Bifurcation Analysis of a Circuit-Related Generalization of the Shipmap

In this paper a bifurcation analysis of a piecewise-affine discrete-time dynamical system is carried out. Such a system derives from a well-known map which has good features from its circuit implementation point of view and good statistical properties in the generation of pseudo-random sequences. The considered map is a generalization of it and the bifurcation parameters take into account some common circuit implementation nonidealities or mismatches. It will be shown that several different dynamic situations may arise, which will be completely characterized as a function of three parameters. In particular, it will be shown that chaotic intervals may coexist, may be cyclical, and may undergo several global bifurcations. All the global bifurcation curves and surfaces will be obtained either analytically or numerically by studying the critical points of the map (i.e. extremum points and discontinuity points) and their iterates. In view of a robust design of the map, this bifurcation analysis should come before a statistical analysis, to find a set of parameters ensuring both robust chaotic dynamics and robust statistical properties.