Direct transition from a stable equilibrium to quasiperiodicity in non-smooth systems

The purpose of this Letter is to show how a border-collision bifurcation in a piecewise-smooth dynamical system can produce a direct transition from a stable equilibrium point to a two-dimensional invariant torus. Considering a system of nonautonomous differential equations describing the behavior of a power electronic DC/DC converter, we first determine the chart of dynamical modes and show that there is a region of parameter space in which the system has a single stable equilibrium point. Under variation of the parameters, this equilibrium may collide with a discontinuity boundary between two smooth regions in phase space. When this happens, one can observe a number of different bifurcation scenarios. One scenario is the continuous transformation of the stable equilibrium into a stable period-1 cycle. Another is the transformation of the stable equilibrium into an unstable period-1 cycle with complex conjugate multipliers, and the associated formation of a two-dimensional (ergodic or resonant) torus.

[1]  Soumitro Banerjee,et al.  Robust Chaos , 1998, chao-dyn/9803001.

[2]  Erik Mosekilde,et al.  Bifurcations and chaos in piecewise-smooth dynamical systems , 2003 .

[3]  Piotr Kowalczyk,et al.  A codimension-two scenario of sliding solutions in grazing–sliding bifurcations , 2006 .

[4]  Erik Mosekilde,et al.  Torus birth bifurcations in a DC/DC converter , 2006, IEEE Transactions on Circuits and Systems I: Regular Papers.

[5]  Alan R. Champneys,et al.  Two-Parameter Discontinuity-Induced bifurcations of Limit Cycles: Classification and Open Problems , 2006, Int. J. Bifurc. Chaos.

[6]  Harry Dankowicz,et al.  Local analysis of co-dimension-one and co-dimension-two grazing bifurcations in impact microactuators , 2005 .

[7]  Harry Dankowicz,et al.  Low-velocity impacts of quasiperiodic oscillations , 2002 .

[8]  James A. Yorke,et al.  Border-collision bifurcations including “period two to period three” for piecewise smooth systems , 1992 .

[9]  Mitrajit Dutta,et al.  Multiple attractor bifurcations: A source of unpredictability in piecewise smooth systems , 1999 .

[10]  Erik Mosekilde,et al.  Birth of bilayered torus and torus breakdown in a piecewise-smooth dynamical system , 2006 .

[11]  Erik Mosekilde,et al.  Border-collision bifurcations in a dynamic management game , 2006, Comput. Oper. Res..

[12]  Fabio Dercole,et al.  Numerical sliding bifurcation analysis: an application to a relay control system , 2003 .

[13]  J. Yorke,et al.  Bifurcations in one-dimensional piecewise smooth maps-theory and applications in switching circuits , 2000 .

[14]  F Dercole,et al.  Coevolution of slow–fast populations: evolutionary sliding, evolutionary pseudo-equilibria and complex Red Queen dynamics , 2006, Proceedings of the Royal Society B: Biological Sciences.

[15]  Tomasz Kapitaniak,et al.  Multiple choice bifurcations as a source of unpredictability in dynamical systems , 1998 .

[16]  Somnath Maity,et al.  Border collision route to quasiperiodicity: Numerical investigation and experimental confirmation. , 2006, Chaos.

[17]  George C. Verghese,et al.  Nonlinear Phenomena in Power Electronics , 2001 .

[18]  C J Budd,et al.  Grazing and border-collision in piecewise-smooth systems: a unified analytical framework. , 2001, Physical review letters.

[19]  R. Leine,et al.  Bifurcations in Nonlinear Discontinuous Systems , 2000 .

[20]  Hiroshi Kawakami,et al.  Bifurcation analysis of switched dynamical systems with periodically moving borders , 2004, IEEE Transactions on Circuits and Systems I: Regular Papers.

[21]  Erik Mosekilde,et al.  Border-collision bifurcations on a two-dimensional torus , 2002 .

[22]  Ott,et al.  Border-collision bifurcations: An explanation for observed bifurcation phenomena. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[23]  B. Brogliato.  Nonsmooth Mechanics: Models, Dynamics and Control , 1999 .