Dynamical systems and games theory

• Main Author: Carvalho, Maria Suzana Balparda de
• Published: University of Warwick 1983
• Subjects: 519.5; QA Mathematics : QH301 Biology
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.524494

This thesis consists of two parts, which deal with different topics in dynamical systems. Part I (DYNAMICS FROM GAMES) is the main scope of the work. There we study a family of flows which are often applied in studies of some game dynamics in animal competition and evolutionary biochemistry. These flows are the solutions, on simplexes, of cubic differential equations determined by "pay-off" matrices. The main result in this part is a proof for a classification of stable flows in this family, in dimension 2, first conjectured by Zeeman in 1979 (stability under small perturbations in the pay-off matrix). We add necessary and sufficient conditions for stability, which decide the exact class for each stable flow in the family. We also give as preliminary properties some simple expressions to calculate eigenvalues at fixed points and prove that hyperbolicity of these is necessary for stability, in all dimensions. In order to complete Zeeman's classification we had to adapt, in dimension 2, some techniques of structural stability for flows not satisfying the usually required transversality condition. We discuss some aspects and difficulties present when one attempts to study cases in dimension i3. One important three-dimensional example, involving a Hopf bifurcation, is discussed in detail. In the final chapter, we present some three-dimensional cases to which a discussion of stability is feasible. Part II (LIAPUNOVF UNCTIONSF OR DIFFEOMORPHISMSh) as as its purpose the construction of Liapunov functions for diffeomorphisms. A local construction is presented in neighbourhood of compact isolated invariant sets. A globalization is obtained for Axiom A diffeos with no cycles.