References
- R. Adler, B. Kitchens, and C. Tresser, Dynamics of non-ergodic piecewise affine maps of the torus, Ergo. The. & Dynam. Sys. 21 (2001), 959-999. https://doi.org/10.1017/S0143385701001468
- E. Akin, The general topology of dynamical systems, Graduate Studies in Mathematics, 1, American Mathematical Society, Providence, RI, 1993. https://doi.org/10.1090/gsm/001
- Z. Artstein and S. V. Rakovic, Feedback and invariance under uncertainty via setiterates, Automatica J. IFAC 44 (2008), no. 2, 520-525. https://doi.org/10.1016/j.automatica.2007.06.013
- P. Ashwin, Non-smooth invariant circles in digital overflow oscillations, in Proceedings of the 4th International Workshop on Nonlinear Dynamics and Electronic Systems, pp. 417-422, Seville, Spain, 1996.
- P. Ashwin, Elliptic behaviour in the sawtooth standard map, Phys. Lett. A 232 (1997), no. 6, 409-416. https://doi.org/10.1016/S0375-9601(97)00455-6
- P. Ashwin, W. Chambers, and G. Petrov, Lossless digital filter overflow oscillations: approximation of invariant fractals, Int. J. Bifur. Chaos Appl. Sci. Eng. 7 (1997), 2603-2610. https://doi.org/10.1142/S021812749700176X
- P. Ashwin, J. H. B. Deane, and X. Fu, Dynamics of a bandpass sigma-delta modulator as a piecewise isometry, in Proc. IEEE Int. Symp. Circuit Sys., pp. 811-814, Sydney, Australia, 2001.
- P. Ashwin, X. Fu, T. Nishikawa, and K. Zyczkowski, Invariant sets for discontinuous parabolic area-preserving torus maps, Nonlinearity 13 (2000), no. 3, 819-835. https://doi.org/10.1088/0951-7715/13/3/317
- F. Blanchini, Set invariance in control, Automatica J. IFAC 35 (1999), no. 11, 1747-1767. https://doi.org/10.1016/S0005-1098(99)00113-2
- J. Buzzi, Piecewise isometries have zero entropy, Ergo. The. & Dynam. Sys. 21 (2001), 1371-1377.
- J. Buzzi and P. Hubert, Piecewise monotone maps without periodic points: rigidity, measures and complexity, Ergodic Theory Dynam. Systems 24 (2004), no. 2, 383-405. https://doi.org/10.1017/S0143385703000488
- A. C. Davis, Nonlinear oscillations and chaos from digital filter overflow, Phil. Trans. Royal Soc. Lond. A-353 (1995), 85-99. https://doi.org/10.1098/rsta.1995.0092
- X.-C. Fu, F.-Y. Chen, and X.-H. Zhao, Dynamical properties of 2-torus parabolic maps, Nonlinear Dynam. 50 (2007), no. 3, 539-549. https://doi.org/10.1007/s11071-006-9179-9
- X.-C. Fu and J. Duan, Global attractors and invariant measures for non-invertible planar piecewise isometric maps, Phys. Lett. A 371 (2007), no. 4, 285-290. https://doi.org/10.1016/j.physleta.2007.06.033
- X.-C. Fu and J. Duan, On global attractors for a class of nonhyperbolic piecewise affine maps, Phys. D 237 (2008), no. 24, 3369-3376. https://doi.org/10.1016/j.physd.2008.07.012
- A. Goetz, Dynamics of a piecewise rotation, Discrete Contin. Dynam. Systems 4 (1998), no. 4, 593-608. https://doi.org/10.3934/dcds.1998.4.593
- A. Goetz, Sofic subshifts and piecewise isometric systems, Ergo. The. & Dynam. Sys. 19 (1999), 1485-1501. https://doi.org/10.1017/S0143385799151964
- A. Goetz and M. Mendes, Piecewise rotations: bifurcations, attractors and symmetries, in Bifurcation, symmetry and patterns (Porto, 2000), 157-165, Trends Math, Birkhauser, Basel, 2003.
- A. Goetz and G. Poggiaspalla, Rotations by π/7, Nonlinearity 17 (2004), no. 5, 1787-1802. https://doi.org/10.1088/0951-7715/17/5/013
- K. Hrbacek and T. Jech, Introduction to Set Theory, second edition, Monographs and Textbooks in Pure and Applied Mathematics, 85, Marcel Dekker, Inc., New York, 1984.
- T. Jech, Set theory, the third millennium edition, revised and expanded., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.
- B. Kahng, Dynamics of symplectic piecewise affine elliptic rotations maps on tori, Ergod. Th. & Dynam. Sys. 22 (2002), 483-505. https://doi.org/10.1017/S0143385702000238
- B. Kahng, Dynamics of kaleidoscopic maps, Adv. Math. 185 (2004), no. 1, 178-205. https://doi.org/10.1016/S0001-8708(03)00170-1
- B. Kahng, The unique ergodic measure of the symmetric piecewise toral isometry of rotation angle θ = kπ/5 is the Hausdorff measure of its singular set, Dyn. Syst. 19 (2004), no. 3, 245-264. https://doi.org/10.1080/14689360410001729595
- B. Kahng, Maximal invariant sets of multiple valued iterative dynamics in disturbed control systems, Int. J. Circ. Sys. Signal Proc. 2 (2008), 113-120.
- B. Kahng, Positive invariance of multiple valued iterative dynamical systems in disturbed control models, in Proceedings of the 17th IEEE Mediterranean Conference on Control and Automation, pp. 664-668, Thessaloniki, Greece, 2009.
- B. Kahng, Redifining chaos: Devaney-chaos for piecewise isometric dynamical systems, International J. Math. Models and Methods in Applied Sciences 4 (2009), 317-326.
- B. Kahng, Singularities of 2-dimensional invertible piecewise isometric dynamics, Chaos 19 (2009), 023115. https://doi.org/10.1063/1.3119464
- B. Kahng, The approximate control problems of the maximal invariant sets of non-linear discrete-time disturbed control dynamical systems: an algorithmic approach, in Proceedings of the 4th International Conference on Control, Automation and Systems, pp. 1513-1518, KINTEX, Gyeonggi-do, Korea, 2010.
- B. Kahng, Multiple valued iterative dynamics models of nonlinear discrete-time control dynamical systems with disturbance, J. Korean Math. Soc. 50 (2013), no. 1, 17-39. https://doi.org/10.4134/JKMS.2013.50.1.017
- B. Kahng, An optimization of maximal invariance in a class of multiple valued iterative dynamics models of nonlinear disturbed control systems, Fractals 24 (2016), 1650044. https://doi.org/10.1142/S0218348X16500444
- B. Kahng, Theory and implementation of the multiple valued iterative dynamics algorithms for a class of singularly disturbed nonlinear control dynamical systems, Appl. Sci. 1 (2019), 1:1061.
- B. Kahng and J. Davis, Maximal dimensions of uniform Sierpinski fractals, Fractals 18 (2010), no. 4, 451-460. https://doi.org/10.1142/S0218348X10005135
- B. Kahng, M. Gomez, and E. Padilla, Visualization algorithms for the steady state sets of a class of singularly disturbed nonlinear control dynamical systems, Int. J. Math. Models Methods 10 (2016), 237-243.
- B. Kahng and M. Mendes, The characterization of maximal invariant sets of non-linear discrete-time control dynamical systems, Discrete Contin. Dyn. Syst. 2013, Dynamical systems, differential equations and applications. 9th AIMS Conference. Suppl., 393-406, 2013. https://doi.org/10.3934/proc.2013.2013.393
- E. Kerrigan, J. Lygeros, and J. M. Maciejowski, A geometric approach to reachability computations for constrained discrete-time systems, in IFAC World Congress, Barcelona, Spain, 2002.
- E. Kerrigan and J. M. Maciejowski, Invariant sets for constrained nonlinear discrete-time systems with application to feasibility in model predictive control, in Proc. 39th IEEE Conf. on Decision and Control, Sydney, Australia, 2000.
- L. Kocarev, C. W. Wu, and L. O. Chua, Complex behavior in digital filters with overflow nonlinearity: analytical results, IEEE Trans. Circuits Systems II 43 (1996), 234-246. https://doi.org/10.1109/82.486469
- T. Lin and L. O. Chua, A new class of pseudo-random generators based on chaos in digital filters, Int. J. Cir. Th. Appl., 21 (1993), 473-480. https://doi.org/10.1002/cta.4490210506
- J. H. Lowenstein, Fixed-point densities for a quasiperiodic kicked oscillator map, Chaos 5 (1995), 566-577. https://doi.org/10.1063/1.166126
- J. H. Lowenstein, Aperiodic orbits of piecewise rational rotations of convex polygons with recursive tiling, Dyn. Syst. 22 (2007), no. 1, 25-63. https://doi.org/10.1080/14689360601028100
- J. Lowenstein, S. Hatjispyros, and F. Vivaldi, Quasi-periodicity, global stability and scaling in a model of Hamiltonian round-off, Chaos 7 (1997), no. 1, 49-66. https://doi.org/10.1063/1.166240
- J. H. Lowenstein, K. L. Kouptsov, and F. Vivaldi, Recursive tiling and geometry of piecewise rotations by π/7, Nonlinearity 17 (2004), no. 2, 371-395. https://doi.org/10.1088/0951-7715/17/2/001
- J. H. Lowenstein, G. Poggiaspalla, and F. Vivaldi, Sticky orbits in a kickedoscillator model, Dyn. Syst. 20 (2005), no. 4, 413-451. https://doi.org/10.1080/14689360500167611
- D. Q. Mayne, M. M. Seron, and S. V. Rakovic, Robust model predictive control of constrained linear systems with bounded disturbances, Automatica J. IFAC 41 (2005), no. 2, 219-224. https://doi.org/10.1016/j.automatica.2004.08.019
- M. Mendes, Dynamics of piecewise isometric systems with particular emphasis to the g map, Ph. D. Thesis, University of Surrey, 2001.
- M. Mendes, On maximal invariant sets, preprint, (August, 2006).
- M. J. Ogorza lek, Chaos and complexity in nonlinear electronic circuits, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, 22, World Scientific Publishing Co. Pte. Ltd., Singapore, 1997. https://doi.org/10.1142/9789812798626
- C. J. Ong and E. G. Gilbert, Constrained linear systems with disturbances: enlargement of their maximal invariant sets by nonlinear feedback, in Proc. Amer. Control Conf. Minneapolis, pp. 5246-5251, MN, 2006.
- M. Pollicott and H. Weiss, The dimensions of some self-affine limit sets in the plane and hyperbolic sets, J. Statist. Phys. 77 (1994), no. 3-4, 841-866. https://doi.org/10.1007/BF02179463
- S. V. Rakovic and M. Fiacchini, Invariant Approximations of the Maximal Invariant Set or "Encircling the Square, in IFAC World Congress, Seoul, Korea, July 2008.
- S. V. Rakovic, E. C. Kerrigan, K. I. Kouramas, and D. Q. Mayne, Invariant approximation of the minimal robust positively invariant set, IEEE Trans. Automat. Control 50 (2005), no. 3, 406-410. https://doi.org/10.1109/TAC.2005.843854
- S. V. Rakovic, E. C. Kerrigan, D. Q. Mayne, and K. I. Kouramas, Optimized robust control invariance for linear discrete-time systems: theoretical foundations, Automatica J. IFAC 43 (2007), no. 5, 831-841. https://doi.org/10.1016/j.automatica.2006.11.006
- S. V. Rakovic, E. C. Kerrigan, D. Q. Mayne, and J. Lygeros, Reachability analysis of discrete-time systems with disturbances, IEEE Trans. Automat. Control 51 (2006), no. 4, 546-561. https://doi.org/10.1109/TAC.2006.872835
- M. Trovati and P. Ashwin, Tangency properties of a pentagonal tiling generated by a piecewise isometry, Chaos 17 (2007), no. 4, 043129, 11 pp. https://doi.org/10.1063/1.2825291