurrence Handle1905204 Occurrence Handle10.1007/s00332-002-0477-1 Occurrence Handle2002JNS....12..207A

    Article  MATH  MathSciNet  ADS  Google Scholar 

  16. Deane, J.H.B. and Hamill, D.C., ‘Analysis, simulation and experimental study of chaos in the buck converter’, Power Electronics Specialists Conference Record, Vol. II, (1990) pp. 491–498.

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  1. Department of Mathematics and Statistics, School of Engineering and Physical Sciences, University of Surrey, Guildford, GU2 7XH, UK

    Jonathan H. B. Deane

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  1. Jonathan H. B. Deane
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Correspondence to Jonathan H. B. Deane.

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Deane, J.H.B. Piecewise Isometries: Applications in Engineering. Meccanica 41, 241–252 (2006). https://doi.org/10.1007/s11012-005-5895-3

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