In the study of finite and infinite
dimensional dynamical systems generated by maps, ordinary differential equations, or partial differential equations, invariant manifolds and foliations are important tools. They provide coordinates in which the systems can be partially decoupled and can be used to track the asymptotic behaviors of the orbits. Therefore, starting with Poincare, Hadamand, Lyapunov, Perron and et al., people have studied extensively their existence, smoothness, and persistence under small perturbations (such as those due to the modelling procedure, small noises, or computational round-off error, etc.) In this series of lectures, we will discuss the
following topic:

1.) ·Local theory: invariant manifolds and foliations near a fixed point of finite dimensional systems-existence and smoothness

2.) Applications of the local theory in local and global bifurcations, blow-up analysis etc.

3.) Global theory: normally hyperbolic invariant manifolds

4.) Applications of normally hyperbolic invariant manifolds: singular perturbations and homoclinic orbits

参考文献:

a.)  《微分动力系统导引》，张锦炎、钱敏，北京大学出版社

b.) Carr, Jack Applications of centre manifold theory. Applied Mathematical Sciences, 35. Springer-Verlag, New York-Berlin, 1981. vi+142
pp. ISBN:  0-387-90577-4

c.) Hirsch, M. W.; Pugh, C. C.; Shub, M. Invariant manifolds. Lecture Notes in Mathematics, Vol. 583. Springer-Verlag, Berlin-New York, 1977. ii+149 pp.

d.) Guckenheimer, John; Holmes, Philip Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Revised and corrected reprint of the 1983 original. Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1990. xvi+459 pp.