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### Bifurcation Diagram and Point of $\dot{x} = rx - x^3.$

You already got the bifurcation point, which is the value of the parameter where the general structure and behavior of the fixed points changes ($r=0$). For $r<0$, the only fixed point is $x=0$, so ...
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Before answering your questions in turn, it's important to know the following. In spherical coordinates, we have a scalar $r$ and a vector $\mathbf{r}$. The vector is defined $$\mathbf{r}=x \,\vec{i}+... • 186 1 vote Accepted ### Completeness of invariant vector fields on a principal bundle Notation Let G be a Lie group and \pi\colon P\to M be a principal G-bundle with principal action denoted by$$\Phi\colon G\times P\to P,\ (g,p)\mapsto \Phi(g,p)\equiv \Phi_g(p)\equiv g.p$$Let ... • 4,847 0 votes ### How do I find the absolute maximum and minimum values of the Lamb-Oseen Vortex? I'm returning a month later with suprising results that I hope serves pedagogical insight. The exact solution to the problem is the Lambert W function, with the substitution of solution r as ... 4 votes Accepted ### Confusion on the vector fields and tangent bundles The tangent bundle TM of a manifold M is the set of pairs (p,\xi) consisting of a point p in the manifold M and a tangent vector \xi \in T_pM in the tangent space at p. If \psi \colon ... • 5,383 0 votes ### Confusion on the vector fields and tangent bundles (1): The tangent bundle TM consists of linear derivations at some point of M (this is the Leibniz rule identity that you wrote down). These are the elements. To see a linear derivation v\in T_mM ... • 101 0 votes ### Why don't I see "vector-valued vector fields"? The Lie derivative of any tensor (even if vector valued) is defined using a vector field and its flow.$$\mathcal L_X T := \left.\frac{d}{dt}\right\rvert_{t=0}\phi_t^*T$$where \phi_t:M\to M is the ... • 1,680 1 vote Accepted ### How to find basis of vector fields? For sake of simplicity let us explain in dimension 2, with two coordinate (x,y), so in an open set of \bf R^2. Let X be a derivation, i.e. an operator on C^{\infty} such that X(f.g)=fX(g)+gX(... • 7,760 1 vote ### Is it possible to show this Integral identity, by assuming the hyposeses I have made? Eventually I reckon this is a possible answer: To show the asymptotic behavior of  e^{h(x)}  as  |x| \to \infty , we need to analyze the integral expression for  h(x)  given by$$ h(x) = - \frac{...
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On the domain of an individual coordinate chart the vectors $\frac{\partial}{\partial x^i}$ ($i=1,\ldots,n$) form a basis in every fibre; but in a different coordinate system, on the intersection of ...
### extension field in the subring $K[u]$
$k$ is an element in the ring $K$, so $k$ can also be viewed as a constant polynomial in the ring of polynomials with coefficients in $K$. In this case, we are just dealing with a function evaluation ...