Non linear phase portrait

Question

Consider the (nonlinear) system, $ \left\{ \begin{array}{l} \dot x = \left| {y\left| {} \right.} \right.\\ \dot y = - x \end{array} \right.$

Sketch the phase portrait of the system. I have tried to sketch for cases $y > 0$ and $y < 0$.

this a decoupled system and i want to think it looks like an absolute value graph with everything moving away there and i have no idea what this looks like can someone give me hint?

I have tried to solve each system separately by splitting the variables and integrating I get x(t) = |y| (t+c) and y(t) = -x(t+c) this doesn't tell me much

Answer 1

The easiest method I have found for graphing phase planes is as follows.

1. Find the nullclines of $x$ and $y$. These are the lines where $x'$ and $y'$ equal zero. They are important because in each region separated by the nullclines, the sign of $x'$ and $y$ doesn't change. Furthermore, on the nullclines, you know that vector field is parallel to one of the coordinate axes on the nullcline. In your case, $|y|=0\Rightarrow y=0$ is the $x$ nullcline, this is where $x'=0$ For the $y$ nullcline, $-x=0\Rightarrow x=0$ is where $y'=0$.

2. In each region, evaluate $x'$ and $y'$ to determine the direction of the vectors in that region. Your nullclines split the graph into 4 regions: $x>0,y>0$; $x>0,y<0$; $x<0,y>0$; and $x<0,y<0$. Checking each of the regions $$ \begin{array}{cccc} x>0&y>0&x'>0&y'<0\\ x>0&y<0&x'>0&y'<0\\ x<0&y>0&x'>0&y'>0\\ x<0&y<0&x'>0&y'>0 \end{array} $$

Here is an image for the final result (on a mobile device, so I can't edit the unnecessary parts out):