Introduction to
Computational Ergodic Theory
Embedded Files
Hybrid (zoom & in person) lecture series, Hokkaido Summer Institute at Research Center of Mathematics for Social Creativity, Research Institute for Electronic Sciences, Hokkaido University
Hybrid (zoom & in person) lecture series, Hokkaido Summer Institute at Research Center of Mathematics for Social Creativity, Research Institute for Electronic Sciences, Hokkaido University
Ergodic theory is a mathematical field that connects dynamical system theory and probability theory, and underlies statistical mechanics and nonlinear physics. However, mathematically, it is difficult to analyze even basic problems such as the uniqueness and existence of invariant densities. One of the purposes of this lectures is to learn how a functional analytic approach can be used for these kind of problems in dynamical systems, giving an efficient method to get information on the invariant measures and on the statistical behavior of the system. The second purpose is to introduce the recently developed validated numerics for computer-assisted proofs in dynamical systems and ergodic theory and to understand the results of proof of existence of noise-induced phenomena in nonlinear physics.
Lecturers:
Stefano Galatolo (University of Pisa, Italy)
Isaia Nisoli (UFRJ, Rio de Janeiro, Brazil)
Yuzuru Sato (Hokkaido University, Japan)
Zin Arai (guest lecturer, Tokyo Institute of Technology, Japan)
Warwick Tucker (guest lecturer, Monash University, Australia)
Date:
14-18 August, 2023
Place:
3-413, Department of Mathematics, Hokkaido University
Registration:
Please register here to join the zoom meeting and to watch the lecture videos.
Installing Julia and VSCode:
The latest version of Julia can be found at Julia Download page
Please download and install the version adapted to your hardware.
Download and install Visual Studio Code from VSCode Download page
Follow the instructions at VSCode Julia Extension installation guide
Now you can download the course notebooks, open them in VSCode, choose the Julia kernel on the right side and run the course notebooks.
-------------------------------------------------------------------------------------------------------------------------------------
Program:
-------------------------------------------------------------------------------------------------------------------------------------
Monday, 14 August 17:00-20:00 (JST) / 10:00-13:00 (CET)
17:00-17:20 Introduction: "Introduction to computational ergodic theory" by Yuzuru Sato
17:20-18:20 Lecture 1: "Introduction to Julia" by Isaia Nisoli and Yuzuru Sato: Jupyter Notebook: Lecture 1
18:30-19:30 Lecture 2 "Numerical experiments in dynamical systems" by Isaia Nisoli and Yuzuru Sato: Jupyter Notebook: Lecture 2
19:20-21:00 Open discussions at spatial chat discussion room by YS, IN.
Tuesday, 15 August 17:00-20:00 (JST) / 10:00-13:00 (CET)
17:00-18:00 Seminar 1: Computer-assisted proofs in dynamical systems by Warwick Tucker
18:20-19:20 Seminar 2: An introduction to set-oriented computations for dynamics and their applications by Zin Arai
19:20-21:00 Open discussions at spatial chat discussion room by YS, SG, IN, WT, ZA.
Wednesday, 16 August 17:00-20:00 (JST) / 10:00-13:00 (CET)
17:00-18:30 Lecture 3: "Chaos, the statistical properties of dynamics: The transfer operator approach" by Stefano Galatolo: Lecture3
18:40-20:00 Lecture 4: "An introduction to interval arithmetics" by IN: Jupyter Notebook: Lecture 4
20:00-21:00 Open discussions at spatial chat discussion room by YS, SG, IN.
Thursday, 17 August 17:00-20:00 (JST) / 10:00-13:00 (CET)
17:00-18:30 Lecture 5: " Quantitative stability results, rigorously validating the numerical approximation" by SG : Lecture5
18:40-20:00 Lecture 6: "Rigorous computations, computing the invariant measure and Lyapunov exponent with a controlled error in deterministic systems" by IN: Jupyter Notebook: Lecture 6
20:00-21:00 Open discussions at spatial chat discussion room by YS, SG, IN.
Download the notebook, complete the julia program, and run it to compute Lyapunov exponents. Send the results to ysato_at_math.sci.hokudai.ac.jp until Friday. If you have questions, please join us for discussion at spatial chat at 5pm (JST) on Friday, 18 August.
Friday, 18 August 17:00-20:00 (JST) / 10:00-13:00 (CET)
17:00-18:30 Lecture 7: "Random dynamical systems, the transfer operator approach and the quantitative statistical stability". by SG: Lecture7
18:40-20:00 Lecture 8: "The approximation of the stationary measure and Lyapunov exponents with a controlled error in the random dynamical system case". by IN: Jupyter Notebook: Lecture 8
20:00-21:00 Open discussions at spatial chat discussion room by YS, SG, IN.
Lecture notes and other resources:
Lecture notes and other resources:
S. Galatolo "Statistical properties of dynamics. Introduction to the functional analytic approach"
Related papers:
Related papers:
S. Galatolo, M. Monge, I. Nisoli Existence of Noise Induced Order, a Computer Aided Proof Nonlinearity, 33(9):4237--4276, (2020).
S. Galatolo, I. Nisoli, B. Saussol. An elementary way to rigorously estimate convergence to equilibrium and escape rates. J. Comput. Dyn., 2015, 2 (1) : 51-64. doi: 10.3934/jcd.2015.2.51
S. Galatolo, I. Nisoli, An elementary approach to rigorous approximation of invariant measures. SIAM J. Appl. Dyn. Syst. 13 (2014), no. 2, 958--985.
S. Galatolo, , I. Nisoli, Rigorous computation of invariant measures and fractal dimension for maps with contracting fibers: 2D Lorenz-like maps. Ergodic Theory Dynam. Systems 36 (2016), no. 6, 1865--1891.
Organizer:
Organizer:
Yuzuru Sato (Hokkaido University) Email: ysato_at_math.sci.hokudai.ac.jp
Y.S is supported by JSPS Grant-in-Aid for Scientific Research (B) No. 17H02861, and (B) No. 21H01002, London Mathematical Laboratory external fellowshp, Hokkaido Summer Institute Program, and Research Center of Mathematics for Social Creativity at Research Institute for Electronic Sciences, Hokkaido University.
Google Sites
Report abuse