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 Mar24 comment When does two curves do not intersect in the phase space Recall the existence and uniqueness of solutions of differential equations. If $F$ is at least $C^1$ in $x$ (one time differentiable with continuous derivative) then the equation $x'=F$ has a unique solution for a given initial condition $x(t_0)$. If two solutions were to intersect, that would mean that to a given initial condition (the point of intersection) correspond two orbits or solution curves. This contradicts unicity, and thus, it cannot happen. Feb4 comment Why are equilibria so important? I guess Poincaré would be the best person to answer. I think he was the person who started to not care about explicit solution, but to look at the behaviour. Indeed, the main problem is that nonlinear systems are rarely solvable. Still, we do not quit in trying to understand them. So the next best thing to do is to study a simplified and local version of the nonlinear problem. For this, we may use the flow-box theorem away from equilibria, and so, it remains to study what happens near equilibria. Jan16 awarded Yearling Oct28 revised Help defining an open cover I have added more explanation. Oct28 comment Help defining an open cover @HagenvonEitzen the maps $\Phi_i$ will be solutions of an ODE. Such ODE is well defined on $\mathbb R^2$ but I have some an interest on making a distinction for initial conditions in the three of the $A_i$'s. Oct28 comment Help defining an open cover Say, I want $U_0$ be an open set a bit bigger than $A_0$, and so on for the other ones. That has the "same cone shape". Oct28 revised Help defining an open cover edited body Oct28 asked Help defining an open cover Sep24 awarded Autobiographer Sep24 accepted How many solutions does the equation $2i+j+3k=l$ have in nonnegative integers? Sep24 comment How many solutions does the equation $2i+j+3k=l$ have in nonnegative integers? thanks a lot for this nice answer! Sep21 accepted self-adjoint operator without eigenvalues? Sep21 asked How many solutions does the equation $2i+j+3k=l$ have in nonnegative integers? Sep18 comment critical points, differential equation To linearise you take derivative, or the so called Jacobian, and then evaluate the equilibrium point $(x,y)=(a,b/a)$. You can try it yourself and then ask if you have problems. Sep18 comment critical points, differential equation there you go then. Sep18 answered critical points, differential equation Sep18 comment Proving that maximal interval of existence exists and that solution is unque Is Gronwall's inequality useful? encyclopediaofmath.org/index.php/Gronwall_lemma Sep15 revised Stability analysis for a system of two differential equations added 1 character in body Sep15 answered Stability analysis for a system of two differential equations Sep12 comment Stability analysis for a system of two differential equations You need to check the eigenvalues, and eigenvectors (at each equilibrium point), they will give you a local picture near each equilibrium point. Then you might be able to draw global conclusions. Of course everything will depend on the choice of the parameters, so you may have several distinct cases to consider, e.g. all parameters are positive, or all parameters are negative, and so on.