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If you want to generalize a potential to a class that's broader than the simple $\frac12 k_2 x^2$, it is tempting as a first step to include a small perturbation of the form $\frac13k_3x^3$. Unfortunately, this drastically changes the structure of the potential, because it becomes unbounded from below. Thus, you might get a slightly perturbed behaviour ...
It is an interesting problem. Usually you would find the infinitesimal oscillations by setting $p_i = b + \delta p_i$, $q_i = a + \delta q_i$ and expanding the Hamiltonian to second order in the $\delta p_i$, $\delta q_i$. Here though, this doesn't work as you just get the same Hamiltonian, $$H = \sum_{i,j} M_{ij} \sqrt{(\delta p_i - \delta p_j)^2 + (\delta ... 2 However, what if the time series is multi-dimensional, indeed of the same dimension as the phase space, to begin with? Well, how would you know that your time series is of the same dimension as the phase space? Usually, because you already know the dynamical equations for your system (as for your pendulum). If you observe a real-life complex system, ... 2 One of the simplest measurements you can do is to characterize the spatial mode coming out of the laser. This can be done with a simple photodiode and a scanning slit (optical chopper or razor blade mounted on a movable stage). I wrote a brief note on making such a measurement here. This isn't the most interesting of measurements if you aren't going to ... 2 You might try measuring the temporal coherence of the laser. (This came to mind 'cause it's closely related to the PhD work I would have done if I didn't quit with an ABD :-( ). Basically, you split the beam with, e.g., a 50-50 splitter, and recombine one way or another to get an intereference pattern. Then use an "optical trombone" (the meaning should ... 2 I can think of a method, although it may require to compute \mathbf{F}_0 for a very large number of test points. It is based on Gauss's law for gravity$$ \frac1{m_0}\oint_{S} \mathbf{F}\cdot\mathrm d\mathbf S=-4\pi GM_S$$where S is a closed surface and M_S is the total mass contained inside it. So the idea would be to use some numerical scheme to ... 1 As a check, for your linear stability matrix I get:$$A = {\left. {\left( {\matrix{ {3{x^2} + a} & 1 \cr 1 & { - 1} \cr } } \right)} \right|_{(x* = 0,y* = 0)}} = \left( {\matrix{ a & 1 \cr 1 & { - 1} \cr } } \right) which is presumably what you have as well. By the way, did you switch notation from, $a \to \alpha$? ...