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I want to draw phase plots.

The differential equations are two coupled second-order non-linear differential equations.

I have the exact particular analytic solutions.

However, the solutions are a function of time (the independent variable) and I am not able to define the variable in such a way that I can find fixed points for the analytic solution in these new variables.

How to draw phase plots when fixed points can't be found.

Which is the best software to plot phase plots for two coupled second-order equations.

Also, are phase plots and phase portraits the same?

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    $\begingroup$ By a 'particular analytic solution' do you mean the solution for only one set of initial conditions? This is obviously helpful for understanding what's going on but since it's just a single curve in a 4-dimensional phase space it's not going to help with the phase plot much. $\endgroup$
    – jacob1729
    Commented Apr 28, 2021 at 12:58
  • $\begingroup$ Also would this make more sense to ask over at Mathematics? $\endgroup$
    – jacob1729
    Commented Apr 28, 2021 at 12:59
  • $\begingroup$ @jacob We have analytically only a particular solution, exactly as you have understood. For other initial conditions we can use Numerical methods, no? $\endgroup$
    – Jose Mathew
    Commented Apr 28, 2021 at 13:06
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    $\begingroup$ What is preventing you from finding the fixed points? Your post makes it sound like you want to find them using the one trajectory you know, but this obviously won't work - the trajectory will join two fixed points (possibly same FP, possibly the one at infinity) together so you can get at most 2 FPs from one trajectory. $\endgroup$
    – jacob1729
    Commented Apr 28, 2021 at 13:12

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how to draw phase plots?

You can solve ODEs numerically. There are many methods that you can program yourself in your language of choice, or you can use one of the solvers available in:

Once you have the solution (for the chosen initial condition), you can plot it using Gnuplot or some of the pieces of software listed above.

Which is the best software

According to which criteria? The first 5 options in the list above are free, powerful, flexible and allow you to both solve the equations and plot the trajectory. Some swear for the (paid) Mathematica and Maple being the easiest to use, but with a template, Python, for instance, is not difficult either and might have more to offer in the long term, given its large number of packages. Sage can do things like symbolic computation, if that's a direction you'd like to go. So, really, there's no absolute best, it depends on your boundary conditions.

are phase plots and phase portraits the same?

Yes, though "phase portrait" is probably more common. Also, often these terms are used to denote not a plot of a single trajectory, but one with a number of different trajectories to characterize the phase space.

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