Feigenbaum's The Nature Of The Equations,

In 70s, Feigenbaum first discovered an astonishing universal metric property of period-doubling bifurcations in the transition to chaotic behavior (i.e. the so-called Feigenbaum phenomenon). For explaining this phenomenon, Feigenbaum proposed a certain number of assumptions, a key one of them is the corresponding functional equation is to determined), (1)has solutions.Many scientists in each field are interested in this equation, and they also have obtained rich fruits in this respect. Including Feigenbaum functional equation has solutions, the construction of many kinds of continuous, differentiable, even smooth solutions, and some dynamical behaviors of Feigenbaum maps, and so on.Considering some recent results of the research on Feigenbaum functional equation, first, we investigate the concavity-convexity of a class of accurate solution of the Feigenbaum functional equation. Then, we investigate the likely limit sets with fractal structure of some types of Feigenbaum maps in high order case.The following are our main points:In Chapter 1, we introduce some concerned concepts and results of dynamical system, fractal geometry and Feiganbaum map.In Chapter 2, we investigate the concavity-convexity of a class of accurate solution of the Feigenbaum functional equation. We obtain the result: if ]) Feigenbaum map f(x) and g(x), and proof that the limit sets are all solenoids, further, the limit set of the product system fxg is A(f)xA(g).