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Invariant circles and depinning transition

Published online by Cambridge University Press:  22 September 2016

WEN-XIN QIN and
YA-NAN WANG
WEN-XIN QIN
Affiliation:
Department of Mathematics, Soochow University, Suzhou 215006, China email qinwx@suda.edu.cn
YA-NAN WANG
Affiliation:
School of Mathematical Sciences, Fudan University, Shanghai 200433, China
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Abstract

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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 $\unicode[STIX]{x1D714}$ .

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 2 , April 2018, pp. 761 - 787
Copyright
© Cambridge University Press, 2016 

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