Many-Valuedness of First Integrals, Their Domains and Many-Valued Solutions

Abstract

Many-valuedness of first integral is studied through the 2-dimension Lotka-Volterra model and it is proved that this many·valuedness brings nonexistence of the Puiseux expansions of the general solutions. In other words, a system with a certain many-valued first integral does not have the weak Painleve property. This impli~s that many-valuedness is necessary for their general solutions. In physical case, we consider the influence of the logarithmic branchings of first integrals on real motion and show that a first integral including logarithmic functions becomes single-valued almost every­ where in the real domain under a certain condition of singularity sets of the first integral. If a dynamical system has a sufficient number of global single-valued first inte­ grals, it is obvious that it has only regular motion. Even if a system has irregular motion including chaos, there is a sufficient number of global first integrals. However they are many~valued globally because they are yielded by the analytic prolongation of local single-valued first integrals which always exist. In other words, many­ valuedness of first integral is a necessary condition for the irregular motion. Then, in this many-valuedness, there must be a root of the difference between two phases of motion. Nevertheless there are few studies of the relations between this many­ valuedness and the behavior of the system as yet, although, there are many analytical or numerical studies 1 )-5) of some relations between many-valued solutions and the behavior of the system. Then we need the considerations of infinitely many­ valuedness of first integral in addition to that of solutions in order to grasp the irregular motion as global structure. In this paper, on two regions in complex phase space, we discuss many-valuedness of first integral. One region is a neighborhood of the dominant behavior to connect this many-valuedness to the singular point analysis of solution. On this region, we show that a many-valued first integral brings many-valued solutions. Another region is a neighborhood of the real part of complex phase space to see the influence of many-valuedness of first integral on physical real motion. On this region, we find that this many-valuedness vanishes under a certain condition of singularity sets of the first integrals. This paper is organized as follows. In § 2, we show that a first integral of the 2-dimension Lotka-Volterra model is infinitely many-valued essentially in the com­ plex phase space and show that this system has a single-valued general solution around its movable singularity if this first integral has no logarithmic branchings around the dominant behavior. In § 3 we generalize this result and present the theorem The Puiseux expansion of general solution is not possible if a many-valued