p-adic Dynamical Systems with Applications to Biology and Social Sciences

Abstract

Let us study dynamical systems in the non-Archimedean number fields [155], [159], [169], [170], [58]. The main results are obtained for the fields of p-adic numbers and complex p-adic numbers. Already the simplest p-adic dynamical systems have very rich structure. There exist attractors, Siegel disks, cycles. There also appear such new structures as ‘fuzzy cycles’. A prime number p plays the role of a parameter of a dynamical system. The behaviour of iterations depends on this parameter very much. In fact, by changing p we can change crucially the behaviour: attractors may become centers of Siegel disks and vice versa, cycles of different length may appear and disappear…. Sometimes we cannot find general laws¹. We illustrate these properties by numerous examples. We are continuing these investigations with the aid of a computer on the basis of the complex of p-adic programs created by De Smedt [57].