Page 388 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
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TOPOLOGICAL METHODS IN DYNAMICAL SYSTEMS (MS-65)
[3] Peixoto M. (1971). On the classification of flows on two-manifolds. Dynamical systems
Proc. Symp. held at the Univ.of Bahia, Salvador, Brasil.
[4] Oshemkov A. A., Sharko V. V. (1998). On the classification of Morse-Smale flows on
2-manifolds (in Russian), Matematicheskiy sbornik 189, No.8, 93-140.
[5] Neumann D., O’Brien T.(1976). Global structure of continuous flows on 2-manifolds, J.
DifF. Eq. 22, 89-110.
[6] Kruglov V. E., Malyshev D. S., Pochinka O. V. (2018). Topological classification of Ω-
stable flows by means of effectively distinguishable multigraphs, Discrete and Continuous
Dynamical Systems 38, No.9, 4305-4327.
[7] Kruglov V. E. (2018). Topological conjugacy of gradient-like flows on surfaces, Dinamich-
eskie sistemy 8(36), No.1, 15-21.
[8] Palis J. (1978). A differentiable invariant of topological conjugacies and moduli of stabil-
ity, Astérisque 51, 335-346.
[9] Kruglov V. E., Pochinka O. V., Talanova G. N. (2020) On functional moduli of surface
flows, Proceedings of the International Geometry Center 13, No.1, 49-60.
Dynamics and bifurcations of a map-based neuron model
Frank Fernando Llovera Trujillo, frankllovera91@gmail.com
Gdansk University of Technology, Poland
Coauthor: Justyna Signerska-Rynkowska
The Chialvo Model is a well known 2D discrete scheme to describe the dynamics of single
neurons. However the existing analytical and numerical results do not yield complete picture of
its dynamics. In some parameters regime this system can be viewed as singularly perturbed
(nonlinear ) discrete system. Observing, that the fast-subsystem is given by the S- unimodal
map, we are able to precisely describe the dynamics of the reduced 1D model
and extend some of these results to the full model. In particular we study flip and saddle-
node bifurcations and chaotic behaviours in the model, interpreting them in terms of Izhikevich-
Hoppensteadt classification of bursting mappings which is important from the point of neuron’s
electrophysiology and excitability properties. Our analytical findings are illustrated numeri-
cally.
Topological estimates of the number of vertices of minimal triangulation
Wacław Marzantowicz, marzan@amu.edu.pl
Adam Mickiewicz University in Poznan´, Poland
Coauthors: Dejan Govc, Petar Pavešic´
From the beginning of the algebraic topology, then also called the combinatorial topology, i.e.
from the beginning of the 20th century, its basic object is the simplicial complex. The represen-
tation of a given topological space X as a simplicial complex (i.e. a homeomorphism with it)
386
[3] Peixoto M. (1971). On the classification of flows on two-manifolds. Dynamical systems
Proc. Symp. held at the Univ.of Bahia, Salvador, Brasil.
[4] Oshemkov A. A., Sharko V. V. (1998). On the classification of Morse-Smale flows on
2-manifolds (in Russian), Matematicheskiy sbornik 189, No.8, 93-140.
[5] Neumann D., O’Brien T.(1976). Global structure of continuous flows on 2-manifolds, J.
DifF. Eq. 22, 89-110.
[6] Kruglov V. E., Malyshev D. S., Pochinka O. V. (2018). Topological classification of Ω-
stable flows by means of effectively distinguishable multigraphs, Discrete and Continuous
Dynamical Systems 38, No.9, 4305-4327.
[7] Kruglov V. E. (2018). Topological conjugacy of gradient-like flows on surfaces, Dinamich-
eskie sistemy 8(36), No.1, 15-21.
[8] Palis J. (1978). A differentiable invariant of topological conjugacies and moduli of stabil-
ity, Astérisque 51, 335-346.
[9] Kruglov V. E., Pochinka O. V., Talanova G. N. (2020) On functional moduli of surface
flows, Proceedings of the International Geometry Center 13, No.1, 49-60.
Dynamics and bifurcations of a map-based neuron model
Frank Fernando Llovera Trujillo, frankllovera91@gmail.com
Gdansk University of Technology, Poland
Coauthor: Justyna Signerska-Rynkowska
The Chialvo Model is a well known 2D discrete scheme to describe the dynamics of single
neurons. However the existing analytical and numerical results do not yield complete picture of
its dynamics. In some parameters regime this system can be viewed as singularly perturbed
(nonlinear ) discrete system. Observing, that the fast-subsystem is given by the S- unimodal
map, we are able to precisely describe the dynamics of the reduced 1D model
and extend some of these results to the full model. In particular we study flip and saddle-
node bifurcations and chaotic behaviours in the model, interpreting them in terms of Izhikevich-
Hoppensteadt classification of bursting mappings which is important from the point of neuron’s
electrophysiology and excitability properties. Our analytical findings are illustrated numeri-
cally.
Topological estimates of the number of vertices of minimal triangulation
Wacław Marzantowicz, marzan@amu.edu.pl
Adam Mickiewicz University in Poznan´, Poland
Coauthors: Dejan Govc, Petar Pavešic´
From the beginning of the algebraic topology, then also called the combinatorial topology, i.e.
from the beginning of the 20th century, its basic object is the simplicial complex. The represen-
tation of a given topological space X as a simplicial complex (i.e. a homeomorphism with it)
386
