# Phase portraits of 2-dimensional systems

I'm trying to plot solutions to two dimensional ordinary differential equations. I found that Sage makes it easy to plot a vector field and, using `ode_solver()`

, I can plot solutions on top of the vector field by specifying an initial condition `y_0`

and some range of time to run (`t_span`

).

However, this method I'm using seems to be quite ad hoc, as I have to choose the right initial conditions and time span / know a lot about my system in order to plot a nice picture. Let's make this more concrete:

Say I want to draw a nice phase portrait for

$\dot{x} = -y$

$\dot{y} = -x$

First I generate the vector field:

```
var('x y')
VF=plot_vector_field([-y,-x],[x,-2,2],[y,-2,2])
```

Then I use `ode_solver()`

to plot solutions with initial conditions going around the unit circle:

```
T = ode_solver()
T.function=lambda t,y:[-y[1],-y[0]]
solutions = []
c = circle((0,0), 1, rgbcolor=(1,0,1))
for k in range(0,8):
T.ode_solve(y_0=[cos(k*pi/4),sin(k*pi,t_span=[0,1],num_points=100)
solutions.append(line([p[1] for p in T.solution]))
```

This generates the following picture:

But if I change run the system for one more unit of time (set `t_span=[0,2]`

), the picture gets skewed:

This makes sense, of course, because the magnitude of the vectors along $y=-x$ get big as you get further away from the origin. Similarly, the trajectory along $y=x$ has trouble getting to the origin because the magnitude of those vectors get very small. Which all makes me think there's a better way to do this. Thoughts?