Find fixed points and give the nullclines of the following system ($\dot{x} = (y)$ and $\dot{y} = \cos(x)$)

Question

Given the following system:

$\dot{x} = \sin(y)$

$\dot{y} = \cos(x)$

1. Find the fixed points and check their stability

2. Give the nullclines

So I thought:

1. fixed points are ($k*\pi+\frac{\pi}{2}$,$n*\pi$) where $n>0$ and $k>=0$ and $\ddot{x} = \cos(y)$ and $\ddot{y} = -\sin(x)$ so filling in the fixed point points out that it is stable for k uneven and n even

2. This right here is difficult. If it was for example $\dot{x} = x-y$ then the x-nullcline was $y = x$ but here we got only $\sin(y)$ so $y = \sin-1(0) = 0$ and y-nullcline is here also $0$ ??

Please help out. Thanks

Answer 1

The nullclines are found by $0=\sin(y)$ so $y = n\pi$ which are horizontal lines and $0 = \cos(x)$ so $x = \pi/2 + k\pi$ which are vertical lines.

We also must remember to consider the domain of arcsine and arccosine so $n = 0$ and $k = 0$.