Say I have a periodic hamiltonian $H: S^1 \times M \longrightarrow M$ defined on a symplectic manifold $M$.
Then, a $1$-periodic Hamiltonian orbit of $H$ is the same thing as a fixed point of the Hamiltonian diffeomorphism: $\phi_H^1$; which is the time-$1$ map of the Hamiltonian flow $\phi_H^t$ (the flow induced by the Hamiltonian vector field, which is itself determined uniquely by the Hamiltonian function and the symplectic form).
So say we have a $1$-periodic orbit $\gamma$, and a point $x \in \gamma$, with $\phi_H^1(x) = x$.
The orbit $\gamma$ is said to be degenerate if the derivative $\mathrm{d}\phi_H^1(x)$ at the point $x$ has $1$ as an eigenvalue. I think the motivation for this definition is so that the action functional from (Hamiltonian) Floer homology be non-degenerate (in the Morse sense), which allows to construct a Floer theory that is very similar to the Morse one.
Here are my questions:
Can you provide me any concrete examples of such degenerate Hamiltonians? Like, computational examples, or visual ones, showing degenerate orbits. Most of the papers I'm reading are always finding some nice, easy-to-satisfy conditions to avoid these degenerate behaviours, but they are precisely what I'm trying to understand.
Actually, one of the things I am trying to understand is how a degenerate orbit can bifurcate into multiple non-degenerate orbits after a slight perturbation of the Hamiltonian (kind of in the same fashion as a branch point, in complex analysis, which is the degeneracy that happens when multiple points "come together"). So if you know of any examples of such behaviours in the literature (or just have a degenerate Hamiltonian at hand to send me), I'd be very grateful! :) Anything in this direction would be appreciated.
Thank you!
Note: I added a "Morse theory" tag, cause I don't think "Floer theory" is an existing one, but feel free to remove it if it's not appropriate!
I have a partial answer to (1):
Starting with a somewhat trivial example: For a Hamiltonian $S^1$ action on a symplectic manifold $(M, \omega)$, generated by $H:M\rightarrow \mathbb{R}$, the time-$1$ flow map $\phi_H^1$ is the identity map by definition. Hence, at every point of $M$ the differential is the identity map, therefore every point in $M$ is critical and degenerate. For the "simplest" example, take the (scaled) height function $H(\theta, h) = 2\pi h$ on the unit sphere $(S^2, d\theta \wedge dh) \subset \mathbb{R}^3$. The Hamiltonian flow is $2\pi \frac{\partial}{\partial \theta}$ which satisfy $\phi_H^1 = Id$.
More generally, for every Hamiltonian $H_t:M \rightarrow \mathbb{R}$ such that $\phi_H^1$ has a fixed submanifold of dimension $> 0$, the points of the fixed submanifolds are degenerate since $d\phi_H^1$ restricted to the submanifold is the identity map. To generate some examples (see the next paragraph for a specific one), take any effective Hamiltonian $S^1$ action with a fixed submanifold, and rescale the Hamiltonian $H$ such that $\phi_{\tilde H}^1$ would be the time-$t$ flow map of the action for some $0 < t < 1$. The resulted Hamiltonian $\tilde H$ would be degenerate, and $\phi_{\tilde H}^1$ would not be the identity like in the example in the first paragraph.
For instance, take $(S^2 \times \mathbb{T}^2, \omega = d\theta \wedge dh + d\theta_1 \wedge d\theta_2)$, and define the Hamiltonian to be the height function on the sphere, i.e. $H(\theta, h, \theta_1, \theta_2) = h$. The Hamiltonian vector field is given by $X_H = \frac{\partial}{\partial \theta}$ and the map $\phi_H^1$ is given by $\phi_H^1(\theta, h, \theta_1, \theta_2) = (\theta + 1, h, \theta_1, \theta_2)$. The set of fixed points of $\phi_H^1$ is $\{s, n\} \times \mathbb{T}^2$, where $s, n$ are the south and north pole. You can check that $d\phi_H^1$ has eigenvalues $1$ in all of the fixed points (the eigenvectors are the directions of the fixed submanifold, i.e. the tangent space of the torus).
I believe it should also be possible to construct an example of a degenerate isolated fixed point. I have the following construction, which I believe should work but I didn't check the details thoroughly: Take $(\mathbb{R}^2, dq\wedge dp)$, and define the Hamiltonian to be $H(q,p) = p^4 - q^4$. Hamilton's equations give $\dot q = 4p^3$ and $\dot p = 4q^3$. I believe that the only 1-periodic solution is $(0,0)$ and that it is a degenerate point (since the derivatives are of high order). I also believe that one can compactify this example and get the same desired behavior around a point.
Remark - All of the Hamiltonians that I defined were time independent. Using time-dependent Hamiltonians gives much more freedom, which can possibly yield much more interesting examples. The downside of using time-dependent Hamiltonians, in my humble opinion, is that it is harder to find a degenerate time dependent Hamiltonian (or at least it is for me!). I guess since generic Hamiltonians are non-degenerate, when looking for non-generic Hamiltonians, it's easier to find them when restricting to the very specific case of autonomous Hamiltonians.