准周期介质中的共振平衡组态：微扰展开
Resonant Equilibrium configurations in quasi-periodic media: perturbative expansions
摘要：
我们考虑一维拟周期Frenkel-Kontorova模型。我们研究的存在的平衡，其频率（即倒数的沉积材料的密度）是共振的频率的底层。我们研究了小势的扰动理论。我们表明，有微扰展开的所有订单的准周期平衡与共振频率。在非常一般的条件下，我们表明，至少有两个这样的摄动展开的平衡参数的小值。我们还开发了这些准周期介质中的平衡的动力学解释。我们证明了平衡点是一个具有非常不寻常性质的动力系统的轨道。我们得到了动力系统的李雅普诺夫指数，即共振准周期平衡态的声子隙。我们表明，即使间隙为零，平衡可以被钉扎。
英文摘要：
We consider 1-D quasi-periodic Frenkel-Kontorova models. We study the existence of equilibria whose frequency (i.e. the inverse of the density of deposited material) is resonant with the frequencies of the substratum. We study perturbation theory for small potential. We show that there are perturbative expansions to all orders for the quasi-periodic equilibria with resonant frequencies. Under very general conditions, we show that there are at least two such perturbative expansions for equilibria for small values of the parameter. We also develop a dynamical interpretation of the equilibria in these quasi-periodic media. We show that equilibria are orbits of a dynamical system which has very unusual properties. We obtain results on the Lyapunov exponents of the dynamical systems, i.e. the phonon gap of the resonant quasi-periodic equilibria. We show that the equilibria can be pinned even if the gap is zero.
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