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 work, we study the dynamics of a three-dimensional, continuous, piecewise smooth map. Much of the nontrivial dynamics of this map occur when its fixed point or periodic orbit hits the switching manifold resulting in the so-called border collision bifurcation. We study the local and global bifurcation phenomena resulting from such borderline collisions. The conditions for the occurrence of nonsmooth period-doubling, saddle-node, and Neimark–Sacker bifurcations are derived. We show that dangerous border collision bifurcation can also occur in this map. Global bifurcations arise in connection with the occurrence of nonsmooth Neimark–Sacker bifurcation by which a spiral attractor turns into a saddle focus. The global dynamics are systematically explored through the computation of resonance tongues and numerical continuation of mode-locked invariant circles. We demonstrate the transition to chaos through the breakdown of mode-locked torus by degenerate period-doubling bifurcation, homoclinic tangency, etc. We show that in this map a mode-locked torus can be transformed into a quasiperiodic torus if there is no global bifurcation.