Iterate And Its Dynamics Of Indeterminate Systems

The purpose of deterministic dynamical system theory is to study rules of change in state which depends on time. Usually there are two basic forms of dynamical systems: continuous dynamical systems described by differential equations and discrete dynamical systems described by iteration of mappings. Many mathematical models in physics, mechanics, biology and astronomy are given in such forms. Iterative equations are the equivalent form that includes iteration of unknown mappings. It has many important applications in the nature. For example, the Feigenbaum phenomena as investigating universality of period-doubling bifurcation cascade, invariant curves and manifolds of a differential equation, invariant tori and curves of Hamiltonian systems and the normal form problem can be reduced to the research of iterative equations. Iterative equation has become an important mathematics equation form which closely links differential equations, difference equations, integral equations and dynamical systems, and attracts attentions of many scholars. In recent decades, many results have been given in this field. In Chapter 1, concepts of iteration and iterative equation as well as relationship between iteration and dynamical systems are introduced. Many known results on iterative equations which are all given for deterministic systems, are summarized. In particular, less results are given for iterative equations with indeterminate systems. For indeterminate systems functions do not have so good properties (e.g., monotonicity, continuity, differentiability, etc) as for deterministic ones. Hence it needs to make some improvement in discussing iteration in indeterminate systems. In Chapter 1, we survey the development of set-valued iterative equations and fuzzy dynamical systems, and give some unsolved problems in these fields.