On determining the homological Conley index of Poincar\'e maps in autonomous systems

DOI10.12775/TMNA.2022.006arXiv2106.14293MaRDI QIDQ6371364

Publication date: 27 June 2021

Abstract: A theorem on computation of the homological Conley index of an isolated invariant set of the Poincar'e map associated to a section in a rotating local dynamical system

p h i

is proved. Let

(N, L)

be an index pair for a discretization

p h i^{h}

of

p h i

, where

h > 0

, and let

S

denote the invariant part of

N s e t m i n u s L

; it follows that the section

S_{0}

of

S

is an isolated invariant set of the Poincar'e 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_{i j}

, the cycles

u_{j}

and

s u m a_{i j} u_{i}

are homologous in the covering pair

(w i d e t i l d e N, w i d e t i l d e L)

of

(N, L)

and the homology relation is preserved in

(w i d e t i l d e N, w i d e t i l d e L)

under the transformation induced by

p h i^{t}

for

t i n [0, h]

then the homological Conley index of

S_{0}

is equal to the Leray reduction of the matrix

[a_{i j}]

. In particular, no information on the values of the Poincar'e 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.

Mathematics Subject Classification ID

Gradient-like behavior; isolated (locally maximal) invariant sets; attractors, repellers for topological dynamical systems (37B35) Index theory for dynamical systems, Morse-Conley indices (37B30)

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