Dimension Reduction of Nonlinear Dynamical Systems and Its Applications

The vast majority of dynamic behavior in natural world involve large degrees of freedom and nonlinearities, hence they are hard to analyze in practice. However, it is frequently observed that the dynamic behavior of functional elements found in natural sciences such as neurons and proteins and their ensembles evolves on a latent low-dimensional structure. Therefore, by exploiting the dominant low-dimensional dynamics of the nonlinear systems, we can drastically simplify the analysis of complex systems such as neuronal networks. This facilitates the analysis not only of natural systems but of engineered systems such as power grids.

In view of the above, we develop a systematic framework, called dimension reduction method, to extract such structures and apply the method to real world problems. We extend the dimension reduction method to new classes of systems such as hybrid dynamical systems and analyze their collective behavior such as synchronization. Also, we develop methods to design cooperative behavior of nonlinear elements, such as optimization methods of synchronization stability, as novel application of the dimension reduction method. Further, we are working on establishing and enhancing links between the dimension reduction method and data-driven science based on operator-theoretic analysis of dynamical systems.