In this paper, we study some nonlinear behaviors in a two-dimensional system defined by a Boost Converter controlled by CPWM (Centered Pulse-Width Modulation) and a ZAD (Zero Average Dynamics) strategy. The dynamics was analyzed using a discrete-time map, which consists of a sampled system at each switching cycle. The structure of the two-parametric space is characterized analytically. This allows proving the existence and stability of an infinite number of codimension-one curves that intersect at the same point in the two-parametric space. This phenomenon has been called a big-bang bifurcation.
In this paper, we provide for the first time rigorous mathematical results regarding the rich dynamics of piecewise linear memristor oscillators. In particular, for each nonlinear oscillator given in [Itoh & Chua, 2008], we show the existence of an infinite family of invariant manifolds and that the dynamics on such manifolds can be modeled without resorting to discontinuous models. Our approach provides topologically equivalent continuous models with one dimension less but with one extra parameter associated to the initial conditions. It is possible to justify the periodic behavior exhibited by three-dimensional memristor oscillators, by taking advantage of known results for planar continuous piecewise linear systems. The analysis developed not only confirms the numerical results contained in previous works [Messias et al., 2010; Scarabello & Messias, 2014] but also goes much further by showing the existence of closed surfaces in the state space which are foliated by periodic orbits. The important role of initial conditions that justify the infinite number of periodic orbits exhibited by these models, is stressed. The possibility of unsuspected bistable regimes under specific configurations of parameters is also emphasized.
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