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

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Thread starter 4 years ago
#1
Dynamical system (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. .

When k is 'even',
which is positive, therefore not asymptotically stable.
And when k is 'odd',
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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4 years ago
#2
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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