Handbook of Dynamical Systems (eBook)

2.3.2 Generic properties of symplectic, Hamiltonian and volume preserving systems

The main reference for the results which we discuss here is [56]. We treat a number of cases separately.

Symplectic diffeomorphisms

We start with the periodic and fixed points of symplectic maps (time set ). Our starting point is a state space which is a manifold with symplectic form , see Section 1.3.1. For a symplectic diffeomorphism :X→X, the 1-jet extension satisfies some extra properties which reflect the fact that has to respect the symplectic form. This is the reason that we define the 1-jet space differently here: ω1(X) is the space consisting of triples x1,x2,L), with 1,x2∈X, and a linear map from x1 to x2 such that ∗(ωx2)=ωx1, i.e. we require to be a linear symplectic map from the tangent space at 1 to the tangent space at 2. Note that this is consistent with what we defined in Section 1.3.1, i.e. a diffeomorphism is symplectic if and only if for each ∈X, the triple x,φ(x),dφx) is in ω1(X) as defined here. The important fact is that with this definition of the 1-jet space, we have, within the class of symplectic diffeomorphisms, the transversalitiy theorem with respect to submanifolds of these jet spaces.

This does however not lead to the same Kupka-Smale theorem we had before: the reason is that, in the group of linear symplectic maps of a vector space to itself, the set of non-hyperbolic maps has interior points. This is related to the fact that for a symplectic automorphism, the eigenvalues have to satisfy some extra conditions: whenever is an eigenvalue of such an automorphism, then so are ―, −1, and ―−1. This implies that a pair of non-real eigenvalues , ― on the unit circle (of multiplicity one) cannot be pushed off the unit circle by a small perturbation. In order to formulate a generic property for fixed points, excluding 1-jets of a certain type (like the non-hyperbolic ones in the non-symplectic case) we need a subset of ℓω(2n), the group of symplectic automorphisms in a vector space of dimension n, which:

1. has no interior points (and is semi-algebraic);

2. is invariant under symplectic conjugations, i.e. independent of the choice of a particular basis;

3. the elements of which cause dynamic complexity.

For this we take the complement of the set of those ∈Gℓω(2n) for which each eigenvalue is either hyperbolic, i.e. in norm different from one, or has norm one, but has only multiplicity one and is not a root of unity (eigenvalues which are a root of unity or have multiplicity greater than one correspond to resonance, and complicated dynamics). Linear symplectic automorphisms which satisfy this last condition are called elementary.1

We say that a fixed point of a symplectic diffeomorphism is elementary if φp is elementary; if is a periodic point of such a diffeomorphism with k(p)=p and i(p)≠p for <i<k, then it is called elementary if φpk is elementary.

In this situation the stable and unstable manifolds of the fixed and periodic points are defined as in the general case. They have however some additional properties. First, on a n-dimensional symplectic manifold, stable and unstable manifolds always have dimension n. This is a consequence of the above mentioned restriction on the eigenvalues of a symplectic automorphism. Second, stable and unstable manifolds are always isotropic, in the following sense:

A sub-manifold of a symplectic manifold with symplectic form is called isotropic if for each ∈Y and ,w∈Tp(Y), (v,w)=0.

Still in this situation the second part of the Kupka-Smale theorem remains unchanged:

The condition of transversal intersections in the last theorem has to be interpreted correctly: if is a periodic or a fixed point, then one has to disregard itself as an intersection of s(p) and u(p): if is not hyperbolic this is indeed a non-transversal intersection. If the non-hyperbolic eigenvalues of such a periodic or fixed point all have multiplicity 1, then, within the symplectic context, it is persistent as a non-hyperbolic fixed or periodic point, and hence the non-transversal intersection cannot be perturbed away.

Volume preserving diffeomorphisms

In the one-dimensional case volume preserving diffeomorphisms are just translations; this is too trivial a case for further consideration. In dimension two, volume preserving diffeomorphisms are just symplectic diffeomorphisms which we have already discussed. In dimensions greater that two, the generic properties of fixed and periodic points are the same as in the general case. The reason is that the only restriction on the derivative of a volume preserving diffeomorphism at a fixed point is that the product of its eigenvalues (the determinant) equals ±1. In dimension three, in the space of linear volume preserving automorphisms, the set of non-hyperbolic ones has no interior points. So all the considerations from the general case carry over.

Hamiltonian flows: the singularities

Here again the state space is a manifold with symplectic form . For a dynamical system with time set there is a generating vector field , i.e. (p)=∂tΦ(p,0). We recall, see Section 1.3.1, that the requirement that the maps t preserve is equivalent to the requirement that the 1-form Zω=ω(Z,⋅) is closed. In many examples, e.g. when is a vector space, this 1-form is even exact so that there is a function, the Hamiltonian function, :X→R such that H=ω(Z,⋅). From now on we consider only the case where there is such a Hamiltonian function.

We recall an important fact from Hamiltonian dynamics: for each evolution (t) of such a system, the function (x(t)) is constant. This means that such a dynamical system decomposes to a 1-parameter family of dynamical systems with state spaces h=H−1(h). This will be especially of importance in the next section on periodic orbits.

We now formulate the first generic property concerning the singularities of a Hamiltonian function. For this we recall that a singularity of a function is a point where its derivative is zero; it is non-degenerate if its second derivative, as a quadratic form, has maximal rank.

We observe that whenever the Hamiltonian vector field is r, a corresponding Hamiltonian function is r+1. The Hamiltonian function is not unique, but any two possible Hamiltonian functions for the same Hamiltonian vector field locally only differ by a constant. So that the statement on the critical points of the Hamiltonian function makes sense for a 1 Hamiltonian vector field and is independent...