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Let $ẋ=f(x)$ be a vector field on the line. Use the existence of a potential function $V(x)$ to show that solutions $x(t)$ cannot oscillate.

I know from the textbook (Nonlinear Dynamics and Chaos, Strogatz) that there are no periodic solutions to $ẋ=f(x)$. I really am not sure how to think or go about this problem. If someone would kindly nudge me in the right direction I would greatly appreciate it–thanks in advance!

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    Try looking at the integral $\int_0^T (\dot x)^2 dt$ over a solution to $\dot x=V'(x)$. – 2017-02-18

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Hint: Let us denote $\gamma$ - a solution of your equation at some starting point $x_0$. What if we are interested in length of $\gamma$ in different times $t$? If we could estimate this lenght such that for any finite time $t < \infty$ this value was finite then this would be a proof to a non-oscillation.

Hint to hint: When you are going to estimate use your potentiality equations for velocity vector $\dot x$.

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    @Khanickus argument can simplify proof, but it is more geometrical argument of mine. Hope you enjoy with this estimation! – 2017-02-18