arXiv subjects

Tomoharu Suda

Publications and source records attributed to Tomoharu Suda.

13 recordsLinked to original sources

arXiv2026

Ergodicity of dynamical systems without uniqueness of orbits

Recently, there has been considerable interest in the study of non-deterministic dynamical systems. To analyze the chaotic behavior of such systems from a measure-theoretic viewpoint, it is desirable to consider ergodicity. However, the classical definition of ergodicity involves invariant sets, whose definition is not unique for non-deterministic dynamical systems. Thus, we are led to the question of which invariance yields an interesting definition of ergodicity. Here, we propose a definition based on the strong backward invariance and show that analogs of classical results hold. We also consider implications of the Birkhoff ergodic theorem for systems without uniqueness of orbits.

math.DS↗

arXiv2026

Joinings in Markov categories

Dynamical systems theory primarily concerns the study of different forms of invariance, such as invariant sets, densities, measures, and observables. Ergodic systems are a special class of dynamical systems which are measure theoretically irreducible. In spite of this specialized property they dominate most discussions on ergodic theory because of the ergodic decomposition theorem. The viewpoint that any dynamical system is a composite of multiple ergodic components enables us to partition the face space into the basins of the different measures. The coexistence and mutual connections between these various coexisting subsystems are illuminated very effectively using the language of Category theory (CT). Some recent advancements have shown how the essence of measure theoretic dynamical systems can be captured through the formalism of Markov categories. This formalism captures the essential features of invariance and ergodicity through the language of limits and colimits. This article continues that formalism by studying the concept of joins using the language of spans and push-outs. A classical result is re-proven which establishes the connection between joins, ergodicity and mixing. In the process, the categorical language is refined to capture the notion of sub sigma-algebras and their invariance.

math.CT↗

arXiv2025

Dynamical systems as enriched functors

This article presents a general description of dynamical systems using the language of enriched functors and enriched natural transformations. This framework is essential to establish the equivalence of three descriptions of dynamics -- a semigroup action on the domain; a parameterized family of endomorphisms; and a transformation of time-space into the collection of endomorphisms. A collection of categorical axioms are presented that provides a complete categorical language to develop dynamical systems theory. None of the assumptions are rooted in specific contexts such as topology and measure spaces. The equivalence of the three descriptions is further used to construct other related notions,such as transfer operators, orbits and shift-spaces. All of these objects are defined by their structural role and universal properties, instead of their usual pointwise definitions.

math.CT↗

arXiv2024

Dynamics, data and reconstruction

The goal of data-driven learning of dynamical systems is to interpret time series as a continuous observation of an underlying dynamical system. This task is not well-posed for a variety of reasons - such as multiple co-existing sub-systems, topologically inter-weaving of these sub-systems; and more importantly, the non-injectivity of the correspondence between dynamical systems and time series. We show how these ambiguities are circumvented if one considers dynamical systems and measurement maps collectively. Dynamical systems, observed dynamical systems, and time series data - each of these three collections have an extensive network of relations within them, which gives them the mathematical structure of a category. One of the new concepts proposed is a rigorous definition of time series data as a chain of measurement sequences with decreasing information content. This definition subsumes the familiar notions of sequences, time series and even subshifts. Using these notions it is shown that the entire process of converting an observed dynamical systems into a time series object is functorial, and passes through a number of phases each bearing its own categorical structure. This discovery sheds new light on the nature of reconstruction algorithms. Under mild conditions of consistency, reconstruction itself is shown to be functorial operation. This provides a new category theoretic perspective on the nature and limits of reconstruction.

math.DS↗

arXiv2024

Effective Kinetics of Chemical Reaction Networks

Chemical reaction network theory is a powerful framework to describe and analyze chemical systems. While much about the concentration profile in an equilibrium state can be determined in terms of the graph structure, the overall reaction's time evolution depends on the network's kinetic rate function. In this article, we consider the problem of the effective kinetics of a chemical reaction network regarded as a conversion system from the feeding species to products. We define the notion of effective kinetics as a partial solution of a system of non-autonomous ordinary differential equations determined from a chemical reaction network. Examples of actual calculations include the Michaelis-Menten mechanism, for which it is confirmed that our notion of effective kinetics yields the classical formula. Further, we introduce the notion of straight-line solutions of non-autonomous ordinary differential equations to formalize the situation where a well-defined reaction rate exists and consider its relation with the quasi-stationary state approximation used in microkinetics. Our considerations here give a unified framework to formulate the reaction rate of chemical reaction networks.

q-bio.MN↗

arXiv2024

Dynamical properties in the axiomatic theory of ordinary differential equations

The axiomatic theory of ordinary differential equations, owing to its simplicity, can provide a useful framework to describe various generalizations of dynamical systems. In this study, we consider how dynamical properties can be generalized to this setting. First, we study the generalizations of dynamical properties, such as invariance and limit sets and show that compact weakly invariant sets can be characterized in terms of the corresponding shift-invariant sets in the function space. This result enables us to consider weakly invariant sets in terms of shift-invariant sets. Next, we compare the axiomatic theory of ODE with other formalisms of generalized dynamical systems, namely generalized semiflow and multivalued semigroup. Finally, we introduce the notion of invariant measures and show that we can generalize classical results, such as Poincaré recurrence theorem.

math.DS↗

arXiv2023

On partial maps derived from flows

The first-return map, or the Poincaré map, is a fundamental concept in the theory of flows. However, it can generally be defined only partially, and additional conditions are required to define it globally. Since this partiality reflects the dynamics, the flow can be described by considering the domain and behavior of such maps. In this study, we define the concept of first-exit maps and first-return maps, which are partial maps derived from flows, to enable such analysis. Moreover, we generalize some notions related to the first-return maps. It is shown that the boundary points of an open set can be classified based on the behavior of these maps, and that this classification is invariant under topological equivalence. Further, we show that some dynamical properties of a flow can be described in terms of the types of boundary points. In particular, if the flow is planar and the open set has a Jordan curve as its boundary, a more detailed analysis is possible, and we present results on the conditions which restrict possible forms of the first-exit maps . Finally, as an application of the results obtained, we consider the relationship between flows and a class of hybrid systems.

math.DS↗

arXiv2022

Equivalence of topological dynamics without well-posedness

The notion of topological equivalence plays an essential role in the study of dynamical systems of flows. However, it is inherently difficult to generalize this concept to systems without well-posedness in the sense of Hadamard. In this study, we formulate a notion of "topological equivalence" between such systems based on the axiomatic theory of topological dynamics proposed by Yorke, and discuss its relation with the usual definition. During this process, we generalize Yorke's theory to the action of topological groups.

math.DS↗

arXiv2022

Poincaré maps and suspension flows: a categorical remark

Poincaré maps and suspension flows are examples of fundamental constructions in the study of dynamical systems. This study aimed to show that these constructions define an adjoint pair of functors if categories of dynamical systems are suitably set. First, we consider the construction of Poincaré maps in the category of flows on topological manifolds, which are not necessarily smooth. We show that well-known results can be generalized and the construction of Poincaré maps is functorial, if a category of flows with global Poincaré sections is adequately defined. Next, we consider the construction of suspension flows and its functoriality. Finally, we consider the adjointness of the constructions of Poincaré maps and suspension flows. By considering the naturality, we can conclude that the concepts of topological equivalence or topological conjugacy of flows are not sufficient to describe the correspondence between map dynamical systems and flows with global Poincaré sections. We define another category of flows with global Poincaré sections and show that the suspension functor and the Poincaré map functor form an adjoint equivalence if these categories are considered. Hence, a categorical correspondence between map dynamical systems and flows with global Poincaré sections is obtained. This will enable us to better understand the connection between map dynamical systems and flows.

math.DS↗

arXiv2021

A characterization of the Filippov convention

The Filippov convention is widely used in the literature to define vector fields on a discontinuity set of piecewise-continuous vector fields. The aim of this paper is to give a sufficient and necessary condition for an interpolation scheme of piecewise-continuous vector fields to coincide with the Filippov convention. That is, we show that a map from a space of piecewise-continuous vector fields with two components to the space of vector fields coincides with the Filippov convention where the latter can be applied, if it is sufficiently well-behaved as a generalization of continuous vector fields.

math.DS↗

arXiv2020

Application of Helmholtz-Hodge decomposition to the study of certain vector fields

Smooth vector fields on $\mathbb{R}^n$ can be decomposed into the sum of a gradient vector field and divergence-free (solenoidal) vector field under suitable hypotheses. This is called the Helmholtz-Hodge decomposition (HHD), which has been applied to analyze the topological features of vector fields. In this study, we apply the HHD to study certain types of vector fields. In particular, we investigate the existence of strictly orthogonal HHDs, which assure an effective analysis. The first object of the study is linear vector fields. We demonstrate that a strictly orthogonal HHD for a vector field of the form $\textbf{F}(\textbf{x}) = A \textbf{x}$ can be obtained by solving an algebraic Riccati equation. Subsequently, a method to explicitly construct a Lyapunov function is established. In particular, if A is normal, there exists an easy solution to this equation. Next, we study planar vector fields. In this case, the HHD yields a complex potential, which is a generalization of the notion in hydrodynamics with the same name. We demonstrate the convenience of the complex potential formalism by analyzing vector fields given by homogeneous quadratic polynomials.

math.DS↗

arXiv2019

Construction of Lyapunov functions using Helmholtz-Hodge decomposition

The Helmholtz-Hodge decomposition (HHD) is applied to the construction of Lyapunov functions. It is shown that if a stability condition is satisfied, such a decomposition can be chosen so that its potential function is a Lyapunov function. In connection with the Lyapunov function, vector fields with strictly orthogonal HHD are analyzed. It is shown that they are a generalization of gradient vector fields and have similar properties. Finally, to examine the limitations of the proposed method, planar vector fields are analyzed.

math.DS↗

arXiv2016

Destruction of Magnetic Long-Range Order by Hole-Induced Skyrmions in Two-Dimensional Heisenberg Model

Motivated by the rapid destruction of antiferromagnetic long-range order in hole-doped cuprate high-temperature superconductors, we study the effect of skyrmions on the magnetic long-range order (MLRO). Here we assume that either a skyrmion or antiskyrmion is introduced by a doped hole. Our numerical simulation indicates that in the case of isolated skyrmions, there is an abrupt disappearance of MLRO for doping concentration $x < 1.0\times 10^{-4}$. In the case of skyrmion-antiskyrmion pairs, the critical doping concentration $x_c$ for the suppression of MLRO is given as a function of the separation of the pairs. For a moderate separation of $3 - 4$ lattice constants, we find that the critical doping is consistent with the experimental value.

cond-mat.supr-con↗