ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Zhusubaliev Z. T., Yanochkina O. O. Bifurcations of a two-­dimensional torus in piecewise-­smooth dynamical systems. Izvestiya VUZ. Applied Nonlinear Dynamics, 2009, vol. 17, iss. 6, pp. 86-98. DOI: 10.18500/0869-6632-2009-17-6-86-98

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
(downloads: 161)
Article type: 

Bifurcations of a two-­dimensional torus in piecewise-­smooth dynamical systems

Zhusubaliev Zhanybaj Tursunbaevich, Federal state budgetary educational institution of higher professional education "South-West state University"
Yanochkina Olga Olegovna, Federal state budgetary educational institution of higher professional education "South-West state University"

Considering a set of coupled nonautonomous differential equations with discontinuous right-hand sides, we discuss two different scenarios for torus birth bifurcations in piecewise-smooth dynamical systems. One scenario is the continuous transformation of the stable equilibrium into an unstable focus period-1 orbit surrounded by a resonant or ergodic torus. Another is the transition from a stable periodic orbit to an invariant torus through a border-collision bifurcation in which two complex-conjugate multipliers jump abruptly from the inside to the outside of the unit circle.

  1. Filippov AF. Differential equations with a discontinuous right. Moscow: Nauka; 1985. 224 p. (In Russian).
  2. Feigin MI. Forced oscillations of systems with discontinuous nonlinearities. Moscow: Nauka; 1994. 285 p. (In Russian).
  3. Zhusubaliyev ZhT, Mosekilde E. Bifurcations and Chaos in Piecewise-Smooth Dynamical Systems. Singapore: World Scientific; 2003. 376 p.
  4. Leine RI, Nijmeijer H. Dynamics and Bifurcations of Non-Smooth Mechanical Systems. Berlin: Springer Verlag; 2004. 236 p.
  5. Arnol'd VI, Afraimovich VS, Ilyashenko YuS, Shilnikov LP. Bifurcation theory. Dynamical systems – 5, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr. Moscow: VINITI; 1986. 5–218 p. (In Russian).
  6. Anishchenko VS, Astakhov VV, Neiman AB, Vadivasova TE, Schimansky-Geier L. Nonlinear Dynamics of Chaotic and Stochastic Systems. Tutorial and Modern Development. Berlin: Springer; 2007. 429 p.
  7. Feigin MI. Doubling the period of oscillation at C-bifurcations in piecemeal-continuous systems. Journal of Applied Mathematics and Mechanics. 1970;34(5):861–869.
  8. di Bernardo M, Feigin MI, Hogan SJ, Homer ME. Local analysis of C-bifurcations in n-dimensional piecewise-smooth dynamical systems. Chaos, Solitons and Fractals. 1999;10(11):1881--1908.
  9. Nusse HE, Yorke JA. Border-collision bifurcations including «period two to period three» for piecewise smooth systems. Physica D. 1992;57(1-2):39–57. DOI: 10.1016/0167-2789(92)90087-4.
  10. Banerjee S, Ranjan P, Grebogi C. Bifurcations in two-dimensional piecewise smooth maps – theory and applications in switching circuits. IEEE Trans. Circ. Syst. I. 2000;47(5):633--643. DOI: 10.1109/81.847870.
  11. Zhusubaliyev ZhT, Soukhoterin EA, Mosekilde E. Border-collision bifurcations and chaotic oscillations in a piecewise-smooth dynamical system. Int. J. Bifurcation Chaos. 2001;11(12):2977--3001. DOI: 10.1142/S0218127401003991.
  12. di Bernardo M, Budd CJ, Champneys AR. Grazing bifurcations in n-dimensional piecewise-smooth dynamical systems. Physica D. 2001;160(3-4):222--254. DOI: 10.1016/S0167-2789(01)00349-9.
  13. Banerjee S, Verghese GC. (Eds.) Nonlinear phenomena in power electronics. New York: IEEE Press; 2001. 472 p.
  14. Keener J, Sneyd J. Mathematical Physiology. New York: Springer Verlag; 1998. 767 p.
  15. Laugesen J, Mosekilde E. Border-collision bifurcations in a dynamic management game. Comp. Oper. Res. 2006;33(2):464--478 p. DOI: 10.1016/j.cor.2004.06.016.
  16. di Bernardo M, Budd C, Champneys AR, Kowalczyk P, Nordmark AB, Olivar G, Piiroinen PT. Bifurcations in nonsmooth dynamical systems. SIAM Review. 2008;50(4):629--701. DOI: 10.1137/050625060.
  17. Zhusubaliyev ZhT, Mosekilde E. Torus birth bifurcation in a DC/DC converter. IEEE Trans. Circ. Syst. I. 2006;53(8):1839--1850. DOI: 10.1109/TCSI.2006.879060.
  18. Zhusubaliyev ZhT, Mosekilde E. Birth of bilayered torus and torus breakdown in a piecewise-smooth dynamical system. Phys. Lett. A. 2006;351(3):167--174. DOI: 10.1016/j.physleta.2005.10.080.
  19. Zhusubaliyev ZT, Mosekilde E, Maity S, Mohanan S, Banerjee S. Border collision route to quasiperiodicity: Numerical investigation and experimental confirmation. Chaos. 2006;16(2):023122. DOI: 10.1063/1.2208565.
  20. Kobzev AV. Multi-zone pulse modulation. Novosibirsk: Nauka; 1979. 300 p. (In Russian).
  21. Kobzev AV, Mikhalchenko GYa, Muzychenko NM. Modulation power sources REA. Tomsk: Radio i svyaz; 1990. 335 p. (In Russian).
  22. Lai JS, Peng FZ. Multilevel converters – a new breed of power converters. IEEE Trans. Ind. Appl. 1996;32(3):509--517. DOI: 10.1109/28.502161.
  23. Meynard TA, Foch H, Thomas P, Courault J, Jakob R, Nahrstaedt M. Multicell converters: Basic concepts and industry applications. IEEE Trans. Ind. Electron. 2002;49(5):955--964. DOI: 10.1109/TIE.2002.803174.
  24. Zhusubaliyev ZhT, Mosekilde E. Direct transition from a stable equilibrium to quasiperiodicity in non-smooth systems. Phys. Lett. A. 2008;372(13):2237--2246. DOI: 10.1016/j.physleta.2007.08.077.
  25. Helig AKh, Churilov AN. Vibrations and stability of nonlinear pulse systems. St. Petersburg: St. Petersburg University Press; 1993. 264 p. (In Russian).
  26. Pikovsky A, Rosenblum M, Kurts Yu. Synchronization. A fundamental nonlinear phenomenon. Moscow: Technosphera; 2003. 493 p. (In Russian).
Short text (in English):
(downloads: 117)