Dynamical system f(x)=(x^4)sin(1/x). How to determine the stability

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SpiralV
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Dynamical systemf(x)=(x^4)sin(1/x) (for x not equal to zero) and f(x)=0 (for x=0). How to find equilibrium points, and determining the stability of each equilibrium point?

I found the equilibrium points x=0 and x=1/(kπ), where k is integer. 

 f′(x)=4(x^3)sin(1/x)−(x^4)cos(1/x).

When k is 'even', f′(x=1/k*pi)=1/(pi^4)
which is positive, therefore not asymptotically stable.
And when k is 'odd', f′(x=1/k*pi)=minus1/(pi^4)
which is negative, therefore asymptotically stable.

But f′(x=0)=0, which is not hyperbolic. And this is the part that I can not solve.


Thanks in advance.
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FireGarden
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For a continuous dynamical system, it's pretty strange to have a function rather than an ODE... but anyway. Generally non-hyperbolic equilibria are a pain, but in a one-dimensional case like this, you can use the higher-order derivative test: https://en.wikipedia.org/wiki/Deriva...erivative_test
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