Computational Methods
in Dynamical Systems 2026
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Dynamical systems theory and ergodic theory are mathematics as a basis of statistical and nonlinear physics. However, even simple problems such as invariant density and global statistics, bifurcation and stability are difficult to be rigorously analyzed. The main purpose of this lecture series is to learn how computational science approaches are used to solve problems in chaotic dynamical systems. The second purpose is to introduce the recently developed techniques of arbitrary precision calculation, computer aided proof, and computational topology into dynamical systems theory and ergodic theory. Both are closely related to dynamical systems theory and it is expected that motivated students will use this lecture as an introduction of deterministic and random dynamical systems.
Lecturers:
Stefano Galatolo (University of Pisa, Italy, remote), Isaia Nisoli (UFRJ, Rio de Janeiro, Brazil), Ippei Obayashi (Okayama University), Yoshitaka Saiki (Hitotsubashi University, Japan), Yuzuru Sato (Hokkaido University, Japan), Warwick Tucker (Monash University, Australia, remote)
Date:
31 August - 4 September, 2026
Place:
3-202, Department of Mathematics, Hokkaido University
Registration:
Please register here to join the zoom meeting (registration deadline 21 August, 2026).
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.
For the lecture Lecture 5-1 and 5-2, you need a google account for google colaboratory.
Programme:
Monday, 31 August 16:00-19:00 (JST) / 9:00-12:00 (CET)
16:00-16:50 Lecture 1-1: Stefano Galatolo, Random dynamical systems with state dependent noise.
17:00-17:50 Lecture 1-2: Isaia Nisoli, Introduction to validated computation in ergodic theory with Julia I
18:00-18:50 Lecture 1-3: Warwick Tucker, Computer-assisted proofs in Dynamical Systems
19:00- Open discussions at spatial chat discussion room
Tuesday, 1 September 16:00-19:00 (JST) / 9:00-12:00 (CET)
16:00-16:50 Lecture 2-1: Stefano Galatolo, Recalls about ergodic theory. The transfer operator and its basic properties.
17:00-17:50 Lecture 2-2: Isaia Nisoli, Introduction to validated computation in ergodic theory with Julia II
18:00-18:50 Lecture 2-3: Yuzuru Sato, Julia codes for random dynamical systems
19:00- Open discussions at spatial chat discussion room
Wednesday, 2 September 16:00-18:00 (JST) / 9:00-11:00 (CET)
16:00-16:50 Lecture 3-1: Stefano Galatolo, Finite dimensional reductions, the Ulam method. Quantitative stability and approximation of the invariant measure.
17:00-17:50 Lecture 3-2: Isaia Nisoli, Introduction to validated computation in ergodic theory with Julia III
18:00- Open discussions at churrasco party on campus
Thursday, 3 September 16:00-19:00 (JST) / 9:00-12:00 (CET)
16:00-16:50 Lecture 4-1: Stefano Galatolo, Convergence to equilibrium, approximation of the resolvent and applications.
17:00-17:50 Lecture 4-2: Isaia Nisoli, Introduction to validated computation in ergodic theory with Julia IV
18:00-18:50 Lecture 4-3: Yoshitaka Saiki, TBA
18:00- Open discussions at spatial chat discussion room
Homework: [Download the notebook], complete the julia program, and run it to solve a given problem. Send the results to ysato_at_math.sci.hokudai.ac.jp until Friday, 4 September. If you have questions, please join us for discussion at spatial chat at 7pm (JST) on Thursday, 3 September.
Friday, 4 September 16:00-19:00 (JST) / 9:00-12:00 (CET)
16:00-16:50 Lecture 5-1: Ippei Obayashi, Understanding Data Geometry Through Persistent Homology Part I: Theory and Applications
17:00-17:50 Lecture 5-2: Ippei Obayashi, Understanding Data Geometry Through Persistent Homology Part II: Hands-on Practice with Software Tools (google account )
18:00- Open discussions at spatial chat discussion room
Lecture notes and other resources:
Lecture notes and other resources:
Stefano 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
Isaia Nisoli (UFRJ) Email: nisoli_at_im.ufrj.br
Stefano Galatoro (University of Pisa) Email: stefano.galatolo_at_unipi.it
Y.S. is supported by Hokkaido Summer Institute Program, and Research Center of Mathematics for Social Creativity at Research Institute for Electronic Sciences, Hokkaido University.
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