Titles and abstracts of talks
27th (Sun), Hiroshi Fujiwara Memorial Hall, Collaboration Complex 2F, Hiyoshi Campus, Keio University
Chairs: Hiroki Takahasi (Keio), Yoshitaka Saiki (Hitotsubashi)
- 10:05-11:00 James A. Yorke (Maryland USA): Exploring dynamics phenomena in weather models via tiny models
- Abstract: To set the scene, I will discuss a large whole-Earth model for predicting the weather, and our method for re-initializing such a model, and what aspects of chaos are seen there.
Then I will discuss a couple of related “very simple” maps that tell us a great deal about very complex models. The results on simple models are new, either recently published or even newer than that.
I will discuss the logistic map mx(1-x). Its dynamics can make us rethink climate models.
Also, we have created a piecewise linear map on a 3D cube that is unstable in 2 dimensions in some places and unstable in 1 in others. It has a dense set of periodic points that are 1 D unstable and another dense set of periodic points that are all 2 D unstable.
The talk is based on the following recent projects on simple models.
[1] Yoshitaka Saiki and James A Yorke,
Can the flap of a butterfly’s wings shift a tornado into Texas – without chaos?
MDPI Atmosphere 14, no. 5: 821 (2023)
[2] Yoshitaka Saiki, Hiroki Takahasi, and J A Yorke,
The twisted baker map.
Nonlinearity 36 1776–1788 (2023)
[3] Yoshitaka Saiki, Hiroki Takahasi, and J A Yorke, Piecewise linear maps with heterogeneous chaos.
Nonlinearity 34 5744-5761 (2021)
- 11:15-12:00 Hiroshi Kokubu (Kyoto): Reservoir computing for dynamics
- Abstract: Reservoir computing (RC) is a kind of machine learning using a recurrent neural network. In this talk I will present our recent work studying the learning mechanism of RC for dynamical systems, such as logistic maps or hyperbolic dynamics. I will present some numerical results mainly for the logistic maps, and then discuss some thoughts about the learning mechanism of RC from a theoretical viewpoint. This is a joint work with Masato Hara (Kyoto U).
- 14:00-14:45 Paul Glendinning (Manchester UK): The boundary of positive topological entropy for some maps of the interval with plateaus and the boundary of positive Hausdorff dimension of survivor sets for maps with holes.
- Abstract: Piecewise continuous maps with two monotonic components, a discontinuity and plateaus are a natural simplification for the study of general questions about discontinuous maps. An obvious question here is how families of such maps move from zero entropy to positive entropy. It turns out that this is equivalent to the question about the boundary of positive entropy for survivor sets of some maps with holes, also called open maps (the survivor set is the set of points which never fall into the hole under iteration). I will describe the boundaries in a natural parameter space and in particular characterise the codimension one transitions. It turns out that there are precisely two classes of such transitions. One is abrupt (from a finite number of periodic orbits on the boundary to an infinite number) and the other is via an infinite cascade of bifurcations (NOT period multiplying: they are codimension two!).
Parts of this is joint (but separate) work with Nikita Sidorov and Clement Hege.
- 15:00-15:45 Kuo-Chang Chen (National Tsing Hua Taiwan): Chaotic orbits of the n-center problem via variational methods
- Abstract: The N-center problem with N ≥ 3 is a classic example of chaotic system. By regularizing collisions, one can associate the dynamics with a symbolic dynamical system which yields infinitely many periodic and chaotic orbits, possibly with collisions. It is a nontrivial task to construct chaotic orbits without any collision. In this talk we consider the special case of planar N-center problem with collinear centers, and show how variational methods can be used to construct collision-free heteroclinic orbits. This is a joint work with Guowei Yu.
- 16:00-16:45 Warwick Tucker (Monash Australia): Lower bounds on the Hausdorff dimensions of Julia sets
- Abstract: We present an algorithm for a rigorous computation of lower bounds on the Hausdorff dimensions of Julia sets for a wide class of holomorphic maps. We apply this algorithm to obtain lower bounds on the Hausdorff dimension of the Julia sets of some infinitely renormalizable real quadratic polynomials, including the Feigenbaum polynomial. This is joint work with Artem Dudko and Igors Gorbovickis.
28th (Mon), Building 14, West Wing 2F, seminar room 1, Yagami Campus, Keio University
Chairs: Kenichi Bannai (Keio/RIKEN), Yi-Chiuan Chen (Academia Sinica Taiwan)
- 10:00-10:45 Suddhasattwa Das (Texas Tech. USA): Learning theory for dynamical systems
- Abstract: The task of modelling and forecasting a dynamical system is one of the oldest problems, and it remains challenging. Broadly, this task has two subtasks- extracting the full dynamical information from a partial observation; and then explicitly learning the dynamics from this information. These two subtasks can be combined into a single mathematical framework using the language of spaces, maps, and commutations. The framework also unifies two of the most common learning paradigms- delay coordinates and reservoir computing. This framework provides a platform for two other investigations of the reconstructed system- its dynamical stability; and the growth of error under iterations. We show that these questions are deeply tied to more fundamental properties of the underlying system the behavior of matrix cocycles over the base dynamics, its non-uniform hyperbolic behavior, and properties of its Koopman operator.
Co-authors : Tyrus Berry (George Mason University)
Paper
- 10:55-11:40 Yuka Hashimoto (NTT): Koopman spectral analysis of skew-product dynamics on Hilbert C*-modules
- Abstract: Operator-theoretic methods have been extensively studied for analyzing dynamical systems. Koopman operator is a linear operator defined by composition, and it plays a central role in the operator-theoretic methods. We introduce a linear operator on a Hilbert C*-module for analyzing skew-product dynamical systems. Hilbert C*-module is a generalization of Hilbert space by means of C*-algebra. The proposed operator is defined by composition and multiplication. We show that it admits a decomposition in the Hilbert C*-module, called eigenoperator decomposition, that generalizes the concept of the eigenvalue decomposition. This decomposition reconstructs the Koopman operator of the system in a manner that represents the continuous spectrum through eigenoperators.
- 14:00-14:45 Zin Arai (Tokyo Tech.): On the parameter space of the Hénon map
- Abstract: We study the parameter space of the real and complex Hénon map using complex analytic techniques and computer-assisted proofs. We first discuss how and when the hyperbolic horseshoe of the map breaks down, and show that the hyperbolic horseshoe locus and the maximal entropy locus are connected and simply connected. With rigorous algorithms to verify the connectedness and disconnectedness of the Julia set, we also obtain a certain topological property of the connectedness locus, which clearly contrasts with that of the one-dimensional conventional Mandelbrot set. This is a joint work with Yutaka Ishii (Kyushu Univerisity).
- 14:55-15:40 Konstantin Mischaikow (Rutgers USA): TBA
- Abstract: TBA
- 16:10-16:55 Masayuki Asaoka (Doshisha): Recent progress on surgery of 3-dimensional Anosov flows
- Abstract: A variety of non-classical 3-dimensional Anosov flow is provided by Goodman-Fried surgery
along periodic orbits. It is a long standing open problem whether all 3-dimensional topologically
transitive Anosov flows can be obtained by surgery from classical Anosov flows or not. In this talk,
we survey recent progress on this problem, including a result by the speaker, Bonatti, and Marty
on surgery into R-covered Anosov flows.
- 17:05-17:50 Santiago Ibáñez Mesa (Oviedo Spain):
Two-dimensional invariant manifolds in a four-dimensional reversible Hamiltonian system: the tentacle-like geometry
- Abstract: A one parameter family of four-dimensional reversible Hamiltonian vector fields is considered. This family has a hyperbolic bifocus for all parameter values in a specific interval, and it is well known from the literature that there exists a plethora of homoclinic orbits. We are interested in the study of the entangled geometry exhibited by the two-dimensional stable and unstable invariant manifolds when they meet a given Poincaré section.
29th (Tue), Building 14, West Wing 2F, seminar room 1, Yagami Campus, Keio University
Chair: Jason Atnip (Queensland Australia)
- 10:00-10:45 David Croydon (RIMS): Generalized hydrodynamic limit for the box-ball system
- Abstract: I will talk about a joint work with Makiko Sasada (University of Tokyo) in which we obtain a generalized hydrodynamic limit for the box–ball system of Takahashi and Satsuma. This explains how the densities of solitons (i.e. solitary waves) of different sizes evolve asymptotically under Euler space-time scaling. To describe the limiting soliton flow, we introduce a continuous state space analogue of the soliton decomposition, namely we relate the densities of solitons of given sizes in space to corresponding densities on a scale of ‘effective distances’, where the dynamics are linear. For smooth initial conditions, we further show that the resulting evolution of the soliton densities in space can alternatively be characterised by a partial differential equation, which naturally links the time-derivatives of the soliton densities and the ‘effective speeds’ of solitons locally. Although our arguments and main result are essentially deterministic, they are inspired by work of Ferrari, Nguyen, Rolla and Wang (and Ferrari and Gabrielli) for stationary random configurations, which I also plan to describe.
- 10:55-11:40 Johannes Jaerisch (Nagoya): Growth rates and amenability for the geodesic flow on hyperbolic surfaces
- Abstract: We use multifractal analysis to investigate the long-term behavior of growth rates associated with the geodesic flow on surfaces of constant negative curvature. The growth rates we consider are given by the number of windings around cusps, the number of crossings of sides of a fundamental domain, and the distance travelled on the surface. This talk is based on joint work with Hiroki Takahasi (Keio University) and Manuel Stadlbauer (Universidade Federal do Rio de Janeiro).
- 11:50-12:35 Yuto Nakajima (Keio): Transversal family of non-autonomous conformal iterated function systems
- Abstract: We study Non-autonomous Iterated Function Systems (NIFSs) with overlaps. A NIFS is a sequence of collections of uniformly contracting maps. In comparison to usual iterated function systems, we allow the contractions applied at each step to depend on the step. In this talk, we focus on a family of parameterized NIFSs. Here, we do not assume any separating conditions. We show that if a family of such systems satisfies the transversality condition, then for almost every parameter value the Hausdorff dimension of the limit set is the minimum of the dimension of the ambient space and the Bowen dimension.
30th (Wed), Building 14, West Wing 2F, seminar room 1, Yagami Campus, Keio University
Chairs: Wei Hao Tey (IRCN Tokyo), Takayuki Watanabe (Chubu)
- 10:00-10:45 Alex Blumenthal (Georgia Tech. USA): TBA
- Abstract: TBA
- 10:55-11:40 Jeroen Lamb (ICL UK): TBA
- Abstract: TBA
- 14:00-14:45 Kenichiro Yamamoto (Nagaoka):
Heterochaos baker maps and the Dyck system: maximal entropy
measures and a mechanism for the breakdown of entropy approachability
- Abstract: We consider piecewise affine maps on [0,1]^2 and [0,1]^3,
called the Heterochaos baker maps, which were introduced and
investigated by Saiki, Takahasi and Yorke, as minimal models of the
unstable dimension variability in multidimensional dynamical systems.
We show that natural coding spaces of these maps coincide with the
Dyck system that has come from the theory of languages. Based on this
coincidence, we start to develop a complementary analysis on their
invariant measures. As a first attempt, we show the existence of two
ergodic measures of maximal entropy for the heterochaos baker maps. We
also clarify a mechanism for the breakdown of entropy approachability.
- 14:55-15:40 Shintaro Suzuki (Tokyo Gakugei): An explicit formula for invariant densities of random dynamical systems of
beta-maps
- Abstract: We consider i.i.d. random dynamical systems generated by a class of beta-maps on the unit interval, which includes the case that the cardinality of maps randomly chosen is uncountable. For such a random dynamical system, we see that there is a unique stationary measure and give an explicit formula for its density function under the assumption that it is mean expanding. Furthermore, we establish an explicit formula for invariant densities in the case of non-i.i.d random dynamical systems of beta-maps satisfying a certain strong expanding condition.
- 16:10-16:55 Dmitry Turaev (ICL UK): TBA
- Abstract: TBA
- 17:05-17:50 Yuzuru Sato (Hokkaido): Anomalous diffusion and intermittency in random dynamical systems
- Abstract: This talk includes a brief review of classical noise-induced phenomena
such as noise-induced synchronization, stochastic resonance, noise-induced chaos,
and noise-induced order from a viewpoint of random dynamical systems theory.
In particular, we discuss about anomalous diffusion and intermittency observed
in random dynamical systems and compare them with those in deterministic dynamical systems.
31st (Thu), Building 14, West Wing 2F, seminar room 1, Yagami Campus, Keio University
Chairs: Hisayoshi Toyokawa (Kitami), Juho Leppänen (Tokai)
- 10:00-10:45 Yushi Nakano (Tokai): Existence and non-existence of Lyapunov exponents for (random) non-hyperbolic dynamical systems
- Abstract: In this talk, I consider the problem of whether the set of points at which Lyapunov exponent fails to exist, called the Lyapunov irregular set, has positive Lebesgue measure. First, I show that surface diffeomorphisms with a robust homoclinic tangency, as well as other several known nonhyperbolic dynamics, have the Lyapunov irregular set of positive Lebesgue measure (joint work with S. Kiriki, X. Li, T. Soma). Such a positive Lebesgue measure set can be constructed both as the time averages exist and do not exist on it. Next, I show that for any smooth dynamical systems, under a certain type of noise, including absolutely continuous additive noise, the Lyapunov irregular set has zero Lebesgue measure and the number of such Lyapunov exponents is finite (joint work with F. Nakamura, H. Toyokawa).
- 10:55-11:40 Masato Tsujii (Kyushu): Analysis of transfer operators from the viewpoint of microlocal analysis
- Abstract: Recently there are some developments in the study of spectral studies of transfer operators for hyperbolic and partially hyperbolic dynamical systems from a viewpoint of microlocal analysis. Here “micro-local” implies localization not only with respect to location in the space, but also with respect to cotangent space directions at a given point. (See Wikipedia.) Naturally, we may use the techniques of wave-packet (or wavelet) transform. We will review a few of the relevant developments and discuss how far we can proceed with the idea.
- 14:00-14:45 Maria José Pacifico (UFRJ Brazil): Uniqueness of Equilibrium States for Lorenz Attractors
- Abstract: Ever since its discovery in 1963 by Lorenz, the Lorenz attractor has been playing a central role in the research of singular flows, i.e., flows generated by smooth vector fields with singularities.
In this talk I will present my newest result obtained with Fan Yang and Jiagang Yang, establishing that a Lorenz like attractor in any dimension has a unique equilibrium state.
- 14:55-15:40 Akira Shudo (Tokyo Metropolitan): Ergodicity of complex dynamics and quantum tunneling in the ultra-near integrable system
- Abstract: The instanton approximation is a widely used approach to construct of the semiclassical theory of tunneling.
The instanton path bridges the regions that are not connected by classical dynamics, but the connection can only be achieved if the two regions have the same energy. This is a major obstacle when applying the instanton method to nonintegrable systems. Here we show that the ergodicity of complex orbits in the Julia set ensures the connection between arbitrary regions and thus provides an alternative to the instanton path in the nonintegrable system. This fact is verified using the ultra-near integrable system in which none of the visible structures inherent in nonintegrability exist in the classical phase space, yet non-monotonic tunneling tails emerge in the corresponding wave functions. The simplicity of the complex phase space allows us to explore the origin of the non-trivial tunneling tails in terms of semiclassical analysis in the time domain. In particular, it is shown that not only the imaginary part but also the real part of the classical action plays a role in creating the characteristic step structure of the tunneling tail that appears as a result of the quantum resonance.
- 16:15-17:00
Sandro Vaienti (CPT France): Statistics of recurrence for random dynamical systems
- Abstract:
We give new results for the determination of extreme value theory and compound Poisson distribution for dynamical systems perturbed in a quenched way. We finally extend such results to stochastic differential equations.
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