arXiv subjects

Tomoo Yokoyama

Publications and source records attributed to Tomoo Yokoyama.

At least 19 recordsLinked to original sources

arXiv2026

Beyond Poincar{é} recurrence: on a topological trichotomy of orbits and on the dependence of limit sets for flows on surfaces

This paper investigates the global structure of orbits for continuous flows on surfaces. One of our main results provides a topological trichotomy of orbits of flows with finitely many connected components of the singular point set on surfaces with finite genus and finite ends: each orbit is either recurrent, wandering, or corresponds to a circuit with wandering holonomy or a strictly limit circuit, two new concepts that we introduce in this paper. These concepts capture non-recurrent behaviors lying beyond the reach of classical tools such as the Poincar{é} recurrence theorem and the Poincar{é}-Bendixson theorem. Moreover, the non-recurrent non-wandering behaviors reveal the dependency between the existence of infinitely many non-degenerate $ω$-limit sets and $α$-limit sets, and provide a necessary and sufficient condition for a flow to be minimal on (possibly non-compact) surfaces.

math.DS↗

arXiv2026

Multi-parameter persistence in dynamical systems for maximizing effects of control inputs

We introduce a new topological method to naturally extend a partial function $h \colon X \rightharpoonup [-\infty, \infty]$ on a ``generalization'' of a metric space $X$ equipped with a dynamical system $f \colon X \rightharpoonup X$, to a function $h_f^{\varepsilon\text{-}\ell^p} \colon X \to [-\infty,\infty]$ with parameters $\varepsilon,p$, which allows us to apply existing topological data analysis techniques to functions defined on the whole space. Moreover, given a function $h$ that evaluates the ``quality'' of points within $\mathop{\mathrm{dom}}h$, using this extended function, one can construct a sufficient condition for the existence of an optimal $\varepsilon$-perturbation path from any point into $\mathop{\mathrm{dom}}h$ that minimizes the value of $h$ under the condition $X = \mathop{\mathrm{dom}} f \sqcup \mathop{\mathrm{dom}}h = \bigsqcup_{n = 0}^\infty f^{-n}(\mathop{\mathrm{dom}}h)$. In addition, if the domain $X$ is finite, then the function $h_f^{\varepsilon\text{-}\ell^p} \colon X \to [-\infty,\infty]$ can be computed recursively. As an application, for a given partial evaluation function on a space equipped with a dynamical system, one can construct a three-parameter filtration associated with its extension, which naturally identifies minimal paths. This clarifies the relationship among three factors: the evaluation of the cost norm, the strength of control, and the resulting value.

math.DS↗

arXiv2026

Filtrations Indexed by Attracting Levels and their Applications

We introduce a new class of filtrations indexed by attracting levels in dynamical systems, providing novel inputs for persistent homology and related methods in topological data analysis. These filtrations quantify, in a forward direction, the sensitivity of trajectories with respect to attractors under perturbations and, in a backward direction, the perturbation magnitude at which attraction breaks down. The construction applies not only to maps on metric spaces but also to general partial maps with cost functions, yielding a filtration-theoretic framework with connections to algebraic topology. This generality ensures complementary filtrations when terminal states are good or bad, inducing natural decompositions of the underlying space. As an illustration, we apply the framework to ensemble forecasts of tropical cyclones, where the filtrations identify regions of heightened sensitivity, demonstrating the potential of our approach as a new tool for topological data analysis applied to dynamical systems.

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arXiv2026

Topological flow data analysis for transient flow patterns: a graph-based approach

We introduce a method of time series analysis for two-dimensional transient flow patterns based on Topological Flow Data Analysis (TFDA), a new approach to topological data analysis. TFDA identifies local topological flow structures from an instantaneous streamline pattern and describes their global connections as a unique planar tree and its string representation. With TFDA, the evolution of two-dimensional flow patterns is reduced to a discrete dynamical system represented as a transition graph between topologically equivalent streamline patterns. We apply this method to study the lid-driven cavity flow for Reynolds numbers from $Re=14000$ to $16000$, a benchmark problem in the analysis of fluid dynamics. Our approach can extract some physical information from the lid-driven cavity flow: transition of the flow from periodic to quasi-periodic and chaotic; estimation of the period of periodic dynamics; relation between variations in energy and enstrophy and topological changes in flow patterns; statistical properties of intricate flow evolution at higher Reynolds number. In addition, we perform an observational causal inference to analyse changes in local flow patterns in the cavity corner. This work demonstrates the potential of TFDA-based time series analysis to uncover complex dynamical behaviours in fluid flow data from a topological perspective.

physics.flu-dyn↗

arXiv2026

On Discrete Morse-Bott Theory

This paper shows that discrete Morse-Bott theory can be developed as a natural extension of R. Forman's discrete Morse theory by improving the definition of the discrete Morse-Bott function introduced by S. Yaptieu. To this end, we demonstrate that the combinatorial structure of critical cells can be extended to critical sets intuitively. Furthermore, we establish the discrete Morse-Bott inequalities, providing a unified view that extends both the discrete Morse inequalities and the continuous Morse-Bott inequalities.

math.AT↗

arXiv2025

Coarse chain recurrence, Morse graphs with finite errors, and persistence of circulations

This paper provides a unified framework connecting dynamical systems with tools from topological data analysis and geometric topology and inspires new interactions among dynamical systems, topology, and nonlinear analysis. To this end, we introduce a one-parameter family of ``chain recurrences'' that generalizes chain recurrence and induces a natural filtration on the underlying metric space of a dynamical system. In particular, the forward directions of the filtrations characterize the level of control required to return to the original position, and the backward directions capture the robustness of the recurrence. The resulting filtrations yield potentials and bifurcation diagrams of dynamical systems that encode the evolution of recurrent sets under bounded total or stepwise perturbations. In addition, we extend Morse graphs to one-parameter families of ``coarse Morse graphs,'' which evolve through vertex collapses reflecting coarse recurrence transitions. These constructions not only refine Conley's decomposition but also reveal singular limit behaviors as the perturbation level vanishes. Furthermore, we establish analogous filtrations for difference equations to bridge the theoretical framework with numerical analysis.

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arXiv2025

Combinatorial structures of the space of gradient vector fields on compact surfaces

Gradient vector fields are fundamental objects from both theoretical and practical perspectives, since various phenomena can be modeled within this framework. The ``moduli space'' of such vector fields provides the foundation for describing these phenomena. However, little is known about the topology of the space of gradient vector fields. For instance, it remains unknown whether a connected component of this space can fail to be simply connected. This paper aims to lay the foundation for describing the possible generic time evolution of gradient vector fields on surfaces, with or without constraints, under the assumption that no creation or annihilation of singular points occurs, by using combinatorics and simple homotopy theory. In fact, the space of gradient vector fields on a closed annulus contains a non-contractible connected component, which is weakly homotopy equivalent to a bouquet of two two-dimensional spheres.

math.DS↗

arXiv2025

Coarse non-wandering sets and their filtration

This paper investigates recurrence properties of dynamical systems under the restriction that control is available only through inputs and outputs. We introduce the concept of ``coarse non-wandering'', a generalization of the classical non-wandering concept, and construct an associated filtration based on levels that quantify the closeness of recurrence behavior under input/output-only control. The forward direction of this filtration describes how the level of control relates to recurrence properties, whereas the backward direction captures the robustness of such behaviors and, in particular, guarantees controllability through control applied only at the observation points when the observational noise is sufficiently small. Furthermore, we demonstrate that the existence of a wandering domain is equivalent to the presence of an orbit reachable within finite error but unable to return within any slightly enlarged error bound.

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arXiv2025

Topological vortex identification for two-dimensional turbulent flows in doubly periodic domains

The dynamics and statistical properties of two-dimensional (2D) turbulence are often investigated through numerical simulations of incompressible, viscous fluids in doubly periodic domains. A key challenge in 2D turbulence research is accurately identifying and describing statistical properties of its coherent vortex structures within complex flow patterns. This paper addresses this challenge by providing a classification theory for the topological structure of particle orbits generated by instantaneous Hamiltonian flows on the torus $\mathbb{T}^2$, which serves as a mathematical model for 2D incompressible flows. Based on this theory, we show that the global orbit structure of any Hamiltonian flow can be converted into a planar tree, named a partially Cyclically-Ordered rooted Tree (COT), and its corresponding string expression (COT representation). We apply this conversion algorithm to 2D energy and enstrophy cascade turbulence. The results show that the complex topological structure of turbulent flow patterns can be effectively represented by simple trees and sequences of letters, thereby successfully extracting coherent vortex structures and investigating their statistical properties from a topological perspective.

math.DS↗

arXiv2025

Relations among Hamiltonian, area-preserving, and non-wandering flows on surfaces

This paper gives a topological characterization of Hamiltonian flows with finitely many singular points on compact surfaces, using the concept of ``demi-caractéristique'' in the sense of Poincaré. Furthermore, we describe the relationships and distinctions among the Hamiltonian, divergence-free, and non-wandering properties for continuous flows, which gives an affirmative answer to the problem posed by Nikolaev and Zhuzhoma under the assumption of finitely many singular points.

math.DS↗

arXiv2025

Structural stability and generic transitions of "incompressible" line fields on surfaces

Various line fields naturally arise on surfaces in both physical and biological contexts, and generic singularities frequently appear in the form of 1-prong (thorn-like) and 3-prong (tripod-like) configurations, which can be modeled by partial differential equations with specific parameter values. However, it remains open under which topologies such line fields are structurally stable and form an open dense subset. In this paper, we propose a new topological framework for describing line fields and their evaluations on surfaces that is suitable from both theoretical and applied perspectives. Specifically, we demonstrate that, under a topology defined by a ``cone'' structure, line fields with 1-prong and 3-prong singularities are generic when an ``incompressibility condition'' holds. We also introduce representations of complete invariants for generic line fields and their generic transitions. These representations enable the evolution of ``incompressible'' line fields -- such as those observed in active nematics -- to be encoded as walks on transition graphs, providing a combinatorial framework for their analysis.

math.DS↗

arXiv2025

Decompositions of surface flows

Flows on surfaces are one of the most fundamental and classical objects in dynamical systems, and are studied from various areas (e.g. integrable systems, differential equations, fluid mechanics). Though hyperbolic flows and recurrent flows on surfaces are classified and characterized using various topological invariants, no complete finite invariants captured both hyperbolicity and recurrence. Moreover, no topological frameworks described even generic time evaluations of gradient flows or incompressible flows (e.g. flows around a circular cylinder placed in uniform flow, solutions of Euler equations and incompressible Navier-Stokes equations). In this paper, to construct a foundation for describing fluid phenomena and capturing hyperbolicity and recurrence, under regularity for the singular point set and tameness of genus and ends, we construct complete finite invariants of flows on (possibly non-compact) surfaces by reconstructing surfaces by gluing five kinds of invariant open subsets, which are trivial flow boxes, transverse/periodic annuli, periodic M{ö}bius bands, and locally dense Q-sets. Such invariants imply a topological framework that can convert various time evaluations of fluids into walks in graphs without losing topological information, and provide a new tool for analyzing fluid phenomena and differential equations through combinatorics and topological data analysis. Furthermore, such invariants partially revive ``Markus-Neumann theorem''.

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arXiv2024

Length averages for codimension one foliations

In this paper we study geometrical and dynamical properties of codimension one foliations, by exploring a relation between length averages and ball averages of certain group actions. We introduce a new mechanism, which relies on the group structure itself, to obtain irregular behavior of ball averages for certain non-amenable group actions. Several geometric realization results show that any such groups can appear connected with the topology of leaves which are connected sums of plugs with a special geometry, namely nearly equidistant boundary components. This is used to produce the first examples of codimension one $\mathcal C^\infty$ regular foliations on a compact Riemannian manifold $M$ for which the length average of some continuous function does not exist on a non-empty open subset of $M$.

math.DS↗

arXiv2024

A Poincaré-Bendixson theorem for flows with arbitrarily many singular points

The Poincaré-Bendixson theorem is one of the most fundamental tools to capture the limit behaviors of orbits of flows. It was generalized and applied to various phenomena in dynamical systems, differential equations, foliations, group actions, translation lines, and semi-dynamical systems. On the other hand, though the no-slip boundary condition is a fundamental condition in differential equations and appears in various fluid phenomena, and Lakes of Wada attractors naturally occur in discrete and continuous real dynamical systems and complex dynamics, no generalizations of the Poincaré-Bendixson theorem can be applied to any differential equations with no-slip boundary condition on surfaces with boundary and flows with Lakes of Wada attractors. To analyze them, we generalize the Poincaré-Bendixson theorem into one for flows with arbitrarily many singular points on possibly non-compact surfaces by introducing some concepts to describe limit behaviors and using methods of foliation theory and general topology.

math.DS↗

arXiv2023

Recurrence for semi-decompositions

This paper constructs a foundation to analyze semi-group actions, group actions, filtrations, and decompositions in a unified manner. In fact, though the studies of decomposition can be applied to foliated spaces and group actions, they can not be applied to semi-group actions and filtrations in general because filtration and the set of orbits of a semi-group need not be decompositions of the base spaces. To analyze these concepts in a unified manner, we introduce a concept of a semi-decomposition which is a natural generalization of these concepts because similar relations among recurrence and their variants to group actions and decompositions hold for semi-decompositions. On the other hand, we demonstrate the difference between the recurrent concepts for group actions and those even for semi-group actions.

math.DS↗

arXiv2023

Combinatorial structures of the space of Hamiltonian vector fields on compact surfaces

In the time evolution of fluids, the topologies of fluids can be changed by the creations and annihilations of singular points and by switching combinatorial structures of separatrices. In this paper, to describe the possible generic time evolution of Hamiltonian vector fields on surfaces with or without constraints, we study the structure of the ``moduli space'' of such vector fields under the non-existence of creations and annihilations of singular points. In fact, we describe the relations of bifurcations between Hamiltonian vector fields to construct foundations of descriptions of the time evaluations of fluid phenomena. Moreover, we show that the space of topologically equivalence classes of such vector fields has non-contractible connected components and is a disjoint union of finite abstract cell complexes such that the codimension of a cell corresponds to the instability of a Hamiltonian vector field by using combinatorics and simple homotopy theory. In particular, there is a connected component of the space that is weakly homotopic to a three-dimensional sphere.

math.DS↗

arXiv2023

Dependency of the positive and negative long-time behaviors of flows on surfaces

Long-time behavior is one of the most fundamental properties in dynamical systems. The limit behaviors of flows on surfaces are captured by the Poincaré-Bendixson theorem using the $ω$-limit sets. This paper demonstrates that the positive and negative long-time behaviors are not independent. In fact, we show the dependence between the $ω$-limit sets and the $α$-limit sets of points of flows on surfaces, which partially generalizes the Poincaré-Bendixson theorem. Applying the dependency result to solve what kinds of the $ω$-limit sets appear in the area-preserving (or, more generally, non-wandering) flows on compact surfaces, we show that the $ω$-limit set of any non-closed orbit of such a flow with arbitrarily many singular points on a compact surface is either a subset of singular points or a locally dense Q-set. Moreover, we show the wildness of surgeries to add totally disconnected singular points and the tameness of those to add finitely many singular points for flows on surfaces.

math.DS↗

arXiv2022

Discrete representations of orbit structures of flows for topological data analysis

This paper shows that the topological structures of particle orbits generated by a generic class of vector fields on spherical surfaces, called {\it the flow of finite type}, are in one-to-one correspondence with discrete structures such as trees/graphs and sequence of letters. The flow of finite type is an extension of structurally stable Hamiltonian vector fields, which appear in many theoretical and numerical investigations of 2D incompressible fluid flows. Moreover, it contains compressible 2D vector fields such as the Morse--Smale vector fields and the projection of 3D vector fields onto 2D sections. The discrete representation is not only a simple symbolic identifier for the topological structure of complex flows, but it also gives rise to a new methodology of topological data analysis for flows when applied to data brought by measurements, experiments, and numerical simulations of complex flows. As a proof of concept, we provide some applications of the representation theory to 2D compressible vector fields and a 3D vector field arising in an industrial problem.

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