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Keywords:
dynamical systems; topological dynamics; topological equivalence; axiomatic theory of ordinary differential equations
dynamical systems; topological dynamics; topological equivalence; axiomatic theory of ordinary differential equations
Summary:
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].
References:
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[3] Filippov, V.V.: The topological structure of solution spaces of ordinary differential equations. Russ. Math. Surv. 48 (101) (1993), 101–154, Translated from Uspekhi Mat. Nauk. 48, no. 1, 103–154. DOI 10.1070/RM1993v048n01ABEH000986
[4] Filippov, V.V.: Basic Topological Structures of Ordinary Differential Equations. Kluwer Acad., Dortrecht, 1998.
[5] Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory. second ed., Springer-Verlag, New York, 1998. Zbl 0914.58025
[6] Suda, T.: Equivalence of topological dynamics without well-posedness. Topology Appl. 312 (2022), 25 pp., Paper No. 108045. DOI 10.1016/j.topol.2022.108045 | MR 4387928
[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.
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