Study On Feigenbaum Constant In Period-doubling Bifurcation

In the past few centuries,complex physical phenomena and their related dynamics have received extensive attention,and considerable cognitive progress has been achieved through various theoretical advances.However,there are still a lot of practical difficulties in the research of complex physics and chaos phenomena,one of which is the interpretation of physical image corresponding to Feigenbaum constant in period-doubling bifurcation.Some mathematical maps whose random behavior of representation is regulated by a single linear parameter will form a bifurcation at the position of a particular parameter value during the process of parameter magnification.After studying the period-doubling bifurcation in logistic mapping,Feigenbaum pointed out that the difference ratio between the parameters corresponding to the bifurcation position tends to a constant,that is,Feigenbaum constant.He further proved that this conclusion is generally applicable to all one-dimensional maps with a single maximum and even to other mathematical functions.Due to the existence of such universal relation,every kind of chaotic system which meets the conditions will bifurcate at the same rate of parameter change.Feigenbaum constant is obviously of great significance to a wide and full understanding of chaos phenomena,and can also provide necessary Pointers for the relevant mathematical analysis.But unfortunately,the physical image of Feigenbaum constant is still unclear so far.The main work of this paper is to find such a physical image.This paper first analyzes the geometric characteristics of period-doubling bifurcation and describes the period doubling caused by bifurcation with the help of the concept of terrain-phase and topological level.The introduction of fractal geometry and the corresponding self-similar theory describe the concept of dysphoria and explain the phenomenon of dysphoria multiplication and bifurcation synchronization.Then,a class of mathematical sequences describing the iterative properties of period-doubling bifurcation process are fully demonstrated and constructed.Finally,physical images are given to the sequence by means of the extension of Rayleigh criterion which reflects the resolution power in optics.The final results show that such iterative sequences can be simplified to obtain a universal constant independent of the specific function form.This not only shows the origin of scale invariance and universality,but also points out that Feigenbaum constant is closely related to the inter-dimensional resolution of adjacent dimensions in infinite dimensional space.In other words,Feigenbaum constant has the same physical significance with the universality coefficient of n dimension(n is relatively large)or even Rayleigh criterion in infinite dimension.