# Formal non-integrability of resonant systems

I’m coming back here to an old but still open (as far as I know) problem:

Show that a generic Hamiltonian system with a fixed point and whose resonance degree at that point is at least 2 is not formally integrable.

Recall that, if the system is non-resonant then it’s formally integrable due to the nondegeneracy of its Birkhoff normal form. In the case with a simple resonance (i.e. when the resonance degree is equal to 1) the system is still integrable

So to be formally non-integrable the resonance degree must be at least 2. The conjecture says that the converse is true: i.e. almost all systems whose degree of resonance is at least 2 are formally non-integrable.

There seem to be not many results on this problem. The first result is probably fue to Duistermaat:

J.J. Duistermaat, (1984), Non-integrability of the 1:1:2-resonance, Ergodic Theory and Dynamical Systems 4, pp. 553-568.

There are also some other papers:

E. Van der Aa and F. Verhulst, (1984) Asymptotic integrability and periodic solutions of a Hamiltonian system in 1:2:2$// -resonance, SIAM J. Math. Anal. 15, pp. 890–911.

I. Hoveijn and F. Verhulst (1990), Chaos in the 1:2:3$// Hamiltonian normal form, Physica D, 44, pp. 397–406.

The above papers treated special cases of resonance. What we want to have is a general method which can treat any type of resonance.

Earlier papers:

Ford (1975): 1:2:3 resonance, lacks 1 integral : Fund. Problelms in Stat. Mechanics III (1975), p. 215.

Contopoulos (1978): Degree of resonance = 2 in 3D, such that H3 has only 2 trigonometric terms (this condition exludes the 1:2:1 resonance case) –> in general no other 1st integral exists.  Disappearance of integrals in systems of more than two degrees of freedomCelestial Mechanics 17 (1978), p167. The paper of Coutopoulos is available online here.

Ill try to reproduce here Duistermaat’s idea of the proof of his result (about non-integrability of 1:1:2 resonance), because it seems to be promising for the general case.

One writes the Hamiltonian $H$ in Birkhoff normal form:

$H = H_2 + H_3 + \hdots$

where

$H_2 = (p_1^2 + q_1^2) + (p_2^2 + q_2^2) + 2(p_3^2 + q_3^2)$

and

$\{H_2, H_3\} = 0$

Denote by $G$ the group of symplectic linear transformations which leave $H_2$ invariant. After a conjugation with an element in $G$, on can further normalizes $H_3$ as:

$H_3 = q_3 (\beta_1 (q_1^2 - p_1^2) + \beta_2(q_2^2 - p_2^2)) + 2 p_3(\beta_1 p_1q_1 + \beta_21 p_2 q_2)$

Observation: the orbits of $X_3 = X_{H_3}$ on $\{H_3 = 0\}$ are periodic, and the period tends to infinity when the point tends to $\{q_1=q_2=0, (p_1,p_2) \neq 0\}$. This will create some infinite branching, which leads to non-integrability … (Moe details later — this paper is probably the most technical among the above cited papers)