I have a program which aims to simulate a Josephson Bifurcation Amplifier. I am currently trying to obtain a plot of the probability of bifurcation as a function of the ratio between the driving and natural frequency of the system.

The (non-dimensionalised) equation of the system is

$$\ddot{x}(t) + \frac{1}{Q} \dot{x}(t) + \sin(x) = \eta \sin(\Delta t) + N(t)$$

For a driving amplitude, $\eta$ of 0.12 and a frequency ratio, $\Delta = \omega_d / \omega_n$, of 0.81 and a quality factor, $Q$, of 100, I output graphs like this for the response of the oscillator:

enter image description here

I have been advised that this isn't an indication of bifurcation; that there is something wrong with the sudden jump - we should expect a change in amplitude and phase when a bifurcation occurs but not like this.

EDIT: Apparently due to the sinusoidal driving and a high $\eta$ the oscillator jumps out of the cosine potential. Each jump, like the one in the graph, is a factor of $2\pi$ and indicates a jump to the "next" potential.

I have tested the system for much smaller values of $\eta = [0.001, 0.12]$ and I still cannot obtain any bifurcations for the expected range of $\Delta = [0.9, 0.95]$. In other words I still seem to obtain the same probability distribution. What else could be happening here?

I am staring at my code and I cannot for the life of me figure out what is wrong. I ran a hefty amount of simulations to obtain a probability of bifurcation for a range of $\Delta$; the error is large but I only did this to obtain a quick measure of where I can expect to see the typical S-Curves. Instead I get:

enter image description here

When I expected to get something like this:

enter image description here

Any ideas? I guess what I am asking is: What does the bifurcation look like for a non-linear driven oscillator with background noise?


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