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!
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 $\dot x$.