On the geometry of orientation-preserving planar piecewise isometries

Journal of Nonlinear Science

Volumn 12, Issue 3, 2002, Pages 207-240

  Ashwin, P ^a   Fu, X C ^a  

^a UNIVERSITY OF EXETER   (United Kingdom)

Author keywords

Discontinuous dynamics; Nonhyperbolic dynamics; Piecewise isometry

Indexed keywords

EID: 0043205100     PISSN: 09388974     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00332-002-0477-1     Document Type: Article

References (28)

1. R. L. Adler, B. Kitchens, and C. Tresser, Dynamics of nonergodic piecewise affine maps of the torus. Ergod. Th. & Dyn. Sys. 21 (2001) 959-999.
2. D. Aharonov, R. L. Devaney, and U. Elias, The dynamics of a piecewise linear map and its smooth approximation. Intl. J. Bifn. Chaos 7 (1997) 351-371.
3. P. Ashwin, Nonsmooth invariant circles in digital overflow oscillations, Proc. 4th Intl. Workshop on Nonlinear Dynamics of Electronic Systems, Sevilla (1996) 417-422.
4. P. Ashwin, Elliptic behaviour in the sawtooth standard map. Phys. Lett. A 232 (1997) 409-416.
5. P. Ashwin, J. H. B. Deane, and X.-C. Fu, Properties of the invariant disk packing, in a model bandpass sigma-delta modulator. To appear in Intl. J. Bifn. Chaos (2002).
6. P. Ashwin and X.-C. Fu, Tangencies in invariant circle packings for certain planar piecewise isometries are rare. Dynamical Systems 16, 4 (2001) 333-345.
7. M. Boshernitzan and I. Kornfeld, Interval translation mappings. Ergod. Th. & Dyn. Sys. 15 (1995) 821-832.
8. J. Buzzi, Piecewise isometries have zero topological entropy. Ergod. Th. & Dyn. Sys. 21 (2001) 1371-1377.
9. L. O. Chua and T. Lin, Chaos in digital filters. IEEE Trans. CAS 35 (1988) 648-658.
10. A. C. Davies, Nonlinear oscillations and chaos from digital filter overflow. Phil. Trans. R. Soc. Lond. A 353 (1995) 85-99.
11. O. Feely and D. Fitzgerald, Nonideal and chaotic behaviour in bandpass sigma-delta modulators, Proc. 4th Intl. Workshop on Nonlinear Dynamics of Electronic Systems, Sevilla (1996).
12. M. Field, Lectures on bifurcations, dynamics and symmetry, Pitman's Research Notes in Mathematics 356, Addison-Wesley (1996).
13. A. Goetz, Perturbation of 8-attractors and births of satellite systems. Intl. J. Bifn. Chaos 8 (1998) 1937-1956.
14. A. Goetz, Dynamics of a piecewise rotation. Continuous and Discrete Dynamical Systems 4 (1998) 593-608.
15. A. Goetz, Stability of piecewise rotations and affine maps. Nonlinearity 14 (2001) 205-219.
16. A. Goetz, Dynamics of a piecewise isometries. Illinois J. Math. 44 (2000) 465-478.
17. A. Goetz, A self-similar example of a piecewise isometric attractor. Dynamical Systems (Luminy-Marseille, 1998), 248-258, World Sci. Publishing, River Edge, NJ, (2000).
18. E. Gutkin and H. Haydn, Topological entropy of generalized polygon exchanges. Bull. Amer. Math. Soc. 32 (1995) 50-57.
19. B. Kahng, Dynamics of symplectic piecewise affine elliptic rotation maps on tori. To appear in Ergod. Th. & Dyn. Sys. (2002).
20. A. Katok and B. Hasselblat, Introduction to Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge 1995.
21. L. Kocarev, C. W. Wu, and L. O. Chua, Complex behaviour in digital filters with overflow nonlinearity: Analytical results. IEEE Trans CAS-II 43 (1996) 234-246.
22. H. T. Levine, Singularities of differentiable manifolds; notes of lectures by R. Thom, In Proceedings of Liverpool Singularities - Symposium I, ed. C. T. C. Wall, Springer Lecture Notes in Mathematics 192 (1970).
23. J. H. Lowenstein and F. Vivaldi, Embedding dynamics for round-off errors near a periodic orbit. Chaos 10 (2000) 747-755.
24. M. Mendes, Quasi-invariant attractors in piecewise isometries. Preprint, Univ. of Surrey (2001).
25. G. Poggiaspala, unpublished DEA thesis, University of Marseille-Luminy (2000).
26. A. J. Scott, C. A. Holmes, and G. J. Milburn, Hamiltonian mappings and circle packing phase spaces. Physica D 155 (2001) 34-50.
27. S. Tabachnikov, On the dual billiard problem. Adv. Math. 115 (1995) 221-249.
28. F. Vivaldi and A. V. Shaidenko, Global stability of a class of discontinuous dual billiards. Commun. Math. Phys. 110 (1987) 625-640.