arXiv subjects

Hayato Chiba

Publications and source records attributed to Hayato Chiba.

At least 19 recordsLinked to original sources

arXiv2026

On the degeneration of Kovalevskaya exponents of Laurent series solutions of quasi-homogeneous vector fields

A structure of families of Laurent series solutions of a quasi-homogeneous vector field is studied, where a given vector field is assumed to have a commutable vector field. For an $m$ dimensional vector field, a family of Laurent series solutions is called principle if it includes $m$ arbitrary parameters, and called non-principle if the number is smaller than $m$. Starting from a principle Laurent series solutions, a systematic method to obtain a non-principle Laurent series solutions is given. In particular, from the Kovalevskaya exponents of the principle Laurent series solutions, which is one of the invariants of quasi-homogeneous vector fields, the Kovalevskaya exponents of the non-principle Laurent series solutions are obtained by using the commutable vector field.

math.DS↗

arXiv2025

Reservoir computing with the Kuramoto model

Reservoir computing aims to achieve high-performance and low-cost machine learning with a dynamical system as a reservoir. However, in general, there are almost no theoretical guidelines for its high-performance or optimality. Therefore, this paper aims to propose the new concept {\it the edge of bifurcation} for designing a high-performance reservoir, and provide a mathematical justification for it. This concept is a generalization of the famous criterion {\it the edge of chaos}. For this purpose, this paper focuses on the reservoir computing with the Kuramoto model and theoretically reveals its approximation ability. The main result provides an explicit expression of the dynamics of the Kuramoto reservoir by using the order parameters. Thus, the output of the reservoir computing is expressed as a linear combination of the order parameters. As a corollary, sufficient conditions on hyperparameters are obtained so that the set of the order parameters gives the complete basis of the Lebesgue space. This implies that the Kuramoto reservoir has a universal approximation property. Furthermore, the edge of bifurcation is also discussed from the viewpoint of its approximation ability. It is numerically demonstrated by prediction tasks.

math.DS↗

arXiv2025

Quasi-Homogeneous Integrable Systems: Free Parameters, Kovalevskaya Exponents, and the Painlev\'e Property

This paper investigates quasi-homogeneous integrable systems by analyzing their Laurent series solutions near movable singularities, motivated by patterns observed in Kovalevskaya exponents of four-dimensional Painlev\'e-type equations. We introduce a parameter space encoding the free coefficients in these expansions and study its deformation under a commuting quasi-homogeneous vector field. Within this framework, we derive lower indicial loci from the principal one and establish an arithmetic resonance condition on Kovalevskaya exponents that governs the emergence of fractional powers and the breakdown of the Painlev\'e property. Moreover, we construct a Frobenius manifold structure on the parameter space via the initial value map, which becomes conformal when all weights coincide. In the Hamiltonian context, we demonstrate that the induced flow on the parameter space preserves a symplectic form and yields a natural pairing of Kovalevskaya exponents. These findings unify analytic and geometric aspects of quasi-homogeneous integrable systems and offer new insights into their deformation theory and singularity structures. Our results provide a comprehensive framework applicable to the classification and analysis of Painlev\'e-type equations and related integrable models.

nlin.SI↗

arXiv2024

Bifurcation Mechanism of Quasi-Halo Orbit from Lissajous Orbit

This paper presents a general analytical method to describe the center manifolds of collinear libration points in the Restricted Three-body Problem (RTBP). It is well-known that these center manifolds include Lissajous orbits, halo orbits, and quasihalo orbits. Previous studies have traditionally treated these orbits separately by iteratively constructing high-order series solutions using the Lindstedt-Poincaré method. Instead of relying on resonance between their frequencies, this study identifies that halo and quasihalo orbits arise due to intricate coupling interactions between in-plane and out-of-plane motions. To characterize this coupling effect, a novel concept, coupling coefficient $η$, is introduced in the RTBP, incorporating the coupling term $ηΔx$ in the $z$-direction dynamics equation, where $Δ$ represents a formal power series concerning the amplitudes. Subsequently, a uniform series solution for these orbits is constructed up to a specified order using the Lindstedt-Poincaré method. For any given paired in-plane and out-of-plane amplitudes, the coupling coefficient $η$ is determined by the bifurcation equation $Δ= 0$. When $η$ = 0, the proposed solution describes Lissajous orbits around libration points. As $η$ transitions from zero to non-zero values, the solution describes quasihalo orbits, which bifurcate from Lissajous orbits. Particularly, halo orbits bifurcate from planar Lyapunov orbits if the out-of-plane amplitude is zero. The proposed method provides a unified framework for understanding these intricate orbital behaviors in the RTBP.

astro-ph.EP↗

arXiv2024

Analytical computation of bifurcation of orbits near collinear libration point in the restricted three-body problem

A unified analytical solution is presented for constructing the phase space near collinear libration points in the Circular Restricted Three-body Problem (CRTBP), encompassing Lissajous orbits and quasihalo orbits, their invariant manifolds, as well as transit and non-transit orbits. Traditional methods could only derive separate analytical solutions for the invariant manifolds of Lissajous orbits and halo orbits, falling short for the invariant manifolds of quasihalo orbits. By introducing a coupling coefficient η and a bifurcation equation, a unified series solution for these orbits is systematically developed using a coupling-induced bifurcation mechanism and Lindstedt-Poincaré method. Analyzing the third-order bifurcation equation reveals bifurcation conditions for halo orbits, quasihalo orbits, and their invariant manifolds. Furthermore, new families of periodic orbits similar to halo orbits are discovered, and non-periodic/quasi-periodic orbits, such as transit orbits and non-transit orbits, are found to undergo bifurcations. When η = 0, the series solution describes Lissajous orbits and their invariant manifolds, transit, and non-transit orbits. As η varies from zero to non-zero values, the solution seamlessly transitions to describe quasihalo orbits and their invariant manifolds, as well as newly bifurcated transit and non-transit orbits. This unified analytical framework provides a more comprehensive understanding of the complex phase space structures near collinear libration points in the CRTBP.

math-ph↗

arXiv2023

Bifurcations and patterns in the Kuramoto model with inertia

In this work, we analyze the Kuramoto model (KM) with inertia on a convergent family of graphs. It is assumed that the intrinsic frequencies of the individual oscillators are sampled from a probability distribution. In addition, a given graph, which may also be random, assigns network connectivity. As in the original KM, in the model with inertia, the weak coupling regime features mixing, the state of the network when the phases (but not velocities) of all oscillators are distributed uniformly around the unit circle. We study patterns, which emerge when mixing loses stability under the variation of the strength of coupling. We identify a pitchfork (PF) and an Andronov-Hopf (AH) bifurcations in the model with multimodal intrinsic frequency distributions. To this effect, we use a combination of the linear stability analysis and Penrose diagrams, a geometric technique for studying stability of mixing. We show that the type of a bifurcation and a nascent spatiotemporal pattern depend on the interplay of the qualitative properties of the intrinsic frequency distribution and network connectivity.

nlin.PS↗

arXiv2021

Stability and bifurcation of mixing in the Kuramoto model with inertia

The Kuramoto model of coupled second order damped oscillators on convergent sequences of graphs is analyzed in this work. The oscillators in this model have random intrinsic frequencies and interact with each other via nonlinear coupling. The connectivity of the coupled system is assigned by a graph which may be random as well. In the thermodynamic limit the behavior of the system is captured by the Vlasov equation, a hyperbolic partial differential equation for the probability distribution of the oscillators in the phase space. We study stability of mixing, a steady state solution of the Vlasov equation, corresponding to the uniform distribution of phases. Specifically, we identify a critical value of the strength of coupling, at which the system undergoes a pitchfork bifurcation. It corresponds to the loss of stability of mixing and marks the onset of synchronization. As for the classical Kuramoto model, the presence of the continuous spectrum on the imaginary axis poses the main difficulty for the stability analysis. To overcome this problem, we use the methods from the generalized spectral theory developed for the original Kuramoto model. The analytical results are illustrated with numerical bifurcation diagrams computed for the Kuramoto model on Erdős--Rényi and small-world graphs. Applications of the second-order Kuramoto model include power networks, coupled pendula, and various biological networks. The analysis in this paper provides a mathematical description of the onset of synchronization in these systems.

math.DS↗

arXiv2021

Generalized eigenvalues of the Perron-Frobenius operators of symbolic dynamical systems

The generalized spectral theory is an effective approach to analyze a linear operator on a Hilbert space $\mathcal{H}$ with a continuous spectrum. The generalized spectrum is computed via analytic continuations of the resolvent operators using a dense locally convex subspace $X$ of $\mathcal{H}$ and its dual space $X'$. The three topological spaces $X \subset \mathcal{H} \subset X'$ is called the rigged Hilbert space or the Gelfand triplet. In this paper, the generalized spectra of the Perron-Frobenius operators of the one-sided and two-sided shifts of finite types (symbolic dynamical systems) are determined. A one-sided subshift of finite type which is conjugate to the multiplication with the golden ration on $[0,1]$ modulo $1$ is also considered. A new construction of the Gelfand triplet for the generalized spectrum of symbolic dynamical systems is proposed by means of an algebraic procedure. The asymptotic formula of the iteration of Perron-Frobenius operators is also given. The iteration converges to the mixing state whose rate of convergence is determined by the generalized spectrum.

math.DS↗

arXiv2020

Bifurcation of the neuronal population dynamics of the modified theta model: transition to macroscopic gamma oscillation

Interactions of inhibitory neurons produce gamma oscillations (30--80 Hz) in the local field potential, which is known to be involved in functions such as cognition and attention. In this study, the modified theta model is considered to investigate the theoretical relationship between the microscopic structure of inhibitory neurons and their gamma oscillations under a wide class of distribution functions of tonic currents on individual neurons. The stability and bifurcation of gamma oscillations for the Vlasov equation of the model is investigated by the generalized spectral theory. It is shown that as a connection probability of neurons increases, a pair of generalized eigenvalues crosses the imaginary axis twice, which implies that a stable gamma oscillation exists only when the connection probability has a value within a suitable range. On the other hand, when the distribution of tonic currents on individual neurons is the Lorentzian distribution, the Vlasov equation is reduced to a finite dimensional dynamical system. The bifurcation analyses of the reduced equation exhibit equivalent results with the generalized spectral theory. It is also demonstrated that the numerical computations of neuronal population follow the analyses of the generalized spectral theory as well as the bifurcation analysis of the reduced equation.

q-bio.NC↗

arXiv2020

Weights, Kovalevskaya exponents and the Painlevé property

Weighted degrees of quasihomogeneous Hamiltonian functions of the Painlevé equations are investigated. A tuple of positive integers, called a regular weight, satisfying certain conditions related to singularity theory is classified. Each polynomial Painlevé equation has a regular weight. Conversely, for $2$ and $4$-dim cases, it is shown that there exists a differential equation satisfying the Painlevé property associated with each regular weight. Kovalevskaya exponents of quasihomogeneous Hamiltonian systems are also investigated by means of regular weights, singularity theory and dynamical systems theory. It is shown that there is a one-to-one correspondence between Laurent series solutions and stable manifolds of the associated vector field obtained by the blow-up of the system. For $4$-dim autonomous Painlevé equations, the level surface of Hamiltonian functions can be decomposed into a disjoint union of stable manifolds.

math.CA↗

arXiv2020

Normal Forms of $C^\infty$ Vector Fields based on the Renormalization Group

The normal form theory for polynomial vector fields is extended to those for $C^\infty$ vector fields vanishing at the origin. Explicit formulas for the $C^\infty$ normal form and the near identity transformation which brings a vector field into its normal form are obtained by means of the renormalization group method. The dynamics of a given vector field such as the existence of invariant manifolds is investigated via its normal form. The $C^\infty$ normal form theory is applied to prove the existence of infinitely many periodic orbits of two dimensional systems which is not shown from polynomial normal forms.

math.DS↗

arXiv2020

Instability of mixing in the Kuramoto model: From bifurcations to patterns

We study patterns observed right after the loss of stability of mixing in the Kuramoto model of coupled phase oscillators with random intrinsic frequencies on large graphs, which can also be random. We show that the emergent patterns are formed via two independent mechanisms determined by the shape of the frequency distribution and the limiting structure of the underlying graph sequence. Specifically, we identify two nested eigenvalue problems whose eigenvectors (unstable modes) determine the structure of the nascent patterns. The analysis is illustrated with the results of the numerical experiments with the Kuramoto model with unimodal and bimodal frequency distributions on certain graphs.

nlin.CD↗

arXiv2018

Bifurcations in the Kuramoto model on graphs

In his classical work, Kuramoto analytically described the onset of synchronization in all-to-all coupled networks of phase oscillators with random intrinsic frequencies. Specifically, he identified a critical value of the coupling strength, at which the incoherent state loses stability and a gradual build-up of coherence begins. Recently, Kuramoto's scenario was shown to hold for a large class of coupled systems on convergent families of deterministic and random graphs. Guided by these results, in the present work, we study several model problems illustrating the link between network topology and synchronization in coupled dynamical systems. First, we identify several families of graphs, for which the transition to synchronization in the Kuramoto model starts at the same critical value of the coupling strength and proceeds in practically the same way. These examples include Erdős-Rényi random graphs, Paley graphs, complete bipartite graphs, and certain stochastic block graphs. These examples illustrate that some rather simple structural properties such as the volume of the graph may determine the onset of synchronization, while finer structural features may affect only higher order statistics of the transition to synchronization. Further, we study the transition to synchronization in the Kuramoto model on power law and small-world random graphs. The former family of graphs endows the Kuramoto model with very good synchronizability: the synchronization threshold can be made arbitrarily low by varying the parameter of the power law degree distribution. For the Kuramoto model on small-world graphs, in addition to the transition to synchronization, we identify a new bifurcation leading to stable random twisted states. The examples analyzed in this work complement the results in [Chiba, Medvedev, The mean field analysis for the Kuramoto model on graphs (parts I and II), arxiv].

math.DS↗

arXiv2018

The mean field analysis of the Kuramoto model on graphs II. Asymptotic stability of the incoherent state, center manifold reduction, and bifurcations

In our previous work [Chiba, Medvedev, arXiv:1612.06493], we initiated a mathematical investigation of the onset of synchronization in the Kuramoto model (KM) of coupled phase oscillators on convergent graph sequences. There, we derived and rigorously justified the mean field limit for the KM on graphs. Using linear stability analysis, we identified the critical values of the coupling strength, at which the incoherent state looses stability, thus, determining the onset of synchronization in this model. In the present paper, we study the corresponding bifurcation. Specifically, we show that similar to the original KM with all-to-all coupling, the onset of synchronization in the KM on graphs is realized via a pitchfork bifurcation. The formula for the stable branch of the bifurcating equilibria involves the principal eigenvalue and the corresponding eigenfunctions of the kernel operator defined by the limit of the graph sequence used in the model. This establishes an explicit link between the network structure and the onset of synchronization in the KM on graphs. The results of this work are illustrated with the bifurcation analysis of the KM on Erdős-R{\' e}nyi, small-world, as well as certain weighted graphs on a circle.

math.DS↗

arXiv2017

Multi-Poisson Approach to the Painlevé Equations: from the Isospectral Deformation to the Isomonodromic Deformation

A multi-Poisson structure on a Lie algebra $\mathfrak{g}$ provides a systematic way to construct completely integrable Hamiltonian systems on $\mathfrak{g}$ expressed in Lax form $\partial X_λ/\partial t = [X_λ, A_λ]$ in the sense of the isospectral deformation, where $X_λ, A_λ\in \mathfrak{g}$ depend rationally on the indeterminate $λ$ called the spectral parameter. In this paper, a method for modifying the isospectral deformation equation to the Lax equation $\partial X_λ/\partial t = [X_λ, A_λ] + \partial A_λ/\partial λ$ in the sense of the isomonodromic deformation, which exhibits the Painlevé property, is proposed. This method gives a few new Painlevé systems of dimension four.

math.CA↗

arXiv2016

The mean field analysis for the Kuramoto model on graphs I. The mean field equation and transition point formulas

In his classical work on synchronization, Kuramoto derived the formula for the critical value of the coupling strength corresponding to the transition to synchrony in large ensembles of all-to-all coupled phase oscillators with randomly distributed intrinsic frequencies. We extend the Kuramoto's result to a large class of coupled systems on convergent families of deterministic and random graphs. Specifically, we identify the critical values of the coupling strength (transition points), between which the incoherent state is linearly stable and is unstable otherwise. We show that the transition points depend on the largest positive or/and smallest negative eigenvalue(s) of the kernel operator defined by the graph limit. This reveals the precise mechanism, by which the network topology controls transition to synchrony in the Kuramoto model on graphs. To illustrate the analysis with concrete examples, we derive the transition point formula for the coupled systems on Erdős-Rényi, small-world, and $k$-nearest-neighbor families of graphs. As a result of independent interest, we provide a rigorous justification for the mean field limit for the Kuramoto model on graphs. The latter is used in the derivation of the transition point formulas.

math.DS↗

arXiv2016

A center manifold reduction of the Kuramoto-Daido model with a phase-lag

A bifurcation from the incoherent state to the partially synchronized state of the Kuramoto-Daido model with the coupling function $f(θ) = \sin (θ+α_1) + h\sin 2(θ+α_2)$ is investigated based on the generalized spectral theory and the center manifold reduction. The dynamical system of the order parameter on a center manifold is derived under the assumption that there exists a center manifold on the dual space of a certain test function space. It is shown that the incoherent state loses the stability at a critical coupling strength $K=K_c$, and a stable rotating partially synchronized state appears for $K>K_c$. The velocity of the rotating state is different from the average of natural frequencies of oscillators when $α_1 \neq 0$.

math.DS↗

arXiv2016

A Hopf bifurcation in the Kuramoto-Daido model

A Hopf bifurcation in the Kuramoto-Daido model is investigated based on the generalized spectral theory and the center manifold reduction for a certain class of frequency distributions. The dynamical system of the order parameter on a four-dimensional center manifold is derived. It is shown that the dynamical system undergoes a Hopf bifurcation as the coupling strength increases, which proves the existence of a periodic two-cluster state of oscillators.

math.DS↗