國立清華大學 數學系

Abstract
      We study a discrete non-autonomous system whose autonomous counterpart admits a saddle-node bifurcation, and in which the bifurcation parameter slowly changes in time and is characterized by a sweep rate constant ε. We show that, in contrast to its autonomous counterpart, when the time mesh size Δt is less than the order O(ε), there is a bifurcation delay as the bifurcation time-varying parameter is varied through the bifurcation point, and the delay is proportional to the two-thirds power of the sweep rate constant ε. On the other hand, when the time mesh size Δt is larger than the order o(ε), the dynamical behavior of the solution is dramatically changed before the bifurcation point. Therefore, the dynamical behavior of the system strongly depends on the time mesh size.
    Random matrix projection is widely applied to reduce the dimensionality of data. Its convenient application and rapid computational properties allow it to efficiently reduce dimensionality. We consider jointly applying random projection and PCA in data processing. Compared to using PCA alone, this can make computations faster and provide a good approximation of the principal components.