Arnold diffusion

Remarks

The point is that $\delta$ does not depend on $\varepsilon$: the actions drift by an amount of order one, while for $\varepsilon = 0$ they are constant. The time $T$ needed tends to infinity as $\varepsilon \to 0$.

Example: Arnold's 1964 example is the time-periodic Hamiltonian $$H(I, \theta, t) = \tfrac{1}{2}(I_1^2 + I_2^2) + \varepsilon (\cos \theta_1 - 1)\big(1 + \mu (\sin \theta_2 + \cos t)\big)$$ with two angles and periodic time. Arnold proved that for suitable small $\mu$ depending on $\varepsilon$ there are orbits along which $I_2$ changes by an amount of order one.

Non-example: an autonomous real-analytic system with $n = 2$ degrees of freedom and isoenergetically nondegenerate $H_0$ has no Arnold diffusion. By the KAM theorem, each $3$-dimensional energy level contains invariant $2$-tori that cut it into regions. The action of an orbit is trapped between two such tori, so the drift tends to $0$ with $\varepsilon$.

For real-analytic $H_\varepsilon$ with steep $H_0$, the Nekhoroshev estimates show that the drift time $T$ grows at least exponentially in a power of $1/\varepsilon$.