Transition from a stable node equilibrium to quasiperiodicity in piecewise-smooth systems
Periodic Control Systems, Volume # 3 | Part# 1
Zhanybai T. Zhusubaliyev; Sergey Chevychelov; Erik Mosekilde
Digital Object Identifier (DOI)
nonlinear theory,control system analysis,stability analysis,chaos,equilibrium
Considering a two-dimensional system of nonautonomous differential equations with discontinuous right-hand sides describing the behavior of a DC/DC converter with pulse-width modulated control, the paper demonstrates how a two-dimensional invariant torus can arise from a stable equilibrium point. We determine the chart of dynamical modes and show that there is a region of parameter space in which the system has a single stable node equilibrium point. Under variation of the parameters, this equilibrium may collide with a discontinuity boundary between two smooth regions in the phase space. When this happens, one can observe a variety of different bifurcation scenarios. One scenario is the continuous transformation of the stable equilibrium into a stable period-1 focus. A second is the transformation of the stable node equilibrium into an unstable period-1 focus, and the associated formation of a two-dimensional (ergodic or resonant) torus.
 Banerjee, S. and Verghese, G. C., Eds. (2001). Nonlinear Phenomena in Power Electronics. IEEE Press. New York, USA.  Banerjee, S., P. Ranjan and C. Grebogi (2000). Bifurcations in two-dimensional piecewise smooth maps -- Theory and applications in switching circuits. IEEE Trans. Circ. Syst. I: Fund. Theory and Appl. 47(5), 633-643.  Dankowicz, H., P. Piiroinen and A. B. Nordmark (2002). Low-velocity impacts of quasiperiodic oscillations. Chaos, Solitons & Fractals 14(2), 241-255.  di Bernardo, M., C. J. Budd and A. R. Champneys (2001). Grazing and border-collision in piecewise-smooth systems: A unified analytical framework. Phys. Rev. Lett. 86(12), 2553- 2556.  di Bernardo, M., M. I. Feigin, S. J. Hogan and M. E. Homer (1999). Local analysis of C-bifurcations in n-dimensional piecewisesmooth dynamical systems. Chaos, Solitons and Fractals 10(11), 1881-1908.  Dutta, M., H. E. Nusse, E. Ott, J. A. Yorke and G-H. Yuan (1999). Multiple attractor bifurcations: A source of unpredictability in piecewise smooth systems. Phys. Rev. Lett. 83, 4281-4284.  Feigin, M. I. (1970). Doubling of the oscillation period with C-bifurcations in piecewise continuous systems. Prikl. Mat. Mekh. 34(5), 861- 869. in Russian.  Feigin, M. I. (1994). Forced Oscillations in Systems with Discontinuous Nonlinearities. Nauka Publ. Moscow. in Russian.  Kapitaniak, T. and Yu. L. Maistrenko (1998). Multiple choice bifurcations as a source of unpredictability in dynamical systems. Phys. Rev. E 58(4), 5161-5163.  Kowalczyk, P., di Bernardo, A. R. Champneys, S. J. Hogan, M. Homer, Yu. A. Kuznetsov and A. B. Nordmark (2006). Two-parameter discontinuity-induced bifurcations of limit cycles: Classification and open problems. Int. J. Bifurcat. Chaos 16(3), 601-629.  Leine, R. I., D. H. Van Campen and B. L. Van De Vrande (2000). Bifurcations in non-linear discontinuous systems. Nonlinear Dynamics 23, 105-164.  Nusse, H. E. and J. A. Yorke (1992). Border-collision bifurcations including "period two to period three" for piecewise smooth systems. Physica D 57, 39-57.  Nusse, H. E., E. Ott and J. A. Yorke (1994). Border-collision bifurcations: An explanation for observed bifurcation phenomena. Phys. Rev. E 49, 1073-1076.  Zhusubaliyev, Zh. T. and E. Mosekilde (2003). Bifurcations and Chaos in Piecewise-Smooth Dynamical Systems. World Scientific. Singapore.  Zhusubaliyev, Zh. T. and E. Mosekilde (2006). Torus birth bifurcation in a DC/DC converter. IEEE Trans. Circ. Syst. I: Fund. Theory and Appl. 53(8), 1839-1850.  Zhusubaliyev, Zh. T., E. Mosekilde, S. M. Maity, S. Mohanan and S. Banerjee (2006). Border collision route to quasiperiodicity: Numerical investigation and experimental confirmation. Chaos 16, 023122.