【摘要】：單調(diào)回復(fù)關(guān)系決定了高維柱面上的一類動(dòng)力系統(tǒng).這類動(dòng)力系統(tǒng)可視為二維柱面上單調(diào)扭轉(zhuǎn)映射的推廣.單調(diào)回復(fù)關(guān)系的解又對(duì)應(yīng)了Frenkel-Kontoroval(F-K)模型的平衡點(diǎn).Aubry-Mather理論指出:對(duì)任意的ω∈R,存在單調(diào)回復(fù)關(guān)系的以ω為旋轉(zhuǎn)數(shù)的Birkhoff最小解.所有以ω為旋轉(zhuǎn)數(shù)的Birkhoff最小解能否構(gòu)成葉狀結(jié)構(gòu)的問題類似于在單調(diào)扭轉(zhuǎn)映射中,是否存在以ω為旋轉(zhuǎn)數(shù)的不變圓周.本文中我們給出最小葉狀結(jié)構(gòu)存在性的判斷準(zhǔn)則,并討論此準(zhǔn)則關(guān)于ω的連續(xù)性.依賴于旋轉(zhuǎn)數(shù)ω的脫釘力Fd(ω)是使得F-K模型存在Birkhoff平衡點(diǎn)時(shí)粒子所受外力的臨界值.當(dāng)外力大于此臨界值時(shí),系統(tǒng)不存在以ω為旋轉(zhuǎn)數(shù)的Birkhoff平衡點(diǎn),從而是滑動(dòng)的.我們將證明,當(dāng)ω是無理數(shù)時(shí),以ω為旋轉(zhuǎn)數(shù)的最小能量構(gòu)型組成的集合構(gòu)成葉狀結(jié)構(gòu)當(dāng)且僅當(dāng)Fd(ω)=0.若ω=p/q為有理數(shù),則Fd(p/q)=0當(dāng)且僅當(dāng)(p,q)-周期的Birkhoff最小解組成的集合形成葉狀結(jié)構(gòu).進(jìn)一步,我們將證明,Fd(ω)在無理點(diǎn)處連續(xù),在丟番圖點(diǎn)處H?lder連續(xù).最后,我們將證明,脫釘力對(duì)局部勢(shì)能函數(shù)具有連續(xù)依賴性.由此,我們可以得到,所有不能生成葉狀結(jié)構(gòu)的勢(shì)能函數(shù)是C2-拓?fù)湎碌拈_集.
[Abstract]:The monotone recovery relation determines a class of dynamic systems on the high-dimensional cylindrical surface. This kind of dynamic system can be regarded as a generalization of monotone torsion mapping on a two-dimensional cylinder. The solution of monotone recovery relation corresponds to the equilibrium point of Frenkel-Kontoroval model. Aubry-Mather theory points out that for any 蠅 鈭，

本文編號(hào)：2439732