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Transition From a Stable Node Equilibrium to Quasiperiodicity in Piecewise-Smooth Systems

Zhanybai Zhusubaliyev, Sergey Chevychelov, Erik Mosekilde
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.
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