Currently I'm studying (qualitatively) the behaviour of systems of nonlinear, autonomous ODEs, e.g.:
$$ \begin{array}{lcl} \dot{x_1} & = & x_1-x_1^2-x_1x_2 \\ \dot{x_2} & = & 3x_2-x_1x_2-x_2^2 \end{array} $$
Generally I'm well aware how this is done (Find the steady states, linearize, investigate the Jacobian of each steady state etc.), however sometimes I "miss" one or two steady-states since they are not always easy to find/see. Therefore: Is there a way to know (a priori) how many steady states a system of nonlinear, autonomous ODEs has?