In chapter 6, with the results of above chapters, we discuss the optimal control problems for weak-smooth system and draw a conclusion coinciding with that of smooth system, i.e., the Pontryagin's maximum principle (see Th 6.2.7).
around the bifercation point of the doubling period from period 1 to period 2,the orbit of period 2 is not perpendsculor to the path of period 1, and the chaos is reached by jumping from orbit 6 directly.
Catastrophe theory can effectively solve catastrophe problem that can possible appear in continuum system. The mechanism of coal and gas delay outburst is studied with catastrophe theory. It is concluded that the delay time of coal and gas delay outburst is produced by the combination of geostresses gas inside coal, physics-mechanics characters of coal and external action (explosion). According to the above research, the way of preventing coal and gas delay outburst is presented.