Dissecting shrimps: results for some one-dimensional physical models

This paper describes how certain shrimp-like clusters of stability organize themselves in the parameter space of dynamical systems. Clusters are composed of an infinite affine-similar repetition of a basic elementary cell containing two primay noble points, a head and a tail, defining an axis of approximate symmetry. Knowledge of the axis and the skewness of the k-periodic main cell of k×2n cluster is enough to define the orientation of the whole cluster in space. Peculiarly simple directions along which shrimp-like clusters align are formed by the locus of doubly degenerate saddle zero multipliers corresponding to the main shrimp head. In addition, we report a family of models having the boundaries of all isoperiodic domains of stability totally degenerate and describe different aspects of their mathematical arrangement and some of their consequences for example, that shrimps are diffeomorphic copies of shrimps.