By direct calculation show that (using polar coordianted) that $$ \dot x=x-y-x(x²+y²)\\ \dot y=x+y-y(x²+y²) $$ has a limit cycle.
I need help understanding how to test whether there is a limit cycle or not. Do we check whether $$ f_x+g_y=0? $$ I am not sure whether this should be done once it is converted to polar series or before. Nor can I find a proper source about "how to test whether a system has a limit cycle" on the web.