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Taiwan Mathematics School: Dynamical System: Deterministic and Chaotic Behavior of Model Systems
13:50-15:50 Every Mon., September 9, 2019 - January 6, 2020
R440, Astronomy-Mathematics Building, NTU

Vladimir Zykov (Max Planck Institute)
Masayasu Mimura (Hiroshima University & MIMS, Meiji University)
Chih-Hung Chang (National University of Kaohsiung)
Jia-Yuan Dai (NCTS)

Jung-Chao Ban (National Chengchi University)
Je-Chiang Tsai (National Tsing Hua University)

一、 課程背景與目的:
This course has a twofold purpose: In the first part we provide various models which come from biological mathematics, physics and dynamical systems. Interesting problems arise when we investigate the solutions of those models: 1. How about the structure of solutions? 2. What is the asymptotic behavior of solutions? And 3. How to characterize a given chaotic system? The aim of the second part is to answer the above problems. To do this, we first introduce elementary theories of dynamical systems in one dimensional domain. Next, we will introduce the singular perturbation theory which is a useful tool in investigating the qualitative behavior of solutions of a parametrized model when a small parameter involved in the model. Finally, we introduce the chaotic behavior of the well-known Lorenz system. The useful technique, namely, the symbolic dynamics theory, which is an efficient method to characterize the chaotic behavior of a dynamical system, will also be presented.
二、 課程之大綱與講者:
Lecturer: Dr. Jia-Yuan Dai 
NCTS, Taiwan
Title: Spiral waves in Ginzburg-Landau equations
Abstract: I will introduce a functional setting for the resulting spiral wave equation, and then establish the existence of Ginzburg-Landau spiral waves. Moreover, I will explain the three steps in my proof: global bifurcation analysis, perturbation arguments, and determination on types of pattern.
Date: Sep. 9, Sep. 16, and Oct. 14, 2019 (3 weeks in total)
Week 1: Ignite talk (introduction of spiral waves); Global bifurcation analysis
Week 2: Perturbation arguments; Determination on types of pattern; Design of spiral Ansatz
Week 6: Hyperbolicity 
Lecturer: Prof. Vladimir Zykov 
Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany
Title: Spatio-temporal pattern in excitable media
Abstract: Rotating spiral waves arise naturally in various chemical systems and biological processes. Specific examples of spiral waves can be found in the Belousov_Zhabotinskii reaction, aggregating slime-mould cells, and cardiac muscle tissue. This course provides an introduction to the theory of spiral waves in excitable media.
Date: Sep. 23, 2019 – Oct. 11, 2019 (3 weeks in total)
Week 3: Mathematical modeling and control of wave processes in excitable media (Dissipative structures; Examples of excitable media; Mathematical models of excitable media)
Week 4: Selection mechanism for rigidly rotating waves (Velocity-curvature relationship; Wave patterns rotating within a disk; Generalized dispersion relation)
Week 5: Spatio-temporal parametric control of spiral waves (Resonant drift of a spiral wave; Discrete feedback control; Continuous feedback control) 
Lecturer: Prof. Masayasu Mimura 
Department of Mathematical and Life Sciences, Hiroshima University, Japan
Title: Singular perturbation theory 
Abstract: In the realistic problems, the model contains parameters which mimic the situations that change the nature of the underlying problem. Specifically, the solution of the problem with small but nonzero parameter ε is qualitatively different from that of the limiting problem where εis equal to zero. This short course will introduce the singular perturbation theory to explore such a class of problems.    
Date: Oct. 21, 2019 – Nov. 4, 2019 (3 weeks in total)
Week 7: Asymptotic expansions.
Week 8: Boundary problems.
Week 9: Applications to a class of PDE. 
Lecturer: Prof. Chih-Hung Chang (chchang@go.nuk.edu.tw)
Department of Applied Mathematics, National University of Kaohsiung
Title: Chaotic dynamical systems
Abstract: Along with the unveiling of high-speed computers, numerical approximations and graphical results of differential equations are widely available nowadays. The discovery of complicated dynamical systems such as the horseshoe map and the Lorenz system and their mathematical analysis reveal that simple stable motions such as periodic solutions are not the most important behavior of differential equations. This course is devoted to the chaotic behavior of higher dimensional systems via the Lorenz system of differential equations. We reduce the problem to the dynamics of a discrete dynamical system, discussing along the way how symbolic dynamics may be used to investigate certain chaotic systems. Finally, we return to nonlinear differential equations to apply these techniques to other chaotic systems that arise when homoclinic orbits are present.
Date: Nov. 11, 2019 – Jan. 6, 2020 (9 weeks in total)
Week 10: Lorenz system
Week 11: Model for Lorenz attractor
Week 12: Chaotic attractor
Week 13: Attractors and repellers of Logistic map
Week 14: Period doubling bifurcation
Week 15: Schwarzian derivatives and basin of attraction
Week 16: Leaking systems and Cantor sets
Week 17: Symbolic dynamics
Week 18: Sharkovskii’s theorem
Every Monday 14:10-16:10
First 6 weeks in Lecture Room B, 4th Floor, The 3rd General Building, NTHU
Last 10 weeks in C02 R408, National University of Kaohsiung (Live streaming)

Contact: murphyyu@ncts.ntu.edu.tw

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 (C) 2019 National Center for Theoretical Sciences