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 Yearling
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  • 36 votes cast
May
13
accepted Poincaré lemma and conservative vector fields
May
13
comment Poincaré lemma and conservative vector fields
Of course! So silly of me, thank you. So should I thought as if $X$ satisfies $\frac{\partial X_i}{\partial x_j}=\frac{\partial X_j}{\partial x_i}$, then it is conservative?
May
13
asked Poincaré lemma and conservative vector fields
Mar
24
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.
Feb
4
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.
Jan
16
awarded  Yearling
Oct
28
revised Help defining an open cover
I have added more explanation.
Oct
28
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.
Oct
28
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".
Oct
28
revised Help defining an open cover
edited body
Oct
28
asked Help defining an open cover
Sep
24
awarded  Autobiographer
Sep
24
accepted How many solutions does the equation $2i+j+3k=l$ have in nonnegative integers?
Sep
24
comment How many solutions does the equation $2i+j+3k=l$ have in nonnegative integers?
thanks a lot for this nice answer!
Sep
21
accepted self-adjoint operator without eigenvalues?
Sep
21
asked How many solutions does the equation $2i+j+3k=l$ have in nonnegative integers?
Sep
18
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.
Sep
18
comment critical points, differential equation
there you go then.
Sep
18
answered critical points, differential equation
Sep
18
comment Proving that maximal interval of existence exists and that solution is unque
Is Gronwall's inequality useful? encyclopediaofmath.org/index.php/Gronwall_lemma