An Example of Superstable Quadratic Mapping of the Space

The superstability of a dynamical motion is defined with existence of a minus infinity Lyapunouv exponent, this mean that this motion is attractive. There are several methods for constructing 1-D polynomial mappings with attracting cycles or superstable cycles [1,2] based on Lagrange and Newton interpolations. Superstable phenomena in some 1-D maps embedded in circuits and systems are studied in [3, 4], these maps are obtained from the study of nonautonomous piecewise constant circuit and biological models [5–10]. Rich dynamical behaviors can be seen in the presence of superstability [5, 8, 9], especially, the attractivity of the motion that guarantees its stability. The essential motivation of the present work is to prove rigorously that a family of 3-D quadratic mappings is superstable in the sense that all its behaviors are superstable, i.e. they have a munis infinity Lyapunouv exponent for all bifurcation parameters in a specific region. This property of superstability is probably rare in n-D dynamical systems with n ≥ 2. Also, superstability is a local property in the space of bifurcation parameters, i.e., in general, not all the behaviors of the