Equivalence of ill-posed dynamical systems

Tomoharu Suda

Archivum Mathematicum (2023)

  • Issue: 1, page 133-140
  • ISSN: 0044-8753

Abstract

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The problem of topological classification is fundamental in the study of dynamical systems. However, when we consider systems without well-posedness, it is unclear how to generalize the notion of equivalence. For example, when a system has trajectories distinguished only by parametrization, we cannot apply the usual definition of equivalence based on the phase space, which presupposes the uniqueness of trajectories. In this study, we formulate a notion of “topological equivalence” using the axiomatic theory of topological dynamics proposed by Yorke [7], where dynamical systems are considered to be shift-invariant subsets of a space of partial maps. In particular, we study how the type of problems can be regarded as invariants under the morphisms between systems and how the usual definition of topological equivalence can be generalized. This article is intended to also serve as a brief introduction to the axiomatic theory of ordinary differential equations (or topological dynamics) based on the formalism presented in [6].

How to cite

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Suda, Tomoharu. "Equivalence of ill-posed dynamical systems." Archivum Mathematicum (2023): 133-140. <http://eudml.org/doc/298977>.

@article{Suda2023,
abstract = {The problem of topological classification is fundamental in the study of dynamical systems. However, when we consider systems without well-posedness, it is unclear how to generalize the notion of equivalence. For example, when a system has trajectories distinguished only by parametrization, we cannot apply the usual definition of equivalence based on the phase space, which presupposes the uniqueness of trajectories. In this study, we formulate a notion of “topological equivalence” using the axiomatic theory of topological dynamics proposed by Yorke [7], where dynamical systems are considered to be shift-invariant subsets of a space of partial maps. In particular, we study how the type of problems can be regarded as invariants under the morphisms between systems and how the usual definition of topological equivalence can be generalized. This article is intended to also serve as a brief introduction to the axiomatic theory of ordinary differential equations (or topological dynamics) based on the formalism presented in [6].},
author = {Suda, Tomoharu},
journal = {Archivum Mathematicum},
keywords = {dynamical systems; topological dynamics; topological equivalence; axiomatic theory of ordinary differential equations},
language = {eng},
number = {1},
pages = {133-140},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Equivalence of ill-posed dynamical systems},
url = {http://eudml.org/doc/298977},
year = {2023},
}

TY - JOUR
AU - Suda, Tomoharu
TI - Equivalence of ill-posed dynamical systems
JO - Archivum Mathematicum
PY - 2023
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
IS - 1
SP - 133
EP - 140
AB - The problem of topological classification is fundamental in the study of dynamical systems. However, when we consider systems without well-posedness, it is unclear how to generalize the notion of equivalence. For example, when a system has trajectories distinguished only by parametrization, we cannot apply the usual definition of equivalence based on the phase space, which presupposes the uniqueness of trajectories. In this study, we formulate a notion of “topological equivalence” using the axiomatic theory of topological dynamics proposed by Yorke [7], where dynamical systems are considered to be shift-invariant subsets of a space of partial maps. In particular, we study how the type of problems can be regarded as invariants under the morphisms between systems and how the usual definition of topological equivalence can be generalized. This article is intended to also serve as a brief introduction to the axiomatic theory of ordinary differential equations (or topological dynamics) based on the formalism presented in [6].
LA - eng
KW - dynamical systems; topological dynamics; topological equivalence; axiomatic theory of ordinary differential equations
UR - http://eudml.org/doc/298977
ER -

References

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  1. Aubin, J.P., Cellina, A., Differential Inclusions, Springer-Verlag, Berlin, 1984. (1984) Zbl0538.34007
  2. Ball, J.M., Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, Mechanics: from theory to computation, Springer, New York, 2000, pp. 447–474. (2000) 
  3. Filippov, V.V., 10.1070/RM1993v048n01ABEH000986, Russ. Math. Surv. 48 (101) (1993), 101–154, Translated from Uspekhi Mat. Nauk. 48, no. 1, 103–154. (1993) DOI10.1070/RM1993v048n01ABEH000986
  4. Filippov, V.V., Basic Topological Structures of Ordinary Differential Equations, Kluwer Acad., Dortrecht, 1998. (1998) 
  5. Kuznetsov, Y.A., Elements of Applied Bifurcation Theory, second ed., Springer-Verlag, New York, 1998. (1998) Zbl0914.58025
  6. Suda, T., 10.1016/j.topol.2022.108045, Topology Appl. 312 (2022), 25 pp., Paper No. 108045. (2022) MR4387928DOI10.1016/j.topol.2022.108045
  7. Yorke, J.A., Spaces of solutions, Mathematical Systems Theory and Economics I/II, (Proc. Internat. Summer School, Varenna, 1967). Lecture Notes in Operations Research and Mathematical Economics, Vols. 11, 12, Springer, Berlin, 1969, pp. 383–403. (1969) 

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