The Research On Superconvergent Bifurcations And Universality In One-Dimensional Maps

With the constant discovery of chaotic phenomena in mathematical, physical, chemical, celestial, biological, and even social systems, chaos research has greatly deepened and become a very important aspect in the study of complex systems in modern science. Chaotic phenomena show that there is intrinsic randomness in deterministic nonlinear systems. However, in the random background there is also strong regularity, such as Feigenbaum’s universality. Following the discovery of universality and the development of bifurcation control and chaos control, chaos research has been theoretically advanced from qualitative analysis and index de-scription to obtaining universal laws, establishing and solving renormalization group equations, and even being verified and applied in real systems. Among these, the study of regularity in bifurcation processes is particularly crucial.In this dissertation, researches are mainly focused on non-associative bifurca-tion processes in one-dimensional continuous maps, which exhibit superconvergence and break Feigenbaum’s universality. On the one hand, using symbolic dynamics as an effective tool, this dissertation proposes a new perspective for analyzing bi-furcations by shifting and interlacing coexisting orbits to form new orbits under the variation of parameters of maps. This perspective is intuitive in geometry, which is helpful to form the composition rules of period-multiplying and period-adding bi- furcations, and generally applicable to arbitrary one-dimensional continuous maps. On the other hand, to study the superconvergent behavior in bifurcation processes, this dissertation introduces the GMP Arithmetic Library of arbitrary precision and MPI parallel computing environment. By improving the word-lifting technique in multimodal maps, high precision and fast computations of bifurcation parameters are achieved with the precision of over1700decimal digits, which provides a very important foundation and technical support for exploring the hyperfine structure and regularity in various iterative systems. The main results of this dissertation include the following four aspects:1. Matrix analysis method for period-multiplying bifurcations. From the perspec-tive of interlacing similar parallel orbits to form new orbits, a new method for analyzing period-multiplying bifurcations (arbitrary period-p-tupling bifurca-tions), i.e., the direct product method of vertex shift matrices, is given, which is completely different from the traditional star product method of symbolic sequences. This method directly decomposes and embeds vertex shift matrices to generate a compound matrix. Comparing with the traditional method of star products, it reflects more intuitively the self-similarity and the impact of orbit disturbance in period-multiplying bifurcation processes.2. Superconvergent universality in non-associative period-multiplying bifurca-tions. For trimodal maps, through high precision parallel computation, the superconvergent phenomena induced by non-associativity (described by right-associative star products) in period-multiplying bifurcation processes, which break Feigenbaum’s universality, are studied. A new superconvergent univer-sality in double exponential form is obtained, which is also verified in other non-associative period-multiplying bifurcation processes including special star products in bimodal maps and non-associative star products in quadrumodal maps. This indicates that it is a new universality in multimodal maps.3. New algebraic composition rules for period-adding bifurcations. For unimodal maps, the generating conditions and methods of (class I) period-adding bifur- cations, induced by shifting and interlacing different periodic orbits around the extreme point (critical point), are given. The construction of period-adding bi-furcation processes with period growing as Fibonacci numbers is also discussed. Then the generating conditions and methods of (class Ⅱ) period-adding bifur-cations, induced by shifting and interlacing different periodic orbits of similar height between different monotonic branches, are also given. Finally, the ex-tension and application of both class I and class II period-adding bifurcations in other maps are discussed.4. Superconvergent universality in Fibonacci period-adding bifurcations. For uni-modal maps, three forms of superconvergent universality are obtained.