Research On Two Kinds Of Discrete Dynamical Systems

Dynamical system is a discipline on studying the patterns of evolution over time,which plays an important role on many disciplines such as physics, mechanics,chemistry,biology and economics and so on. Cellular automata and Boolean networks which arerapidly developed in recent decades are two such typical ones. Cellular Automata(CA),introduced by J.v.Neumann, are a class of spatially and temporally discrete, deterministicmathematical systems. Because the simple structure of CA can exhibit complex dynam-ical behaviours, CA have attracted lots of researchers over the past few decades. CA arenot only restricted to analysis of math and physics, but also have been widely applied tothe fields of sociology, computer science, biology and so on. Boolean networks are firstlyintroduced by S.A.Kaufman to explain the origin of life. In fact, Boolean networks area more general class of discrete dynamical systems than Cellular automata. Althoughtheir structures are simple, they have been demonstrated to play an important role in lifesciences, financial science, and study of some other typical complex systems.In chapter2, the dynamical behaviours of CA rule106are investigated in the frame-work of bi-infinite symbolic sequence space. Firstly, a subsystem of rule106is obtained,and the dynamical behaviours on it are studied, such as positive topological entropy, topo-logical mixing. Then it proves that there are infinitely many disjoint chaotic subsystemsin this chaotic subsystem via releasing transformation. Moreover, rule106is topologicalmixing and possesses the positive topological entropy on each subsystem.In chapter3, the problems on dynamical classification of Boolean networks are stud-ied. Firstly, the relationship between the dynamics of Boolean network and the charac-teristic polynomial of linearized matrix corresponding to Boolean network is discussed.Then dynamical classification of Boolean networks is investigated based on the connec-tion between Boolean network and a special nonlinear Diophantine equation, so that eachclass has the same amount of limit set. Meanwhile, a recursive formula for the number ofnonnegative integer solution to the Diophantine equations is proposed, by which one canexpress the classification numbers of all the n-nodes Boolean network.In chapter4, the problems about the applications of Boolean networks are researched.In the beginning, the dynamics of two biological systems are explored according to the theory of the third chapter, moreover the limit sets of the systems are obtained. Then, thedynamics of two CA rules are investigated from the point of Boolean networks. Mean-while three invariant subsystems of rule106and the long Isles of Eden of rule45areobtained.In chapter5, one makes a brief summary on this thesis, and prospects for futurestudies.