# Phase Diagram for a System of Ordinary Differential Equations

In an economics paper, there is this system of first-order ordinary differential equations:

The author then plots its phase diagram for $$i(t) = 0$$ for $$t \leq T$$ and reaching $$(0,0)$$ at $$t = T$$:

My question is, how can I generate such a plot? I use MATLAB and it does have functions such as quiver but it doesn't give what I have in the picture above. Could you please help? I am looking for something which plots the system of ODEs, together with direction arrows in the four quadrants and the arrows on solution trajectories. Is it possible to do it in MATLAB? I searched MATLAB Central and it seems there isn't anything that can produce such a diagram. Do I need to use another application?

• Powerpoint? To me it seems like the author analyzed the equations by hand and deduced this graph. – eranreches Jun 15 at 6:27
• Have you looked at PPLANE? math.rice.edu/~dfield – Hans Lundmark Jun 15 at 6:35
• @eranreches, I am not sure but I have seen similar phase diagrams in at least one more economics paper newyorkfed.org/medialibrary/media/research/staff_reports/…. The original paper from which I took the phase diagram in the question is here economics.mit.edu/files/7558. – V Kahn Jun 15 at 9:04
• @HansLundmark, I discovered PPLANE yesterday. I checked the link you mentioned and also looked at the 3rd edition of the manual the authors mention on their site. But nothing appears there which shows a phase diagram like the one I want. I downloaded codes (which are in MATLAB) for another paper which has phase diagrams with direction arrows princeton.edu/~moll/ODP.pdf (see, page 153, figure 1, for instance). But that diagram doesn't have arrows on trajectories. I want to be able to exactly reproduce the diagrams in the two papers that I linked to in my previous comment. – V Kahn Jun 15 at 9:09
• @Moo, do you mean I should 'draw' such a phase diagram in a graphics application after I know about behaviour of the system (for example by plotting it through quiver function in MATLAB)? So I will be doing it manually, instead of a software doing it, right? – V Kahn Jun 15 at 12:09