On Some Topological Properties of the Structural Graph of Dynamical Systems

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-24-2026-267
Published: Aug 24, 2026
Cite as:
Bouchra Tarhzout, Ali Ouadfel, Fouad Zinoun, On Some Topological Properties of the Structural Graph of Dynamical Systems, Int. J. Anal. Appl., 24 (2026), 267.

Main Article Content

Bouchra Tarhzout, Ali Ouadfel, Fouad Zinoun

Abstract

Within the scope of the qualitative study of dynamical systems via graph-based methods, we extend the notion of structural graph to dynamical systems generated by actions of monoids, as abstract algebraic structures. The setting considers the action of an ordered monoid (not necessarily \(\mathbb{N},\ \mathbb{Z}\) or \(\mathbb{R}\)) on a metric space in a general sense, and therefore includes a wider class of dynamical systems than those described solely by ordinary differential or difference equations. We then investigate several fundamental properties of the structural graph, such as connectedness and degeneracy. In particular, it is shown that in the finite case, the structural graph associated to a \(C^{1}\)-vector field on a closed differential manifold is necessarily connected. As illustrative examples, we consider the classical van der Pol oscillator on the Poincaré sphere, the mathematical pendulum on the phase plane and the nonanalytic case of the center-focus. And as a perspective, we suggest work about what could be considered as a new form of linearization of dynamical systems via their structural graph, and address the question about a possible use for investigating global qualitative properties such as structural stability.

Article Details

References

  1. D.F. Anderson, P.S. Livingston, The Zero-Divisor Graph of a Commutative Ring, J. Algebra 217 (1999), 434–447. https://doi.org/10.1006/jabr.1998.7840.
  2. F. Balibrea, J.L.G. Guirao, M. Lampart, A Note on the Definition of $alpha$-Limit Set, Appl. Math. Inf. Sci. 7 (2013), 1929–1932. https://doi.org/10.12785/amis/070530.
  3. I. Beck, Coloring of Commutative Rings, J. Algebra 116 (1988), 208–226. https://doi.org/10.1016/0021-8693(88)90202-5.
  4. G.D. Birkhoff, Nouvelles Recherches sur les Systèmes Dynamiques, Mem. Pontif. Acad. Sci. Novi Lyncaei (3) 1 (1935), 85–216.
  5. P. Cermelli, G. Indelicato, R. Twarock, Nonicosahedral Pathways for Capsid Expansion, Phys. Rev. E 88 (2013), 032710. https://doi.org/10.1103/physreve.88.032710.
  6. C. Conley, Isolated Invariant Sets and the Morse Index, American Mathematical Society, 1978. https://doi.org/10.1090/cbms/038.
  7. R.L. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd ed., Addison-Wesley, 1989.
  8. H. Dulac, Sur les Cycles Limites, Bull. Soc. Math. Fr. 51 (1923), 45–188. https://doi.org/10.24033/bsmf.1031.
  9. M. Giunti, C. Mazzola, Dynamical Systems on Monoids: Toward a General Theory of Deterministic Systems and Motion, in: Methods, Models, Simulations and Approaches Towards a General Theory of Change, World Scientific, pp. 173–185, (2012). https://doi.org/10.1142/9789814383332_0012.
  10. B. Giunti, V. Perri, Dynamical Systems on Graphs through the Signless Laplacian Matrix, Ric. Mat. 67 (2017), 533–547. https://doi.org/10.1007/s11587-017-0326-z.
  11. J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer New York, 1983. https://doi.org/10.1007/978-1-4612-1140-2.
  12. J. Hadamard, Les Surfaces à Courbures Opposées et Leurs Lignes Géodésiques, J. Math. Pures Appl. (5) 4 (1898), 27–73.
  13. M.W. Hero, Special $alpha$-Limit Points for Maps of the Interval, Proc. Am. Math. Soc. 116 (1992), 1015–1022. https://doi.org/10.1090/s0002-9939-1992-1100653-4.
  14. J. Kotus, M. Krych, Z. Nitecki, Global Structural Stability of Flows on Open Surfaces, Mem. Am. Math. Soc. 37 (1982), no. 261. https://doi.org/10.1090/memo/0261.
  15. J. Lambek, Deductive Systems and Categories: I. Syntactic Calculus and Residuated Categories, Math. Syst. Theory 2 (1968), 287–318. https://doi.org/10.1007/bf01703261.
  16. L.S. Lyagina, The Integral Curves of the Equation $y^{prime}=dfrac{ax^2+bxy+cy^2}{dx^2+exy+fy^2}$, Usp. Mat. Nauk 6 (1951), 171–183.
  17. J. Mikram, F. Zinoun, A. El Abdllaoui, Poincaré Code: A Package of Open-Source Implements for Normalization and Computer Algebra Reduction near Equilibria of Coupled Ordinary Differential Equations, Comput. Phys. Commun. 184 (2013), 2204–2213. https://doi.org/10.1016/j.cpc.2013.04.003.
  18. H.M. Morse, A One-to-One Representation of Geodesics on a Surface of Negative Curvature, Am. J. Math. 43 (1921), 33–51. https://doi.org/10.2307/2370306.
  19. V.V. Nemytskii, V.V. Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, 1960.
  20. G. Osipenko, Dynamical Systems, Graphs, and Algorithms, Springer Berlin Heidelberg, 2007. https://doi.org/10.1007/3-540-35593-6.
  21. M.M. Peixoto, Structural Stability on Two-Dimensional Manifolds, Topology 1 (1962), 101–120. https://doi.org/10.1016/0040-9383(65)90018-2.
  22. L. Perko, Differential Equations and Dynamical Systems, Springer, 1996. https://doi.org/10.1007/978-1-4684-0249-0.
  23. H. Poincaré, Mémoire sur les Courbes Définies par une Équation Différentielle, J. Math. Pures Appl. (3) 7 (1881), 375–422.
  24. S. Smale, Diffeomorphisms with Many Periodic Points, in: Differential and Combinatorial Topology, Princeton University Press, pp. 63–80, (1965). https://doi.org/10.1515/9781400874842-006.
  25. S. Walcher, On Transformations into Normal Form, J. Math. Anal. Appl. 180 (1993), 617–632. https://doi.org/10.1006/jmaa.1993.1420.