Invariant manifolds

Here I want to say something about invariant manifolds of flows and diffeomorphisms. There are close connections between the two. I feel a closer attachment to the first (continuous evolution) than the second (discrete evolution) and so I will tend to emphasize it. I have known various things about invariant manifolds and used them in my work for many years. Just recently I was able to add some small things to my knowledge on the subject which has given me the feeling that I have a more global view. A treatment of this subject which I found very helpful is that given in lecture notes of Grant. I am interested here in the situation of a smooth dynamical system on R^n with a stationary point. Let its linearization at that point be denoted by A. Let a be a real number such that no eigenvalue of A has real part a. Then the eigenvalues can be split into the sets with real parts less than and greater than a respectively. The generalized eigenvectors corresponding to eigenvalues in the first set define a subspace E^s_a called the pseudostable subspace. The pseudostable manifold theorem says that if the system is C^k for some k\ge 1 there is a C^k submanifold V^s_a passing through the stationary solution which is invariant and whose tangent space at the stationary point is E^s_a. If a=0 the terminology is simplified by omitting the prefix ‘pseudo’ and this gives rise to the more widely known stable manifold theorem. By reversing the direction of time it is possible to get corresponding statements replacing ‘stable’ by ‘unstable’. If a>0 and there are no eigenvalues whose real parts lie in the interval (0,a) then E^s_a is called a centre-stable manifold V^{cs}. The intersection of a centre-stable and a centre-unstable manifold is called a centre manifold.The pseudostable manifold is uniquely determined in a small neighourhood of the stationary point if a<0. The other invariant manifolds are in general not unique. These results for continuous time dynamical systems have analogues for a diffeomorphism (which by iteration defines a discrete dynamical system). It is merely necessary to replace the additive inequalities on the real part of the eigenvalues by multiplicative inequalities on the modulus of the eigenvalues.

There are two common methods to prove the stable manifold theorem. The first is called the Lyapunov-Perron method and is analytical in flavour while the second, called the graph transform and due to Hadamard, is more geometrical. The first method starts by writing down an integral equation. It is then proved that any solution of this integral equation is a solution of the dynamical system which lies on the unstable manifold. The stable manifold is obtained as a union of solutions of this type. I found the proof of these statements rather easy to follow. What disturbed me was that that I did not at all see where the integral equation comes from. Fortunately in his lecture notes Grant gives an elementary step by step description of how to get to that integral equation, starting from the solution formula for an inhomogeneous linear ODE (Duhamel’s formula). The second method represents the stable manifold as a graph over the stable subspace. It defines an iteration for the function describing this graph. The map from one iterate to the next is given by the time-one flow of the system. If you think about this by drawing a picture for a saddle point in the two-dimensional case it is very plausible that it works. The actual proof can be done by noting that thetime-one map is a diffeomorphism whose stable manifold is identical with the manifold being sought. So this proof reduces the continuous time case to the discrete time case.

Up to know I have been talking about a single dynamical system. There are useful extensions to systems which depend on a parameter \lambda. There is a trick which I had seen before but never really appreciated the importance of. Suppose we have a system \dot x=f(z,\lambda) for x\in R^n. Augment it by the equation \dot\lambda=0. Then we have a dynamical system on x\in R^{n+1}. If for \lambda=0 the system has a stationary point at x_0 then we can study invariant manifolds for the augmented system about the point (x_0,0). Considering for instance the centre-stable manifold in a situation of this type can give valuable information about the way in which solutions near that point change when \lambda passes through zero.


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