Nondegenerate singularities of integrable non-Hamiltonian systems

Last updated: 07/Apr/2011

The purpose of this note is to study nondegenerate singularities of integrable non-Hamiltonian systems. In particular we want to extend the Vey-Eliasson theorem about the local linearization of nondegenerate singularities of integrable Hamiltonian systems to the non-Hamiltonian case, and show that, in the non-Hamiltonian case, nondegenerate singularities are also rigid and linearizable in a natural sense.

First we have to define what does it mean to be nondegenerate in the non-Hamiltonian case, because, our knowledge, such a definition didn’t exist in the literature.

Recall that, a integrable (non-Hamiltonian) dynamical system on a manifold M^m of dimension m consists of p commuting vector fields (X_1,\hdots,X_p) and q common first integrals F_1,\hdots,F_q, where p \geq 1, q \geq 0 and p+q=m. The pair (p,q) is called the type of the system. For example, integrable Hamiltonian systems on symplectic 2n-dimensional manifolds are of type (n,n), i.e. n first integrals, and n vector fields, which are nothing but Hamiltonian vector fields of the first integrals.

Of course, we will assume that X_1,\hdots,X_p are linearly independent almost everywhere, and that the functions F_1,\hdots,F_q are functionally independent almost everywhere. Instead of choosing an explicit of first integrals F_1,\hdots,F_q, one can simply say that the ring \mathcal{F} of common first integrals of  X_1,\hdots,X_p has functional dimension equal to q = m-p, for the p-tuple X_1,\hdots,X_p of commuting vector fields to be called integrable.

Remark: In general, the ring \mathcal{F} of common first integrals does not admit a free basis, i.e. there does not exist a q-tuple of elements H_1,\hdots, H_q in \mathcal{F} such that any other element F \in \mathcal{F} can be written as F = f(H_1,\hdots,H_q) where f is a smooth function. That’s why, depending on the problem in question, it may be better to consider the whole \mathcal{F} than to pick q elements of it as the q-tuple of common first integrals for the system.

We have the following natural notion of geometric equivalence of two integrable systems of type (p,q):

Definition: Two systems are called geometrically equivalent if, after a diffeomorphism, they will have the same ring of common first integrals, and each vector field of anyone of the two systems can be written as a linear combination of the vector fields of the other system with coefficients in the algebra of common first integrals.

The above definition works in many categories: smooth global, smooth local, analytic local, etc.

A general problem is to classify integrable systems up to geometric equivalence. In this note, we will restrict our attention to the local problem of classification of singularities. Moreover, we will consider only nondegenerate singularities.

Definition: An integrable system of type (p,q) is called singular at a point O if

r =\dim \mathbb{K}(X_1(O),\hdots,X_p(O)) + \dim\{df(O) \ | \ f \in \mathcal{F} \} < m

where \mathcal{F} denotes the ring of common first integrals. The number r is called the rank, and m- r the corank, of the systelm of O. If

The pair (r_1,r_2) = (\dim \mathbb{K}(X_1(O),\hdots,X_p(O)),\dim\{df(O) \ | \ f \in  \mathcal{F} \}) is an invariant of the singular point. If r=0 then we say that O is a completely singular point (a singular point of rank 0).

If r_1= \dim \mathbb{K}(X_1(O),\hdots,X_p(O)) > 0 then there is a local free flow of \mathbb{K}^{r_1} through O, and locally we can write the system as a direct product of this flow with a system of type (p-r_1,q). As q consequence, we can reduce the study of singular points to the case where X_1(O) = \hdots X_p(O) = 0.

If, for example dF_1(O) \neq 0, then we can consider $F_1$ as a parameter, and our system as a 1-dimensional family of systems of type $(p,q-1)$ parametried by $F_1$. In other words, the case with  \dim\{df(O) \ | \ f \in  \mathcal{F} \} > 0 can be considered as a parametrized version of the case with \dim\{df(O) \ | \ f \in  \mathcal{F} \}= 0

Thus, the study of singularities can be reduced to the case of rank 0.

Let us now assume that O is a singular point of rank 0: X_1(0) = \hdots = X_p((O) = 0 and dF(O) = 0 for any first integral F.Consider the linear parts Y_i = X_i{(1)} of the vector fields, and the set $\mathcal{F}^{hom}$ of homogeneous parts of the functions in \mathcal{F} (with respect to some local system of coordinates)? Then it’s clear that the vector fields Y_i commute with each other, and the elements of $\mathcal{F}^{hom}$ are common first integrals of Y_i.

Definition: With the above notations, a singular point O of rank 0 is called nondegenerate if:

a) The linear vector fields Y_i are semisimple and form an abalien subalgebra of dimension p in the Lie algebra of linear vector fields in \mathbb{K}^m (this Lie algebra is isomorphic to gl(m,\mathbb{K})).

b) the functional dimension of $\mathcal{F}^{hom}$ is q.

In particular, the linear part ((Y_1,\hdots,Y_p), \mathcal{F}^{hom}) of a integrable system at a nondegenerate singular point of rank 0 is again an integrable system.

The definition of a nondegenerate singular point of positive rank is similar.

In the analytic case, we have the following rigidity / linearization theorem:

Theorem 1: If O is a nondegenerate singular point of an integrable system of type (p,q), then locally in a neigborhood of O the system is analytically geometrically equivalent to its linear part, i.e. it can be analytically linearized locally.

Remark: The above theorem is an analog of Vey’s theorem about local linearization of analytic integrable Hamiltonian systems near a nondegenerate singular point.

Proof: The proof is based on our study of intrinsic local torus actions for dynamical systems (see our papers “Convergence versus integrability …”, Math Res Lett 2002 & Ann Math 2005). In the nondegenerate case, we have an effective analytic  torus action of dimension p which preserves everything. Due to the existence of q = m-p functionally independent first integrals, the vector fields must in fact be tangent to the orbits of this torus action. The rest follows easily.

Remark: The proof in the case of positive rank is the same, or more precisely it is just a parametrized version of the rank 0 case.

Conjecture: The same result holds in the smooth case.

We believe that the conjecture is true, though we don’t have a proof yet. If true, it will give an non-Hamiltonian analogue of Eliasson’s theorem about smooth linearization of smooth integrable Hamiltonian system near a nondegenerate singular point.

Example: Consider the smooth (1,1) case

X = x \partial_x - 2y \partial_y + \hdots, F = (x^2y)^k + \hdots, with X(F) = 0.

Formally no problems (absolutely similar to the analytic case). How to show that thegeometric linearization can be done not only formally, but also smoothly ?

The (1,1) case:

We have elliptic and hyperbolic singularities:

* elliptic: geometrically equivalent to

X = x \partial_y - y \partial_x, F = x^2 + y^2

* hyperbolic of bi-index (l,k), where l,k are positive and coprime: geometrically equivalent to:

X = lx \partial_y - ky \partial_x, F = x^ky^l

Theorem 2: Up to geometric equivalence, the above list is complete in the smooth nondegenerate (1,1) case.

Remark: The above theorem is an analogue of the Vey-Colin de Verdière’s “lemme isochore” (circa 1979 ?)

Question: What about volume-preserving (isochore) systems ? Can we extend the above results to volume-preserving systems ? In the analytic case, the answer is obviously yes. It’s also probably yes in the smooth case.

Proof of theorem 2:

* The elliptic case is easy: first by taking a root of the first integral, one shows that can choose the first integral which has a Morse singularity of elliptic type. Then take a coordinate system in which this first integral is written as x^2 + y^2.

* The hyperbolic case is more complicated and consists of several steps.

Step 1. Use formal geometric linearization + Borel lemma -> existence of a smooth coordinate system in which

X = (lx \partial_y - ky \partial_x).G + \alpha

and

F = f(x^ky^l) + \beta

where \alpha, \beta are flat functions, and G is a first integral up to a flat function, G(0) \neq 0. Note that Y = X/G also admits F of first integral, and moreover

Y = (lx \partial_y - ky \partial_x) up to a flat term.

Step 2. Smooth linearization results for vector fields (e.g. Sternberg’s theorem ?) implies that Y cvan be smoothly linearized (by a flat diffeomorphism)

Step 3. Once Y is linear, then we can change the first integral to F_1 = x^ky^l. The vector field $X$ has the form

X = gY

where g is a smooth function which does not vanish at 0, but which is not necessarily a first integral. However, g is a first integral up to a flat term, i.e. X(g) is flat.

Step 4. Change the coordinate system in such a way that the orbits of X remain unchanged, and g becomes a true first integral: the same “diagonal trick” as used in other papers (Colin de Verdière, Dufour-Molino-Toulet, Eliasson, …):  fix a diagonal which intersects the orbits, and then reparametrize the flow through the diagonal, and we are done.

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