Phase-plane solutions to some singular perturbation problems
Phase-plane solutions to some singular perturbation problems
复制标题
一些奇异摄动问题的相平面解
DOI：
10.1016/0022-247x(76)90214-6
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发表时间：
1976
影响因子：
1.3
通讯作者：
R. O'Malley
中科院分区：
文献类型：
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作者：
R. O'Malley
我们寻求两点问题f（x）= 0，0 <$t <$1，x（0），x（1）的渐近解，当小参数<$趋于零时.在（x，n x今）相平面中，解通常倾向于保持在对应于势能V（x）= n x f（s）ds的最大值的静止点（x，0）处。当n → 0时，我们（粗略地）发现1。(i)如果V（x（0））或V（x（1））超过V的所有有限最大值（如果有的话），则不存在有界变差解; 2. (ii)如果V在B处具有最大值，其中对于x< B，V（x）< V（B），并且max（x（0），x（1））< B，则存在趋于（0，1）内的B的解; 3. (iii)如果V在x（0）和x（1）之间有有限个最大值，则存在唯一的单调解，该解保持在这些最大值处; 4. (iv)如果x（0）和x（1）包含在V的连续最大值点d 1和d 2之间，V（d 1）= V（d 2），则存在无数个在值d 1和d 2之间切换的解; 5. (v)如果x（0）和x（1）包含在连续的点e1和e2之间，且V（e1）= V（e2），其中e1是V的最大值点，而e2不是，则在（0，1）中存在无数个几乎处处具有极限e1的解.
We seek asymptotic solutions of the two-point problem ϵ 2 x ̈+ f (x)= 0, 0⩽ t⩽ 1, x (0), x (1) prescribed, as the small parameter ϵ tends to zero. In the (x, ϵ x ̇) phase plane, solutions generally tend to remain at rest points (x, 0) corresponding to maxima of the potential energy V (x)=∝ x f (s) ds. As ϵ→ 0, we (roughly) find that 1.(i) no solutions of bounded variation exist if either V (x (0)) or V (x (1)) exceeds all finite maximum values of V (if any); 2.(ii) if V has a maximum at b with V (x)< V (b) for x< b and max (x (0), x (1))< b, a solution tending to b within (0, 1) exists; 3.(iii) if V has a finite number of maxima between x (0) and x (1), there is a unique monotonic solution which remains at these maximum values; 4.(iv) if x (0) and x (1) are contained between successive maximum points d 1 and d 2 of V with V (d 1)= V (d 2), there are denumerably many solutions which switch between the values d 1 and d 2; 5.(v) if x (0) and x (1) are contained between successive points e 1 and e 2 with V (e 1)= V (e 2) where e 1, but not e 2, is a maximum point of V, there are denumerably many solutions with limit e 1 almost everywhere in (0, 1).