Mathmatical study of complex systems

研究了高维分段扩张映射的动力学性质。与一维情形相反，我们证明了C^γ分段扩张映射不一定允许绝对连续的不变测度。我们还证明了，如果我们假设映射是二维的分段解析映射或高维的分段仿射映射，则它总是允许绝对连续的不变测度。我们对10个全局耦合逻辑映射进行了广泛的数值实验，得出了这种情况下所谓的“混沌无能”是由不变子空间上一系列开关间歇引起的。

We study the dynamics of piecewise expanding maps in higher dimension. On the contrary to the one dimensional case, we show that a C^γ piecewise expanding map does not necessarily admit an absolutely continuous invariant measure. Also we show that if we assume that the map is piecewise analytic in 2dim or piecewise affine in higher dimension, it always admits an absolutely continuous invariant measure.We also studied the so-called global lycoupled maps from a mathematical viewpoint. We made an extensive numerical experiment for 10 globally coupled logisitic map and came to the idea that the so-called "Chaotic inenerancy" in this case is caused by a hierarchy of on-off intermittencies on the invariant subspaces.