Abstract booklet of RIMS Symposium (open)
Dynamical Systems: Theory and Applications

RIMS 共同研究（公開型）「力学系の理論と応用」概要集

May 28, 2024
You can get the latest version of the program and the abstract from the webpage
http://saiki.hub.hit-u.ac.jp/rims2024.html[June 3, Monday]
13:30-14:15
Speaker: Motomasa Komuro (University of Tokyo)/小室元政（東京大学）
Title: BIFURCATIONS OF COUPLED SIN-CIRCLE MAPS (6)/結合サインサークル写像の分岐について（6）結合サインサークル写像の不動点近傍の共鳴ストリップの消滅現象を解析する。
14:30-15:15
Speaker: Isao Ishikawa (Ehime University)/石川勳（愛媛大学）
Title: 力学系から定まる局所解析汎函数上の押し出しの有限次元近似について
本講演では、力学系から定まる局所解析汎函数上の押し出し写像の有限次元近似をデータ駆動的行う
理論的な枠組みについて紹介する。この枠組みによって力学系のデータ駆動解析において重要な動的モード分解（Dynamic mode decomposition; DMD）を組織的に取り扱うことが可能となり、DMD のアルゴリズムによって得られる近似行列について数学的に厳密な解釈を与えることが可能となる。応用として、ユークリッド空間上の解析的なベクトル場について、ベクトル場から定まる力学系の軌道上の離散的な点から元のベクトル場を復元する手法についても紹介する。

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15:30-16:15
Speaker: Kohei Ueno (Daido University)/上野康平 (大同大学)
Title: Newton polygons and Bottcher coordinates for skew products: superattracting caseand polynomial caseLet f (z, w) = (p(z), q(z, w)) be a superattracting or polynomial skew product of two complex variables.
Under one or two conditions, we construct a Bottcher coordinate on an invariant region that conjugates f
to a monomial map. For the superattracting case, the closure of the region contains the superattracting
fixed point and is included in the attracting basin. The monomial map and the region are determinedby the order of p and the Newton polygon of q. For the polynomial case, the region is included in theattracting basin of a superattracting fixed or indeterminacy point at infinity, or in the closure of theattracting basins of two points at infinity. The monomial map and the region are determined by the
degree of p and a Newton polygon of q.
16:25-16:50 [short communication]
Speaker: Yanghong Yu (Tokyo Institute of Technology)/余楊鴻（東京工業大学）
Title: Transition Matrix without Continuation in the Conley Index
Theory Given a one-parameter family of flows over a parameter interval Λ, assuming there is a continuation of Morse decompositions over Λ, Reineck defined a singular transition matrix to show the existenceof a connection orbit between some Morse sets at some parameter points in Λ. My presentation aims
to extend the definition of a singular transition matrix in cases where there is no continuation of Morse
decompositions over the parameter interval. This extension will help study the bifurcation associatedwith the change of Morse decomposition from a topological dynamics viewpoint.

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[June 4, Tuesday]
10:00-10:25 [short communication]
Speaker: Masato Hara (Kyoto University)/原誠人（京都大学）
Title: A reservoir computing method for dynamical systems on manifolds/多様体上の力学系のリザバー計算
リザバー計算は機械学習法の一種であり，力学系が生成する時系列の学習に適している．リザバー計
算の学習機構を解明するための数学的研究は近年少しずつ進展しているが，現時点では，構造安定性などのよい性質を仮定して議論を進めることが多い．そして，そのようなよい性質をもつ力学系は，典型的には（Euclid 空間ではなく）球面やトーラスの上で定義される．ところが，通常のリザバー計
算法は専ら Euclid 空間上のデータを扱うことを想定しているために，球面やトーラスの上で定義された力学系を適切に学習することができない．従って，理論の検証に役立つ数値例を作るために，「多様体上の力学系のリザバー計算」法を考えることには意義がある．このショートコミュニケーションでは，講演者が提案する「多様体上の力学系のリザバー計算」法の概略を数値例とともに紹介する．
10:35-10:55 [short communication]
Speaker: Kengo Nakai (Okayama University)/中井拳吾（岡山大学）
Title: Low-period periodic orbits model the Lorenz attractor
We investigate the effect of reservoir computing training data on the reconstruction of a chaotic attractor.
We show that a time series of a few periodic orbits of low periods is sufficient to reconstruct the Lorenz
attractor and that biased training data does not affect the reconstruction’s success. The degree ofreconstruction is much better than that of so-called cycle expansion. It is also shown that a chaotic
saddle can be reconstructed by a few periodic orbits. This is a joint work with Yoshitaka Saiki.
11:05-11:50
Speaker: Tomoharu Suda (Tokyo University of Science)/須田智晴（東京理科大学）
Title: Remarks on the structure of time series data
Determining the rule of time evolution from actual observation data is a practically important problem.
Essentially, time series data is a collection of finite sequences with an additional structure. In this view,reconstructing dynamics from time series data necessarily involves finding an infinite sequence (an orbit)
consistent with the given data. Here, we introduce some general results on this question. We present
dynamical systems and sequences as data as separate categories, and present their inter-dependence using
the categorical language of functors and natural transformations. This talk is based on a joint work with
S. Das.

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13:30-14:50 [special talk]
Speaker: Michio Yamada (Kyoto University)/山田道夫（京都大学）
Title: 流れの遷移と波動現象
境界層流れの乱流への遷移は，通常の線形不安定性では説明困難な過程として知られていたが，今世紀に入ってから多くの研究によって，線形不安定性によらない遷移過程として記述できることが明らかになりつつある．ここではこの描像を述べた後，さらに線形波動を伴う流体系における流れの構造
形成について触れたい．
15:05-16:05 [special talk]
Speaker: Hiroaki Miura (University of Tokyo)/三浦裕亮（東京大学）
Title: 気候系の成立ちとシミュレーションの課題
地球の気候の将来変化を計算するために、気候モデルというプログラムパッケージが使われている。
流体力学の方程式を、放射伝達方程式や水の相変化の方程式と組み合わせて大気の流れを計算し、また、塩分の変化や氷床の方程式と組み合わせて海洋の流れを計算する。さらには、土壌物理や植物の活動を表現する式を立てて陸のエネルルギー収支を評価する。現代においては、天気予報に用いられる気象モデルと気候予測に用いられる気候モデルにはあまり違いがない。本講演では、気候モデルが表現しようとする気候系の概略について紹介するとともに、コンピュータの計算速度や我々の物理的理解の不十分さに起因する気候モデルの怪しさについてお話しする。また、次世代の高解像度気候モデルが表現を目指す現象や、近年の気候モデルの改良の方向性についても（学生の成果を借りながら）紹介する。
16:15-16:55 [special talk]
Speaker: Takuya Jinno (Toyama University)/神野拓哉（富山大学）
Title: 二次元格子モデルを用いた積雲対流の自己組織化現象についての研究
積雲対流は熱帯のさまざまな環境場で自発的にクラスターを形成することが知られており、これは自己組織化現象と呼ばれている。しかし、対流の自己組織化は地表面の熱フラックス、大気放射、大気
境界層内の水蒸気の分布などさまざまな物理過程からの影響を受けており、どのプロセスが主要な役割を果たしているかは明確になっていない。本研究では深い対流の空間分布を表現するような確率過程に基づく格子モデルを開発した。このモデルには雲同士の相互作用として、他の対流の発生を抑制
する放射冷却の効果と、新しい対流の発生を促進する降水性の下降流の効果が導入されている。その結果、放射冷却による効果の強度を変化させることにより、対流がランダムに発生する状態から自発的に集合して生じる状態への遷移が再現されることが示された。これは長波放射のフィードバックが対流の自己組織化を引き起こす主要な過程の一つであることを示唆し、さらに実験領域のサイズ等、いくつかのモデルのパラメーターに対する自己組織化の依存性も明らかにした。

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[June 5, Wednesday]
10:00-10:45
Speaker: Akio Matsumoto (Chuo University)/松本昭夫（中央大学）
Title: Delay Solow Model Revisited
This paper shows how cyclic behavior can emerge in a simple Solow capital accumulation model. For thispurpose, it introduces a production delay (i.e., time-to-build) and a depreciation delay, and the resultant
capital accumulation equation is a delay differential equation having delay-dependent coefficients. Then,it constructs a condition under which the interaction of the delays and the population growth rate can leadto growth cycles that are not unlike those observed in reality. In particular, it is demonstrated that (i)the production delay alone is always harmless; (ii) with a lower population growth rate, the depreciation
delay alone has a destabilizing effect if it is larger than a threshold value; (iii) with a larger population
growth rate, the stable steady state arises regardless of the delay length. It is further demonstrated thatmultiple stability switching can occur in the case of two distinct delays.
11:00-11:45
Speaker: Tomoo Yokoyama (Saitama University)/横山知郎（埼玉大学）
Title: Structural stability of “incompressible” line fields on surfaces and generic intermediate line fields
Though there is a representation that completely describes the topologies of generic flows on compact
surfaces, there is no such a known representation that describes line fields (e.g., fingerprints, nematic
liquid crystals, pinwheel structures in the visual field), which is a generalization of a concept of flow. In
fact, one stated that there is no natural topology of the space of line fields on surfaces. In this talk, we
introduce the topology generated by cone structures in the space of line fields as an analog of the topologyof the space of flows to describe the structural stability of “incompressible” line fields on surfaces. We
will show that, for a line field corresponding to a Hamiltonian flow, the general structure observed in theline field that appears in some phenomena is structurally stable for this topology.
13:30-14:15 [English]
Speaker: Yi-Chiuan Chen (Academia Sinica)
Title: More on the Concept of Anti-integrability for Hénon Maps√
For the family of Hénon maps (x, y) 7→ ( a(1 − x2 ) − by, x) of R2 , the so-called anti-integrable (AI) limit
concerns the limit a → ∞ with fixed Jacobian b. At the AI limit, the dynamics reduces to a subshift offinite type. There is a one-to-one correspondence between sequences allowed by the subshift and the AI
orbits. The theory of anti-integrability says that each AI orbit can be continued to becoming a genuine
orbit of the Hénon map for a sufficiently large (and fixed Jacobian).
In this talk, we assume b is a smooth function of a and show that the theory can extended to inves√tigating the limit lima→∞ b/ a = r̂ for any r̂ > 0 provided that the one dimensional quadratic map
1
x 7→ (1 − x2 ) is hyperbolic.
r̂(Joint work with Zin Arai, Tokyo Institute of Technology.)5

14:30-15:15 [English]
Speaker: Tomoki Kawahira (Hitotsubashi University)/川平友規（一橋大学）
Title: J-stability in complex and non-Archimedean dynamics
In structurally stable dynamical systems, the chaotic locus moves continuously as the parameter changes.
In the dynamics with one complex variable, we have such a continuous motion of the Julia set (chaotic
locus) under a slightly weaker condition than structural stability, called “J-stability”. Indeed, the bifurcation locus in the parameter space is the complement of the set of J-stable parameters.
In this talk I will give a quick survey on J-stability in the dynamics of rational functions with complex
coefficients, following the works of Mañé-Sad-Sullivan (1983) and McMullen-Sullivan (1997). Then we
discuss its non-Archimedean analogues recently developed in the works of Lee (2019), Benedetto-Lee(2021), and in my joint work with Junghun Lee.
15:35-16:20 [English]
Speaker: Yuto Nakajima (Tokai University)/中島由人（東海大学）
Title: On the zeros of power series
We study the set of zeros of power series. We prove that the set of zeros in the unit disk is connected
and locally connected if some graphs on the sets of coefficients are connected. Furthermore, we discuss
examples related to the connectedness locus for a parameter family of iterated function systems.
16:30-16:55 [short communication, English]
Speaker: Kanji Inui (Doshisha University)/伊縫 寛治（同志社大学）
Title: An example of non-ergodic measure preserving systems
In ergodic theory, given a concrete exmaple of the systems, it is sometimes difficult to prove the (non)ergodicity of the systems. Inspired by Esaki’s, Kazukawa’s, and Mitsuishi’s paper (Esaki, Kazukawa,and Mitsuishi (2024)), we can find an example of the non-ergodic systems obtained by a representation
used in Bayesian statistics, and the representation seems especially interesting in ergodic theoretical
viewpoint. In this talk, we show the non-ergodicity of the systems by the representation, and discuss thetechnique obtained by the representation. This is a joint work with Syota Esaki.

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[June 6, Thursday]
10:00-10:25 [short communication, English]
Speaker: Yuki Yayama (Universidad del Bío-Bío)/矢山ゆき（ビオビオ大学）
Title: On Gibbs measures for almost additive potentials
Given an almost additive sequence of continuous functions with bounded variation F = {log fn }∞n=1
on a subshift X over finitely many symbols, we study properties of a function f on X such thatRR
limn→∞ n1 log fn dµ = f dµ for every invariant measure µ on X. Under some conditions we construct a function f on X and study a relation between the property of F and some particular typesof f . We also study the case when F is weakly almost additive. As applications we study images ofGibbs measures for continuous functions under one-block factor maps. In order to study conditions forthe image to be a Gibbs measure for a continuous function, we investigate a relation between the almost
additivity of the sequences associated to relative pressure functions and the fiber-mixing property of afactor map.
10:35-11:45 [special talk, English]
Speaker: Mao Shinoda (Ochanomizu University)/篠田万穂（お茶の水女子大学）
Title: Ergodic optimization for continuous functions on non-Markov shifts
The aim of Ergodic optimization is to describe invariant probablity measures that maximize the integralof a given function. For a wide class of intrinsically ergodic subshifts over a finite alphabet, we show thatthe space of continuous functions splits into two subsets: one is a residula set for which all maximizing
measures have ”small” entropy; the other is contained in the clousure of the set of functions haveing
uncountably many, fully supported ergodic maximizing measures with ”large” entropy.
13:30-14:40 [special talk, English]
Speaker: Renaud Leplaideur (Université de la Nouvelle-Calédonie)
Title: Some advances in the selection problem of measures at zero temperature
The talk will be as self-contained as possible. This is a joined work with Jairo Mengue from UFRGS.
Ergodic theory in dynamical systems aims to describe almost all trajectories of some given dynamicalsystems. As a systems usually have many invariant measures, the thermodynamic formalism aims to
select one of these measures via a variational principle. Such a measure is seen as an equilibrium. Then,ergodic optimization aims to describe how this equilibrium changes as the temperature of the system(seen as an extra-parameter) goes to zero, and what happens at the limit.
Then the talk will present in a non-technical way the question of selection and give some new resultsin that topic.

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14:55-15:20 [short communication, English]
Speaker: Hiroki Takahasi (Keio University)/高橋博樹（慶應義塾大学）
Title: Large deviation principle and distribution of periodic points for the Dyck shift andthe heterochaos baker maps
We pursue the complementary analysis of the dynamics of the heterochaos baker map and that of the Dyck
shift launched in [4,5]. The heterochaos baker map [2] is one of the simplest partially hyperbolic systems,
and expected to provide a hands-on, elementary understanding of complicated dynamical phenomena inhigher dimension. The name ‘heterochaos’comes from its distinctive property that periodic points withdifferent unstable dimensions coexist densely in the same transitive invariant set [2], a phenomenon aka
the unstable dimension variability in physics literature, which was recognized earlier since the 70s as a
C1 robust phenomenon for diffeomorphisms of closed manifolds of dimension at least three. The Dyck
shift [1] is a well-known example of a subshift that has two ergodic measures of maximal entropy (MME).
There is a universal formal language due to W. Dyck. The Dyck shift is the symbolic dynamics generated
by that language.
A simple surprising connection between the heterochaos baker map and the Dyck shift has been discovered in [5], and used in [5] to show that the heterochaos baker map counter-intuitively has exactly two
ergodic MMEs. Little is known on properties of these MMEs. Exponential mixing for the two ergodic
MMEs of the Dyck shift has been established in [4] exploiting the geometric structure of the heterochaosbaker map.
In this talk, we investigate asymptotic distributions of periodic points of these two apparently unrelated
dynamical systems from different origins. On the one hand, exploiting the structure of the Dyck shift
space we establish the level-2 large deviation principles, and as a consequence show that the two ergodic
MMEs of the Dyck shift are represented as asymptotic distributions of the sets of periodic points ofdifferent symbolic types. On the other hand, we transfer this result to the heterochaos baker map toshow that its two ergodic MMEs are represented as asymptotic distributions of the sets of saddles ofdifferent unstable indices. This settles the conjecture in [3].
[1] Krieger, W.: On the uniqueness of the equilibrium state, Math. Systems Theory., 8, 1974/75, 97–104.
[2] Saiki, Y., Takahasi, H., Yorke, J.A.: Piecewise linear maps with heterogeneous chaos. Nonlinearity
34, 5744–5761 (2021)[3] Saiki, Y., Takahasi, H., Yamamoto, K., Yorke, J. A.: The dynamics of the heterochaos baker maps.
arXiv:2401.00836[4] Takahasi, H.: Exponential mixing for heterochaos baker maps and the Dyck system. Journal of
Dynamics and Differential Equations, to appear[5] Takahasi, H., Yamamoto, K.: Heterochaos baker maps and the Dyck system: maximal entropy
measures and a mechanism for the breakdown of entropy approachability. Proc. Amer. Math. Soc., toappear8

15:30-16:50 [special talk]
Speaker: Masaru Inatsu (Hokkaido University)/稲津将（北海道大学）
Title: 中高緯度大気の予測可能性
中高緯度大気はそのカオス的振る舞いのため、気象予報の限界は高々２週間程度とされている。しかし、中高緯度大気であっても予測限界が長い場合と短い場合があることが知られている。その中でも中高緯度の予測可能性に重要な因子となる大気ブロッキングをキーワードに講演者が取り組んだ２つの研究を紹介する。
大気ブロッキングの発達の原因として低周波・高周波の渦フラックス収束の効果があるとされている。しかし、予測データにその診断をしようとすると、未来時間のデータにフィルタを適用する必要がある。研究 [2] ではまず、準地衡系で行われているモード展開法をプリミティブ系に拡張し、モード方程式を導出する。その後、Barriopedro に従い定義したブロッキング事例の合成として、ブロッキン
グを Z500 主成分の線型結合の持続として再定義する。これにより、導出したモード方程式によるブロッキング診断が可能になる。本研究では予測データを使わず、あくまで再解析データでの診断を行う。
次に、大気ブロッキングを含む多様な長周期変動を自己組織化写像で生成された潜在空間上で表現
し、気象予測可能性を診断した [1]。まず、解析データから基準となる潜在空間を生成する。次に、
S2S プロジェクトによる各国気象機関の１か月アンサンブル予報データに射影する。潜在空間上のノードをまとめたクラスタ分析から大気状態を少数に分類することで、予測限界が長い場合と短い場
合が、予報モデルによらず共通していることがわかった。これは物理的に考えられる予測可能性に一致するものと期待される。以上は、講演者が過去に行った主成分で張った相空間上での単一モデルの予測可能性研究 [3] と整合する結果である。
[1] Inatsu, M., M. Matsueda, N. Nakano, and S. Kawazoe, 2023: Prediction skill and practical predictability depending on the initial atmospheric states in S2S forecasts. Journal of Atmospheric Sciences, 80,
1449–1462.
[2] Aikawa, T., M. Inatsu, N. Nakano, and T. Iwano, 2019: Mode-decomposed equation diagnosis foratmospheric blocking development. Journal of the Atmospheric Sciences, 76, 3151–3167.
[3] Inatsu, M., N. Nakano, and H. Mukougawa, 2013: Dynamics and practical predictability of extratropical wintertime low-frequency variability in a low-dimensional system. Journal of the Atmospheric
Sciences, 70, 939–952.

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[June 7, Friday]
10:00-10:45
Speaker: Shousuke Ohmori (National Institute of Technology, Gunma College, Waseda
University)・Yoshihiro Yamazaki (Waseda University)/大森祥輔（群馬工業高等専門学校, 早稲田大学）・山崎義弘 (早稲田大)[共同発表]
Title: Ultradiscrete Bifurcation for low-dimensional ultradiscrete dynamical systems/低次
元超離散力学系の分岐構造
本発表では、分岐現象を有する低次元連続力学系に対する超離散方程式の力学的性質について、これまでの研究に対する全体的な報告を行う。発表の前半では、低次元超離散モデル（max-plus 離散力学
系）が引き起こす分岐やリミットサイクルについて、相平面解析や Poincare 写像の手法を用いた解析
結果を紹介する。特に離散周期軌道がリミットサイクルとなることを述べる。 超離散モデルは、連続モデルにトロピカル差分化 (正値差分化) の手法を適用して差分化した後、超離散化を施すことで得られる。そこで、発表の後半では、このトロピカル差分化によって得られる差分方程式の力学的性質に関する研究を述べる。具体的には、連続モデルからトロピカル差分化する際に起こる力学的変化についての一般論を紹介した後、トロピカル差分化から超離散状態に至る過程でのリミットサイクルの変化についてまとめた結果を、発表前半で述べた研究に紐づけながら報告する。
11:00-11:45
Speaker: Kazuyuki Yagasaki (Kyoto University)/矢ヶ崎一幸（京都大学）
Title: Semiclassical Perturbations of Single-Degree-of-Freedom Hamiltonian Systems: Separatrix Splitting and Nonintegrability
We study semiclassical perturbations of single-degree-of-freedom Hamiltonian systems and provide sufficient conditions for the separatrices to split and for their meromorphic nonintegrability such that the first
integrals depend on the small parameter meromorphically. To obtain the former and latter results, weuse a Melnikov-type approach and a generalization due to Ayoul and Zung of the Morales-Ramis theory,
respectively. We illustrate the theoretical results for bounded potentials.
13:30-13:55 [short communication]
Speaker: Shin-itiro Goto (Chubu University)/後藤振一郎（中部大学）
Title: Derivation of a contact Hamiltonian system from the Fokker-Planck equation
The Fokker-Planck equation is a PDE known in nonequilibrium statistical mechanics, describes relaxation
processes of a distribution function, and is formulated on a Riemannian manifold. Meanwhile contact
Hamiltonian systems are also known to describe relaxation processes for thermodynamic variables. In
this talk, to clarify a relation between these two equations, a class of contact Hamiltonian systems isshown to be derived from a class of the Fokker-Planck equations on Riemannian manifolds.

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14:05-14:50
Speaker: Sogo Murakami (University of Tokyo)/村上聡梧（東京大学）
Title: Multidimensional C 0 transversality and the shadowing property for Axim A diffeomorphisms
In this talk, we introduce a notion of C 0 transversality and prove related results we found. Petrov andPilyugin (2015) generalized a notion of C 0 transversality of Sakai (1995). Their definition involves only
continuous maps from Rn to a manifold, which is a purely topological one. They also provided a sufficientcondition for the C 0 transversality in terms of homological nature. We prove that such a homologicalcondition of Axiom A diffeomorphisms is sufficient for enjoying the shadowing property.

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