Phase plots: The exact particular solution is a function of time, can't find fixed points. Now, in this situation, how to draw phase plots? 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?
 A: 
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:

*

*Python, possibly in a Jupyter notebook

*Sage

*SciLab

*Octave

*Maxima

*Mathematica

*Maple

*Many others
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.
