ISSN 2079-6900 (Print) 
ISSN 2587-7496 (Online)

Middle Volga Mathematical Society Journal

DOI 10.15507/2079-6900.25.202302.37-52

Original article

ISSN 2079-6900 (Print)

ISSN 2587-7496 (Online)

MSC2020 37D15

Bicolor Graph of Morse-Smale Cascades on Manifolds of Dimension Three

E. Y. Gurevich, E. K. Rodionova

National Research University «Higher School of Economics» (Nizhny Novgorod, Russian Federation)

Abstract. The purpose of this study is to single out a class of Morse-Smale cascades (diffeomorphisms) with a three-dimensional phase space that allow topological classification using combinatorial invariants. In the general case, an obstacle to such a classification is the possibility of wild embedding of separatrix closures in the ambient manifold, which leads to a countable set of topologically nonequivalent systems. To solve the problem, we study the orbit space of a cascade. The ambient manifold of a diffeomorphism can be represented as a union of three pairwise disjoint sets: a connected attractor and a repeller whose dimension does not exceed one, and their complement consisting of wandering points of a cascade called the characteristic set. It is known that the topology of the orbit space of the restriction of the Morse-Smale diffeomorphism to the characteristic set and the embedding of the projections of two-dimensional separatrices into it is a complete topological invariant for Morse-Smale cascades on three-dimensional manifolds. Moreover, a criterion for the inclusion of Morse-Smale cascades in the topological flow was obtained earlier.These results are used in this paper to show that the topological conjugacy classes of Morse-Smale cascades that are included in a topological flow and do not have heteroclinic curves admit a combinatorial description. More exactly, the class of Morse-Smale diffeomorphisms without heteroclinic intersections, defined on closed three-dimensional manifolds included in topological flows and not having heteroclinic curves, is considered. Each cascade from this class is associated with a two-color graph describing the mutual arrangement of two-dimensional separatrices of saddle periodic points. It is proved that the existence of an isomorphism of two-color graphs that preserves the color of edges is a necessary and sufficient condition for the topological conjugacy of cascades. It is shown that the speed of the algorithm that distinguishes two-color graphs depends polynomially on the number of its vertices. An algorithm for constructing a representative of each topological conjugacy class is described.

Key Words: Morse-Smale diffeomorphisms, topological classification, structurally stable diffeomorphisms, be-color graph, topological conjugacy

For citation: E. Y. Gurevich, E. K. Rodionova. Bicolor Graph of Morse-Smale Cascades on Manifolds of Dimension Three. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 25:2(2023), 37–52. DOI:

Submitted: 21.03.2023; Revised: 10.05.2023; Accepted: 25.05.2023

Information about the authors:

Elena Ya. Gurevich, Associate Professor, Department of Fundamental Mathematics, Senior Researcher, Laboratory "Dynamical Systems and Applications"(25/12 Bolshaya Pecherskaya St., Nizniy Novgorod 603150, Russia), ORCID:,

Elena K. Rodionova, Student of the Faculty of Informatics, Mathematics and Computer Science, Trainee Researcher, Laboratory "Dynamical Systems and Applications" (25/12 B. Pecherskaya St., Nizhny Novgorod 603150, Russia), ORCID:,

All authors have read and approved the final manuscript.

Conflict of interest: The authors declare no conflict of interest.

Creative Commons Attribution 4.0 International License This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International License.