Does anyone know how to show this? Thanks!
$ \frac{dx}{dt} = x $
$ \frac{dy}{dt} = 2x^2 $
Show that x = 0 and y = $ x^2 - 2 $ , are invariant manifolds for the vector field.
Showing an equation is an invariant manifold?
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ordinary-differential-equations
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0In case you have nothing else in mind, why not solving the equations first? – 2017-01-12
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0i thought you had to show that the manifold is tangent to the vector field – 2017-01-12
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0Exactly, one way (in case you don't know any other, you don't say anything about what you tried... you really should), is what I suggested. – 2017-01-12
1 Answers
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I am no mathematician, but following the Wikipedia entry on an invariant manifold, we can say that $x=0$ is an invariant manifold because $\frac{dx}{dt}$ remains $0$ for $x=0$. As for the parabola $y=x^2-2$, take the time derivative of $y-x^2-2$ and show that it equals to zero. $$\frac{d}{dt}(y-x^2-2)=0$$ I hope this helps.