关于确定自治系统中Poincaré映射的同调Conley指数
On determining the homological Conley index of Poincaré maps in autonomous systems
摘要：
证明了旋转局部动力系统$ϕ$中一个截面上的Poincaré映射的孤立不变集的同调Conley指数的计算定理。令$(N,L)$是$ϕ$的离散化$ϕ^h$的索引对，其中$h>0$，并且令$S$表示$N\setminus L$的不变部分;由此得出$S$的部分$S_0$是庞加莱映射的孤立不变集。该定理断言，如果$N$的部分$N_0$和$L$的部分$L_0$是ANR，则某些循环$u_j$的同调类$[u_j]$形成了$H(N_0,L_0)$的基础，并且对于某些纯量$a_{ij}$，在$(N,L)$的覆盖对$(\widetilde N,\widetilde L)$中，$u_j$和$\sum a_{ij}u_i$是同源的，并且在$ϕ^t$诱导的对$t\in [0,h]$的变换下，同源关系在$(\widetilde N,\widetilde L)$中保持，然后同源$S_0$的Conley指数等于矩阵$[a_{ij}]$的Leray约化。特别是，不需要关于庞加莱映射或其近似值的信息。在一个特殊的情形下，由具有有理$T/h>1$的$T$ -周期非自治常微分方程生成的系统，M. Mrozek，R. Srzednicki和F. Weilandt，SIAM J. Appl. Dyn. 14（2015），1348-1386，并且它激发了用于确定索引的算法的构建。
英文摘要：
A theorem on computation of the homological Conley index of an isolated invariant set of the Poincaré map associated to a section in a rotating local dynamical system $ϕ$ is proved. Let $(N,L)$ be an index pair for a discretization $ϕ^h$ of $ϕ$, where $h>0$, and let $S$ denote the invariant part of $N\setminus L$; it follows that the section $S_0$ of $S$ is an isolated invariant set of the Poincaré map. The theorem asserts that if the sections $N_0$ of $N$ and $L_0$ of $L$ are ANRs, the homology classes $[u_j]$ of some cycles $u_j$ form a basis of $H(N_0,L_0)$, and for some scalars $a_{ij}$, the cycles $u_j$ and $\sum a_{ij}u_i$ are homologous in the covering pair $(\widetilde N,\widetilde L)$ of $(N,L)$ and the homology relation is preserved in $(\widetilde N,\widetilde L)$ under the transformation induced by $ϕ^t$ for $t\in [0,h]$ then the homological Conley index of $S_0$ is equal to the Leray reduction of the matrix $[a_{ij}]$. In particular, no information on the values of the Poincaré map or its approximations is required. In a special case of the system generated by a $T$-periodic non-autonomous ordinary differential equation with rational $T/h>1$, the theorem was proved in the paper M.\,Mrozek, R.\,Srzednicki, and F.\,Weilandt, SIAM J. Appl. Dyn. Syst. 14 (2015), 1348-1386, and it motivated a construction of an algorithm for determining the index.
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