Dynamic system: ODE

Today, I finished the last exam in my 2A the dynamic system. I have to say that it is not easy to prepare for an additional exam, but this time, finally, I got the idea of the ODE and here I would like to say something about it.

The history of ODE is so long that we cannot recall the beginning of this subject and, in fact, the development of this subject is so closed to the analysis. Except the techniques of resolving the specific equation, the existence, unity and continuity are three most important ideas in the development of ODE.  That is Cauchy problem, namely we know the initial condition and how it evolves, we should know the state of every time. This idea is so simple and it influences also the philosophy school - determinism. We believe it not only because we can solve some equations, but also because that we know the law - there is no reason why we cannot fix the state in a given point.

The proof of this theorem - I used to find very hard, but just the general version of Picard theorem. In the course of Polytechnique, we use one course to prove all the version of Picard theorem, the simple version, lip-version and C^k version. By this we derive the implicit function theorem and local inversion theorem, useful weapons in the later proof of stability. The Picard theorem in Banach space gives the famous Cauchy-Lip theorem, that is in Lip condition, the solution exists locally.

Although this is just the base, it is enough. The longtime evolution is just this basic point adding some extension theorem. In fact, the terminal of the evolution is necessarily blow-up and before this case, all the trace cannot intersect. By these two points, we can nearly decide all the evolution of ODE.

When talking about the asymptotic behaviors, the perturbation theory, the resolvant theory, and the idea from the geometry are so useful. In the case of periodic theory, generally we know that the period is stable even though there are errors and sometimes we can make the modification to make the exact periodic theorem. The study of the flot or resolvant, which generalize the evolution to study the diffimorphism between space has a high viewpoint mathematical. And also, we use the first return technique of Poincare. By this, we treat the pendulum problem totally, like in this exam. The period is determined mostly by the force but not its own period.

I believe that there are many other problems to answer, like the chaos theory and ergodic  theory. I will continue some discovery in 3A.