Applied Categorical Structures

, Volume 25, Issue 5, pp 709–745 | Cite as

Prolongations, Suspensions and Telescopes

  • Jaime Martín Fernández Cestau
  • Luis Javier Hernández ParicioEmail author
  • María Teresa Rivas Rodríguez


Autonomous differential equations induced by continuous vector fields usually appear in non-smooth mechanics and other scientific contexts. For these type of equations, given an initial condition, one has existence theorems but, in general, the uniqueness of the solution can not be ensured. For continuous vector fields, the equation solutions do not generally present a continuous flow structure; one particular but interesting case, occurs when under some initial conditions one can ensure existence of solutions and uniqueness in forward time obtaining in this case continuous semi-flows. The discretization and return Poincaré techniques induce the corresponding discrete flows and semi-flows and some inverse methods as the suspension can construct a flow from a discrete flow or semi-flow. The objective of this work is to give categorical models for the diverse phase spaces of continuous and discrete semi-flows and flows and for the relations between these different phase spaces. We also introduce some new constructions such as the prolongation of continuous and discrete semi-flows and the telescopic functors. We consider small Top-categories (weakly enriched over the category Top of topological spaces) and we take as categorical models of the solutions of these differential equations some categories of continuous functors from a small Top-category to the category of topological spaces. Moreover, the processes of discretizations, suspensions, prolongations, et cetera are described in terms of adjoint functors. The main contributions of this paper are the construction of a tensor product associated to a functor between small Top-categories and the interpretation of prolongations, suspensions and telescopes as particular cases of this general tensor product. In general, the paper is focused on the establishment of links between category theory and dynamical systems more than on the study of differential equations using some categorical terminology.


Enriched category Top-category Discrete semi-flow Continuous semi-flow Prolongation Suspension Telescope 

Mathematics Subject Classification (2010)

18D20 18A25 37B99 54H20 


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  1. 1.
    Agarwal, R.P., Lakshmikantham, V.: Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations. World Scientific Publishers, Singapore (1993)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bhatia, N.P., Szegö, G.P.: Stability Theory of Dynamical Systems. Springer, Berlin (1970)CrossRefzbMATHGoogle Scholar
  3. 3.
    Ball, J.M.: Continuity Properties and Global Attractors of Generalized Semi-flows and the Navier-Stokes Equations. Journal of Nonlinear Science 7(5), 475–502 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Birkhoff, G.D.: Dynamical Systems. Amer. Math. Soc. Coll. Publ. 9 (1927)Google Scholar
  5. 5.
    Dubuc, E.J.: Kan Extensions in Enriched Category Theory. Lecture Notes in Mathematics, vol. 145. Springer (1970)Google Scholar
  6. 6.
    Dugundji, J.: Topology. Allyn and Bacon Inc., Boston (1966)zbMATHGoogle Scholar
  7. 7.
    Engelking, R.: General topology. Sigma Series in Pure Mathematics, vol. 6. Heldelmann, Berlin (1989)Google Scholar
  8. 8.
    Extremiana, J.I., Hernández, L.J., Rivas, M.T.: An Approach to Dynamical Systems using Exterior Spaces. In: Contribuciones científicas en honor de Mirian Andrés Gómez. Serv. de Publ. Univ. de La Rioja, Logroño (2010)Google Scholar
  9. 9.
    Filippov, A.F., Differential Equations with Discontinuous Right-Hand Sides. Kluwer Acad. Publish., Dordrecht (1988)CrossRefzbMATHGoogle Scholar
  10. 10.
    García-Calcines, J.M., Hernández, L.J., Rivas, M.T.: Limit and end functors of dynamical systems via exterior spaces. Bulletin of the Belgium Math. Soc.- Simon Stevin 20(5), 937–959 (2013)MathSciNetzbMATHGoogle Scholar
  11. 11.
    García-Calcines, J.M., Hernández, L.J., Rivas, M.T.: A completion construction for continuous dynamical systems. Topological methods in nonlinear analysis 44(2), 497–526 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Haddad, W.M., Nersesov, S.G., Du, L.: Finite-Time stability for Time-Varying nonlinear dynamical systems. In: Advances in nonlinear analysis: Theory methods and applications, pp. 139–150. Cambridge Scientific Publishers (2009)Google Scholar
  13. 13.
    Hector, G., Hirsch, U.: Introduction to the Geometry of Foliations: Foliations of codimension one. F Vieweg and Sohn (1981)Google Scholar
  14. 14.
    Hu, J., Li, W.: Theory of Ordinary Differential Equations. Existence, Uniqueness and Stability. Publications of Department of Mathematics. The Hong Kong University of Science and Technology (2005)Google Scholar
  15. 15.
    Kelly, G.M.: Basic concepts on enriched category theory. Reprints in Theory and Applications of Categories, no. 10 (2005)Google Scholar
  16. 16.
    Mac Lane, S., Moerdijk, I.: Sheaves in geometry and logic: a first introduction to topos theory. Springer (1992)Google Scholar
  17. 17.
    Megrelishvili, M., Schröder, L.: Globalization of confluent partial actions on topological and metric spaces. Topology and Applications 145, 119–145 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Muñoz, V., Pérez Marco, R.: Ergodic solenoids and generalized currents. doi: 10.1007/s13163-010-0050-7
  19. 19.
    Morón, M.A., Ruiz del Portal, F.R.: A note about the shape of attractors of discrete semidynamical systems. Proc. Amer. Math. Soc. 134(7), 2165–2167 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Morón, M.A., Sánchez-Gabites, J.J., Sanjurjo, J.M.R.: Topology and dynamics of unstable attractors. Fund. Math. 197, 239–252 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Poincaré, J.H.: Mémoire sur les courbes définies par les équations différentielles, I. J. Math. Pures Appl., 3. série 7, 375–422 (1881); II. 8, 251–286 (1882); III. 4. série 1, 167-244 (1885); IV. 2, 151–217 (1886)Google Scholar
  22. 22.
    Poincaré, J.H.: Les méthodes nouvelles de la mécanique céleste. Vol. 1,2,3. Gauthier-Villars et fils, Paris (Unknown Month 1892)Google Scholar
  23. 23.
    Porter, T.: Proper homotopy, theory. In: Handbook of algebraic topology, pp. 127–167. Elsevier, Chapter 3 (1995)Google Scholar
  24. 24.
    Ruelle, D.: Elements of differentiable dynamics and bifurcation theory. Elsevier (2014)Google Scholar
  25. 25.
    Sanjurjo, J.M.: Stability, attraction and shape: a topological study of flows. Lecture Notes in Nonlinear Analysis 12, 93–122 (2011)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Jaime Martín Fernández Cestau
    • 1
  • Luis Javier Hernández Paricio
    • 1
    Email author
  • María Teresa Rivas Rodríguez
    • 1
  1. 1.Departamento de Matemáticas y ComputaciónUniversidad de La RiojaLogroñoSpain

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