2021/06/27 by Srzednicki, Roman
#Dynamical Systems (math.DS) #FOS: Mathematics #Primary 37B30 #Secondary 37B35
paper · doi:10.48550/arxiv.2106.14293
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 ϕh of ϕ, where h>0, and let S denote the invariant part of N∖ L; it follows that the section S0 of S is an isolated invariant set of the Poincaré map. The theorem asserts that if the sections N0 of N and L0 of L are ANRs, the homology classes [uj] of some cycles uj form a basis of H(N0,L0), and for some scalars aij, the cycles uj and ∑ aijui 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 ϕt for t∈ [0,h] then the homological Conley index of S0 is equal to the Leray reduction of the matrix [aij]. 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.