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I'm trying to create a bifurcation plot for a driven damped pendulum. In particular, I'm trying to recreate the plot found in Taylor's 'Classical Mechanics' (page 484) for a driving strength $\gamma$ in the range $1.060 \leq \gamma \leq 1.087$:

enter image description here

I believe I have the code down and I can reproduce the main properties of the plot, however for $\gamma \geq 1.081$ the points shift down a couple of units which causes a large discontinuity. I'd appreciate any help on the matter. I've attached an image of my result below.

Bifurcation plot with discontinuity.

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  • $\begingroup$ I guess you mean the discontinuity around 1.081? $\endgroup$ – Stéphane Rollandin Apr 10 '18 at 18:29
  • $\begingroup$ Yes, apologies for that. I've looked at the points at that $\gamma$ and compared it to values for the previous value $\gamma-0.001$ and the results do not align. I've ran the algorithm a couple times and each time the same gap appears. Looks unlikely to be a visual bug. $\endgroup$ – Damian Anslik Apr 10 '18 at 18:32
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    $\begingroup$ Maybe this can help: mathematica.stackexchange.com/q/141653 $\endgroup$ – Stéphane Rollandin Apr 10 '18 at 18:47
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    $\begingroup$ And this: mathematica.stackexchange.com/q/96004 $\endgroup$ – Stéphane Rollandin Apr 10 '18 at 18:49
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    $\begingroup$ It might be better to include the equation you're using and perhaps some pseudocode to outline what you're doing. $\endgroup$ – Kyle Kanos Apr 15 '18 at 12:21
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Without more details it's hard to answer precisely, but two possibilities are:

  • As the maximum amplitude increases, some numerical angle normalization makes a jump with no meaning (e.g., from $2\pi$ to $4\pi$) - that might very well be the case here, if "End Position" in your plot is normalized by $2\pi$.
  • This system exhibits multistability (i.e., it can have more than one stable final state, for a given set of parameter values) and which state is reached depends on which basin of attraction the initial condition belongs to. As $\gamma$ is varied, the boundary between different basins moves in phase space, and the fixed initial condition being used to construct the bifurcation diagram might suddenly find itself on a different basin at a certain $\gamma^*$: in this way, the diagram shows the bifurcations of attractor $A$ for $\gamma<\gamma^*$, and of attractor $B$ afterwards.

Perhaps supporting this second possibility are the following diagrams found in the web:

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