Invariant circles and depinning transition

We associate the existence or non-existence of rotational invariant circles of an area-preserving twist map on the cylinder with a physically motivated quantity, the depinning force, which is a critical value in the depinning transition. Assume that $H:\mathbb{R}^{2}\mapsto \mathbb{R}$ is a $C^{2}$ generating function of an exact area-preserving twist map $\bar{\unicode[STIX]{x1D711}}$ and consider the tilted Frenkel–Kontorova (FK) model: $$\begin{eqnarray}{\dot{x}}_{n}=-D_{1}H(x_{n},x_{n+1})-D_{2}H(x_{n-1},x_{n})+F,\quad n\in \mathbb{Z},\end{eqnarray}$$ where $F\geq 0$ is the driving force. The depinning force is the critical value $F_{d}(\unicode[STIX]{x1D714})$ depending on the mean spacing $\unicode[STIX]{x1D714}$ of particles, above which the tilted FK model is sliding, and below which the particles are pinned. We prove that there exists an invariant circle with irrational rotation number $\unicode[STIX]{x1D714}$ for $\bar{\unicode[STIX]{x1D711}}$ if and only if $F_{d}(\unicode[STIX]{x1D714})=0$. For rational $\unicode[STIX]{x1D714}$, $F_{d}(\unicode[STIX]{x1D714})=0$ is equivalent to the existence of an invariant circle on which $\bar{\unicode[STIX]{x1D711}}$ is topologically conjugate to the rational rotation with rotation number $\unicode[STIX]{x1D714}$. Such conclusions were claimed much earlier by Aubry et al. We also show that the depinning force $F_{d}(\unicode[STIX]{x1D714})$ is continuous at irrational $\unicode[STIX]{x1D714}$.