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Consider the system of differential equations with respect to time $t$:

$\dot{x}=y$

$\dot{y}=x-2x^3+y(x^2-x^4-y^2)$.

Let $L$ be the test function $\displaystyle L(x,y)=\frac{-x^2+x^4}{2}+\frac{y^2}{2}$. Answer the following questions( I'm inserting a pic because there are lot to type)

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So I'm done for part a, but have no idea on c,d,e. At least hints for them are appreciated. For b, the curve which is identically zero is clearly a solution. For the second one I got that $y\dot{y}=\dot{x}[x-2x^3+y(x^2-x^4-y^2)], $ but could not derive the two differential equations separately. Can somebody please help?

EDIT: Note that $-1/8$ is the absolute minimum of $L$.

Extremal
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  • $\alpha $is any real number bigger than or equal to $-1/8$ – Extremal Apr 07 '17 at 02:17
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    Study the sign of $\dot{L}(x, y)$ and use the following theorem: if trajectory (geometrically) in positive time stays in some compact domain, it exists for all $t \geqslant 0$. The usage of test function $L(x, y)$ is similar to this example. – Evgeny Apr 07 '17 at 10:06

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