Mathematical Sciences: Dynamical Systems, Geometry, and Quasiconformal Homeomorphisms

首席研究员将在一个真实变量或一个复杂变量中继续他的动力系统研究。他将研究结构变化，因为一个真实参数是变化的。重整化群方法将用于发展对几个计算机生成的发现的理论理解。无限分岔将通过拟共形变形进行研究。当能量加到许多机械装置上时，系统就开始振荡。当能量进一步增加时，许多简单的非线性器件的振荡频率翻倍。更多的能量会导致另一倍。随着能量的增加，这种连续的倍频“分岔”继续进行，直到刚刚超过费根鲍姆极限，系统陷入混乱。首席研究员将以他的数学理论为基础，准确解释这种频率翻倍是如何在混沌开始时演变的。

The Principal Investigator will continue his study of dynamical systems in one real, or one complex variable. He will investigate structural changes as one real parameter is varied. Renormalization group methods will be used to develop a theoretical understanding of several computer-generated discoveries. Infinite bifurcations will be studied via quasiconformal deformations. As energy is added to many mechanical devices, the system begins to oscillate. As the energy is increased further, many nonlinear yet simple devices double their oscillation frequency. Still more energy results in another doubling. This succession of frequency doubling "bifurcations" continues as more energy is added until, just past the Feigenbaum limit, the system goes chaotic. The Principal Investigator will build on his mathematical theory which explains exactly how such frequency doubling evolves at the onset of chaos.