Quasi-periodicity and border-collision bifurcations in a dc-dc converter with pulsewidth modulation

IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications

Volumn 50, Issue 8, 2003, Pages 1047-1057

  Zhusubaliyev, Zhanybai T ^a   Soukhoterin, Evgeniy A ^a   Mosekilde, Erik ^b  

^a SOUTHWEST STATE UNIVERSITY   (Russian Federation)

^b TECHNICAL UNIVERSITY OF DENMARK   (Denmark)

Author keywords

Border collision bifurcations (BCB); Chaos; Dc dc converters; Piecewise smooth systems; Power electronics; Quasi periodicity

Indexed keywords

BIFURCATION (MATHEMATICS); CHAOS THEORY; HYSTERESIS; POWER ELECTRONICS; PULSE WIDTH MODULATION; RESONANCE;

QUASI-PERIODICITY;

ELECTRIC CONVERTERS;

EID: 0042378939     PISSN: 10577122     EISSN: None     Source Type: Journal    
DOI: 10.1109/TCSI.2003.815196     Document Type: Article

References (46)

1. M.I. Feigin, "Doubling of the oscillation period with C-bifurcations in piecewise continuous systems" (in Russian), Prikl. Mat. Mekh.., vol. 34, no. 5, pp. 861-869, 1970.
2. _, "On the generation of sets of subharmonic modes in a piecewise continuous system" (in Russian), Prikl. Mat. Mekh., vol. 38, no. 5, pp. 810-818, 1974.
3. _, Forced Oscillations in Systems with Discontinuous Nonlinearities (in Russian). Moscow, Russia: Nauka, 1994.
4. _, "The increasingly complex structure of the bifurcation tree of a piecewise-smooth system," J. Appl. Math. Mech., vol. 59, pp. 853-863, 1995.
5. H. E. Nüsse and J. A. Yorke, "Border-collision bifurcations including "period two to period three" for piecewise smooth systems," Physica, vol. D57, pp. 39-57, 1992.
6. _, "Border-collision bifurcation for piecewise smooth one-dimensional maps," Int. J. Bifurcation Chaos, vol. 5, no. 1, pp. 189-207, 1995.
7. H. E. Nusse, E. Ott, and J. A. Yorke, "Border-collision bifurcations: An explanation for observed bifurcation phenomena," Phys. Rev., vol. E49, pp. 1073-1076, 1994.
8. M. di Bernardo, M. I. Feigin, S. J. Hogan, and M. E. Homer, "Local analysis of C-bifurcations in n-dimensional piecewise-smooth dynamical systems," Chaos, Solitons Fractals, vol. 10, no. 11, pp. 1881-1908, 1999.
9. S. Banerjee, J. A. Yorke, and C. Grebogi, "Robust chaos," Phys. Rev. Lett., vol. 80, no. 14, pp. 3049-3052, 1998.
10. S. Banerjee and C. Grebogi, "Border collision bifurcations in two-dimensional piecewise smooth maps," Phys. Rev. E, vol. 59, no. 4, pp. 4052-4061, 1999.
11. S. Banerjee, M. S. Karthik, G. Yuan, and J. A. Yorke, "Bifurcations in one-dimensional piecewise smooth maps - Theory and applications in switching circuits," IEEE Trans. Circuits Syst. I, vol. 47, pp. 389-394, Mar. 2000.
12. S. Banerjee, P. Ranjan, and C. Grebogi, "Bifurcations in two-dimensional piecewise smooth maps - Theory and applications in switching circuits," IEEE Trans. Circuits Syst. I, vol. 47, pp. 633-643, May 2000.
13. M. di Bernardo, C. J. Budd, and A. R. Champneys, "Grazing and border-collision in piecewise-smooth systems: A unified analytical framework," Phys. Rev. Lett., vol. 86, no. 12, pp. 2553-2556, 2001.
14. _, "Corner collision implies border collision," Physica D, vol. 154, no. 3-4, pp. 171-194, 2001.
15. Z. T. Zhusubaliyev, E. A. Soukhoterin, and E. Mosekilde, "Border-collision bifurcations and chaotic oscillations in a piecewise-smooth dynamical system," Int. J. Bifurcation Chaos, vol. 11, no. 12, pp. 2977-3001, 2001.
16. _, "Border-collision bifurcations on a two-dimensional torus," Chaos, Solitons, Fractals, vol. 13, no. 9, pp. 1889-1915, 2002.
17. Z. T. Zhusubaliyev and E. Mosekilde, Bifurcations and Chaos in Piecewise-Smooth Dynamical Systems. Singapore: World Scientific, 2003.
18. G. H. Yuan, S. Banerjee, E. Ott, and J. A. Yorke, "Border-collision bifurcations in the buck converter," IEEE Trans. Circuits Syst. I, vol. 45, pp. 707-716, July 1998.
19. M. di Bernardo and F. Vasca, "Discrete-time maps for the analysis of bifurcations and chaos in dc-dc conveters," IEEE Trans. Circuits Syst. I, vol. 47, pp. 130-143, Feb. 2000.
20. M. di Bernardo and C. K. Tse, "Chaos in power electronics: An overview," in Chaos in Circuits and Systems, G. Chen and T. Ueta, Eds. Singapore: World Scientific, 2002, pp. 317-340.
21. D. C. Hamill and D. J. Jefferies, "Subharmonics and chaos in a controlled switched-mode power converter," IEEE Trans. Circuits Syst., vol. 35, pp. 1059-1061, Aug. 1988.
22. D. C. Hamill, J. H. B. Deane, and D. J. Jefferies, "Modeling of chaotic dc-dc conveters by iterated nonlinear mappings," IEEE Trans. Power Electron., vol. 7, pp. 25-36, Jan. 1992.
23. A. E. Aroudi and R. Leyva, "Quasiperiodic route to chaos in a PWM voltage-controlled dc-dc boost converter," IEEE Trans. Circuits Syst. I, vol. 48, pp. 967-978, Aug. 2001.
24. S. Banerjee and G. C. Verghese, Eds., Nonlinear Phenomena in Power Electronics. New York: IEEE Press, 2001.
25. V. S. Baushev and Z. T. Zhusubaliyev, "Indeterminable states of a voltage regulator with pulsewidth control," Elect. Technol., no. 3, pp. 85-98, 1992.
26. V. S. Baushev, Z. T. Zhusubaliyev, and S. G. Michal'chenko, "Stochastic features in the dynamic characteristics of a pulsewidth controlled voltage stabilizer," Elect. Technol., no. 1, pp. 137-150, 1996.
27. E. Fossas and G. Olivar, "Study of chaos in the buck converter," IEEE Trans. Circuits Syst. I. vol. 43, pp. 13-25, Jan. 1996.
28. A. E. Aroudi, L. Benadero, E. Toribio, and S. Machiche, "Quasiperiodicity and chaos in the dc-dc buck-boost converter," Int. J. Bifurcation Chaos, vol. 10, pp. 359-371, 2000.
29. V. S. Afraimovich and L. P. Shil'nikov, "Invariant two-dimensional tori, their breakdown and stochasticity," Amer. Math. Soc. Transl., vol. 149, no. 2, pp. 201-212, 1991.
30. J. Maurer and A. Libchaber, "Rayleigh-benard experiment in liquid helium: Frequency locking and the onset of turbulence," J. Phys. Lett., vol. 40, pp. L419-L423, 1979.
31. D. G. Aranson, M. A. Chori, G. R. Hall, and R. P. McGenehe, "Bifurcations from an invariant circle for two-parameter families of maps of the plane: A computer-assisted study," Commun. Math. Phys., vol. 83, pp. 303-354, 1982.
32. V. Franceschini, "Bifurcations of tori and phase locking in a dissipative system of differential equations," Physica D, vol. 6, no. 3, pp. 285-304, 1982.
33. J. Stavans, "Experimental study of quasi-periodicity in a hydrodynamical systems," Phys. Rev. A, vol. 35, pp. 4314-4328, 1987.
34. V. S. Anishchenko and M. A. Safonova, "Mechanism of destruction of invariant curve in a model map of the plane" (in Russian), Radiotekh. Elektron., vol. 32, pp. 1207-1216, 1987.
35. P. R. Severns and G. E. Bloom, Modern DC-to-DC Switchmode Power Converter Circuits. New York: Van Nostrand Reinhold, 1985.
36. P. P. K. Chetty, Switch-Mode Power Supply Design. Blue Ridge Summit, PA: TAB Books Inc., 1986.
37. B. K. Bose, Ed., Modern Power Electronics: Evolution, Technology, and Applications. New York: IEEE Press, 1992.
38. Y. Z. Tsypkin, Sampling Systems Theory and Its Application. New York: Macmillan, 1964.
39. A. K. Gelig and A. N. Churilov, Stability and Oscillations of Nonlinear Pulse-Modulated Systems. Boston, : Birkhäuser, 1998.
40. A. V. Kobzev, G. Y. Michal'chenko, and N. M. Muzhitchenko, Modulating Power Supplies of Radioelectronic Devices (in Russian). Tomsk, Russia: Radio i Svyaz, 1990.
41. B.-L. Hao, Elementary Symbolic Dynamics and Chaos in Dissipative Systems. Singapore: World Scientific, 1989.
42. Y. L. Maistrenko, V. L. Maistrenko, S. I. Vikul, and L. O. Chua, "Bifurcations of attracting cycles from time-delayed Chua's circuit," Int. J. Bifurcation Chaos, vol. 5, no. 3, pp. 653-671, 1995.
43. Z. T. Zhusubaliyev, E. A. Soukhoterin, V. N. Rudakov, Y. V. Kolokolov, and E. Mosekilde, "Bifurcations and chaotic oscillations in an automatic control relay system with hysteresis," Int. J. Bifurcation Chaos, vol. 11, no. 5, pp. 1193-1231, 2001.
44. L. E. Christiansen, T. Lehn-Schiøler, E. Mosekilde, P. Gránásy, and H. Matsushita, "Nonlinear characteristics of randomly excited transonic flutter," Math. Camp. Sim., vol. 58, pp. 385-405, 2002.
45. V. I. Arnol'd, "Small denominators. I. Mappings of the circumference into itself," AMS Trans. Series 2, vol. 46, pp. 213-284, 1965.
46. V. I. Arnol'd, Ed., Dynamical Systems V, Encyclopaedia of Mathematical Sciences. Berlin, Germany: Springer-Verlag, 1991, vol. 5.