We present a self contained tour of the Conley index and some applications. The starting point is the fundamental theorem of dynamical systems, passing through the necessary topological background, with a short stop at the basic properties of the Conley index, and arriving at the construction of connection matrices with a panoramic view of the applications: detect heteroclinic orbits arising in delay differential equations, and partial differential equations of parabolic type. The ride will be filled with examples and pictures. 查看全文>>