Mathematical Sciences: Dynamical Systems, Geometry, and Quasiconformal Homeomorphisms

首席研究员将继续研究一个实数或一个复变量的动力系统。他将研究结构变化，因为一个真实的参数是不同的。重整化群方法将被用来发展对几个计算机生成的发现的理论理解。无限分叉将通过拟共形变形来研究。当能量被添加到许多机械设备中时，系统开始振荡。随着能量的进一步增加，许多非线性但简单的器件将其振荡频率翻了一番。更多的能量会导致另一次翻倍。随着更多能量的增加，这种连续的倍频“分叉”继续进行，直到刚刚超过费根鲍姆极限，系统变得混乱。首席调查员将建立在他的数学理论的基础上，该理论准确地解释了这种倍频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.