Interacting bifurcations in switching systems

We study a class of switching systems whose dynamics are characterized by an inner switching feedback loop and an outer continuous control loop. The loops have two widely separated time scales, performing fast-scale and slow-scale dynamics accordingly. Treating the two time scales separately, the stability of the systems can be studied, with results focusing on fast-scale bifurcation and slow-scale bifurcation. In current-mode controlled switching converters, period-doubling has been identified as fastscale bifurcation whereas Hopf type bifurcation has been found as slow-scale bifurcation. However, in practice, the fast-scale and the slow-scale dynamics are interacting because the inner loop that is responsible for the fast-scale dynamics is actually controlled by the slow-scale outer feedback. This paper investigates the coexisting fast-scale and slow-scale bifurcations in simple dc/dc converters under peak current-mode control operating in continuous conduction mode. Boundaries of stable region, slow-scale bifurcation region, fast-scale bifurcation region, coexisting fast and slow-scale bifurcation region are identified.