# Construct differential equation given the phase portrait (non-linear pendulum)

I am curious to know how to recover the differential equation that goes with a phase portrait. I have seen the following posts but the first one was a $y'$ (and easy enough to "do in my head") and the second one lacks the figure.

As an example, I solved (in Mathematica) the non-linear pendulum equation and plotted the phase portrait as below:

$$y''(t)+\frac{y'(t)}{q}+\sin \left(y(t)\right)=g \cos (\omega t)$$

Where $q$ is damping, $g$ is the forcing term while $\omega$ is the frequency of forcing. The phase portrait ($y'$ vs $y$) is plotted: So, now if I were given this phase portrait, how do I derive the autonomous differential equation for it? I ask because I do not know the "language" to search for online.

The reason I posted this question here instead of mathematica.SE is because this is more of a Mathematics question and not so much a Mathematica question.

My Mathematica code, in case interested:

g = 1/10; q = 8; \[Omega] = -0.04; TMax = 40;
pSolNL = NDSolveValue[{y''[t] + (1/q) y'[t] + Sin[y[t]] ==
g  Cos[\[Omega]  t], y == 0, y' == 0}, y, {t, 0, TMax}];

ParametricPlot[{pSolNL[\[Tau]] /. \[Tau] -> t,
D[pSolNL[\[Tau]], \[Tau]] /. \[Tau] -> t}, {t, 0, TMax},
AxesLabel -> {"y(t)", "y'(t)"}]

• The system that produced this portrait is nonautonomous, so it's hard to see how to derive an autonomous equation for it. The phase portrait involves a loss of information (time is implicit). You'd be better off fitting y[t] and y'[t] rather than y'[y]. That said, check this out: science.sciencemag.org/content/324/5923/81.abstract – Chris K Feb 3 '16 at 15:23

• Never from a single trajectory. But you could try to find the equation by adding $t'=1$ and thus rediscovering yours plus $t'=1$, provided that you had many trajectories and looked at the phase portrait as if the curve evolved also along a third direction (according to $t'=1$). Tortuous, and difficult to be rigorous, never seen it really, but feasible in theory. :-) – John B Feb 3 '16 at 0:30
• In principle, you should increase $q$ and decrease $g$, but you surely know that already. Other than this, I would plot many other solutions. Really, to do something like you are asking for a nonautonomous equation is very much equation dependent (and for the same reason parameter dependent). – John B Feb 3 '16 at 1:11