Bifurcations in a piecewise-linear discrete model of the pulse modulated control system

   Purpose. In this paper we study degenerate bifurcations and merging bifurcations of chaotic attractors in a pulse-width modulated control system, the behavior of which is described by a bimodal piecewise linear continuous mapping. It is well known that in piecewise linear maps, classical bifurcations such as period doubling, transcritical and pitchfork, become degenerate, combining the properties of classical smooth bifurcations and border collision bifurcations.

   Methods. First we describe а technique for obtaining of a piecewise linear mapping from a vector field with a discontinuous right-hand side using the method construction of the Poincare map. Then are investigated degenerate period -doubling bifurcations by methods of the theory of critical lines for non-invertible maps.

   Results. We found that the considered mapping has an unusual property, which is as follows. At the flip bifurcation point for a fixed point, an interval I appears, on the boundaries of which two points of the period doubled cycle lie. Moreover, any point of this interval is a periodic point with a period of two. We have proved that periodic points with a period of two lying on the boundaries of this interval coincide with two switching manifolds. As a specific example of a real physical system, we consider a power converter system with pulse width modulated control, which is modeled by a piecewise linear mapping. Moreover, we experimentally show a fixed point, a 2-cycle and chaotic oscillations.

   Conclusion. Finally we have studied degenerate period-doubling bifurcations and merging bifurcations of cyclic chaotic attractors. Such bifurcation is also known as a merging crisis. At the bifurcation point, an unstable fixed point with a negative multiplier collides with the boundaries of a chaotic attractor. It is well known, that the boundaries of a chaotic attractor are formed by the so-called critical points and their images. At the moment of bifurcation, a homoclinic orbit arises. Due to the fact that the considered mapping is piecewise linear, the equations of bifurcation boundaries are obtained analytically, the solutions of which are either analytically or numerically.

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