Appendix A. Stability, the Krein–Moser theorem, and refinements and References

4 Jun 2024


(1) Agustin Moreno;

(2) Francesco Ruscelli.

Appendix A. Stability, the Krein–Moser theorem, and refinements

Now we explain how the GIT sequence topologically encodes the (linear) stability of periodic orbits, and compare it to the basic notions of Krein theory, and the Krein–Moser stability theorem. We will also explain how to obtain Theorem A.

We follow the exposition in Ekeland’s book [Eke90] (see also [Ab01]). Consider a linear symplectic ODE

One can show that the ODE x˙ = JA(t)x is (strongly) stable if and only if R(T) is (strongly) stable [Eke90]. Moreover, stability is equivalent to R(T) being diagonalizable (i.e. all eigenvalues are semi-simple), with its spectrum lying in the unit circle [Eke90].

Now, consider an elliptic pair {λ, λ} of eigenvalues of a symplectic matrix R. Then any other symplectic matrix close to R will also have simple eigenvalues in the unit circle different from ±1 (otherwise an eigenvalue would have to bifurcate into two, as every eigenvalue comes in quadruples, which is not possible if eigenspaces are 1-dimensional). Therefore in this situation, R is strongly stable. The case of eigenvalues with higher multiplicity is dealt with via Krein theory. Whenever two elliptic eigenvalues come together, this gives a criterion for when they cannot possibly escape the circle and transition into a complex quadruple. This works as follows.

if x, y are the corresponding eigenvectors. Moreover, if we consider the generalized eigenspaces

Definition A.2. (Krein-positivity/negativity) If λ is an eigenvalue of the symplectic matrix R with |λ| = 1, then the signature (p, q) of Gλ is called the Krein-type or Krein signature of λ. If q = 0, i.e. Gλ is positive definite, λ is said to be Krein-positive. If p = 0, i.e. Gλ is negative definite, λ is said to be Krein-negative. If λ is either Krein-negative or Krein-positive, we say that it is Krein-definite. Otherwise, we say that it is Krein-indefinite.

If λ is of Krein-type (p, q), then λ is of Krein-type (q, p) [Eke90]. If λ satisfies |λ| = 1 and it is not semi-simple, then it is easy to show that it is Krein-indefinite [Eke90]. Moreover, ±1 are always Krein-indefinite if they are eigenvalues, as they have real eigenvectors x, which are therefore G-isotropic, i.e. G(x, x) = 0. The following, originally proved by Krein in [Kre1; Kre2; Kre3; Kre4] and independently rediscovered by Moser in [M78], gives a characterization of strong stability in terms of Krein theory:

Theorem 3 (Krein–Moser). R is strongly stable if and only if it is stable and all its eigenvalues are Krein-definite.

See [Eke90] for a proof. Note that this generalizes the case where all eigenvalues are simple, different from ±1 and in the unit circle, as discussed above. Now, the way that the GIT sequence ties up with Krein theory is the following.

Proposition A.1 ([FM]). For a Wonenburger matrix, the Krein signature coincides with the B-signature, for elliptic eigenvalues.

Example A.3. As a simple example, to illustrate Proposition A.1, consider the Wonenburger matrices

As a corollary of the Krein–Moser theorem and of Proposition A.1, we obtain the following.

Theorem 4. Let R be a Wonenburger matrix. Then R is strongly stable if and only if it is stable and all its eigenvalues are B-definite.

In higher dimensions, whether or not a given high-multiplicity elliptic or hyperbolic eigenvalue of a Wonenburger matrix can be perturbed to be a complex quadruple is determined by whether or not its B-signature is definite; see e.g. Figure 9 and Remark 5.2. This gives a topological proof of the Krein–Moser theorem in all dimensions, and in fact generalizes it for the hyperbolic case, in the case of Wonenburger matrices, proving Theorem A in the Introduction.


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(A. Moreno) Universität Heidelberg, Mathematisches Institut, Heidelberg, Germany Email address:

(F. Ruscelli) Universität Heidelberg, Mathematisches Institut, Heidelberg, Germany Email address:

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