Narrow Results

In this paper, we propose a qualitative and quantitative dynamical study about the evolution of nonsmooth torus in a Digital Delayed Pulse-Width Modulator (PWM) switched buck converter. We explain the birth and destruction of the torus by successive discontinuity induced bifurcations (DIBs). The Digital-PWM control is based on Zero Average Dynamics (ZAD) strategy and a one-period delay is included in the control law. The control parameter (k_s) of the ZAD strategy can be varied in a large range, ideally (-∞, ∞). On these borders, the dynamical behavior is the same and thus an annulus-like parameter space is considered. Under variation of k_s, the system gets closer to a codimension-two bifurcation point, where two simultaneous border-collision bifurcations (BC) meet. The system leaves the high-order periodic behavior and a high-order band torus appears. Wrinkles and nonsmoothness due to more and more successive border-collisions bifurcations cause the torus destruction in high-order band chaos with tent-map-like structures and Mandelbrot-like sets. Finally, the chaotic bands merge in one-band chaos. The switched converter is modeled as a piecewise linear system where an analytical expression of the Poincaré map is available. Characteristics as duality, symmetry and recurrent behavior are determined using additional numerical methods for the Poincaré map decomposition in periodic sequences.

In this paper, we report Mandelbrot-like bifurcation structures in a one-dimensional parameter space of real numbers corresponding to a dc-dc power converter modeled as a piecewise-smooth system with three zones. These fractal patterns have been studied in two-dimensional parameter space for smooth systems, but for nonsmooth systems has not been reported yet. The Mandelbrot-like sets we found are created in transition from the torus band to chaos band scenarios exhibited by a dc-dc buck power converter controlled by Delayed Pulse-Width Modulator (PWM) based on Zero Average Dynamics (or ZAD strategy), which corresponds to a piecewise-smooth system (PWS). The real parameter is provided by the PWM control strategy, namely ZAD strategy, and it can be varied in a large range, ideally (-∞, +∞). At -∞ and +∞ the dynamical behavior is the same, and thus we will describe the synamics in an ring-like parameter space. Mandelbrot-like borders are built by four chaotic bands, therefore these structures can be thought as instability islands where the state variables cannot be located. Using the Poincaré map approach we characterize the bifurcation structures and we describe recurrent patterns in different scales.