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Taiwan Mathematics School: Dynamics in Network Systems
Every Tuesday, 10:10-12:00, February 17 - June 16, 2020
R440, Astronomy-Mathematics Building, NTU

Chih-hao Hsieh (Institute of Oceanography, National Taiwan University)
Atsushi Mochizuki (Kyoto University & RIKEN)
Bogdan Kazmierczak (Polish Academy of Sciences)
Chao-Ping Hsu (Academia Sinica)
Ching-Cher Yan (Academia Sinica)
Chih-Hung Chang (National University of Kaohsiung)

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

一、 課程背景與目的:

Complex networks arise from physical systems, chemical reactions and biological process. It is believed that physical mechanisms/biological functions arise from the dynamics within such network systems. On the other hand, due to the complexity of network systems and the limited information of kinetics and parameters, the dynamics resulting from such complex network systems are not well understood. In this course, we provide some new approaches for the study of network systems, and give the application of these theories for problems from ecosystem, the central carbon metabolism of the E. coli and system biology. In addition, we introduce some useful techniques developed in discrete dynamical systems that can be used to analyze the topological behavior of the derived network system. Fractal geometry will be included if possible.

Lecturer: Prof. Chih-Hao Hsieh (謝志豪) 
NTU, Taiwan
Title: Spiral Waves in Ginzburg-Landau Equations
Abstract: Natural systems are often complex and dynamic (i.e. nonlinear), and are difficult to understand using linear statistical approaches. Linear approaches are fundamentally based on correlation and are ill posed for dynamical systems, because in dynamical systems, not only can correlation occur without causation, but causation can also occur in the absence of correlation. To study dynamical systems, nonlinear time series analytical methods have been developed in the past decades [1-5]. These nonlinear statistical methods are rooted in State Space Reconstruction (SSR), i.e. lagged coordinate embedding of time series data [6]  (http://simplex.ucsd.edu/EDM_101_RMM.mov). These methods do not assume any set of equations governing the system but recover the dynamics from time series data, thus called Empirical Dynamic Modeling (EDM).
EMD bears a variety of utilities to investigating dynamical systems: 1) determining the complexity (dimensionality) of system [1], 2) distinguishing nonlinear dynamic systems from linear stochastic systems [1], 3) quantify the nonlinearity (i.e. state dependence) [7], 4) determining causal variables [3], 5) tracking strength and sign of interaction [8, 9], 5) forecasting [5], 6) scenario exploration of external perturbation [4], and 7) classifying system dynamics [2, 10]. These methods and applications can be used for mechanistic understanding of dynamical systems and providing effective policy and management recommendations on ecosystem, climate, epidemiology, financial regulation, and much else.
Date: Feb. 17, Feb. 24, 2020 (2 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
Lecturer: Prof. Bogan Kazmierczak 
Institute of Fundamental Technological Research, Polish Academy of Sciences, Poland
Title: A Apatially Extended Model of Kinase-Receptor Interaction
Abstract: TBA
Date: TBA
Lecturer: Prof. Chao-Ping Hsu (許昭萍) and Dr. Ching-Cher Sanders Yan (顏清哲) 
Institute of Chemistry, Academia Sinica
Title: Noise Effect on Network System 
Abstract:  TBA
Date: May 12 – Jun. 16, 2020 (6 weeks in total) 
Week 13: TBA
Week 14: TBA
Week 15: TBA
Week 16: TBA
Week 17: TBA
Week 18: TBA 
Lecturer: Prof. Chih-Hung Chang (張志鴻) 
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: Mar. 3, 10, 17, 24, 31, Apr. 21, 28, and May 5, 2020 (8 weeks in total)
Week 3: Period 3 and Sharkovskii’s theorem
Week 4: Period 3 window and subshift of finite type
Week 5: Period 3 window and subshift of finite type
Week 6: Critical points and basins of attraction
Week 7: Critical points and basins of attraction
Week 10: Introduction of kneading theory
Week 11: Introduction of kneading theory
Week 12: Fractals and iterated function systems
三、 課程詳細時間地點以及方式:

Every Tuesday 10:10-12:00, except for Week 1& 2(Monday)

  1. Lecture Room R440, Astronomy-Mathematics Building, NTU
  2. Lecture Room B, 4th Floor, The 3rd General Building, NTHU (Live streaming)
  3. C02 R408, National University of Kaohsiung (Live streaming)
四、 學分數:
Credit: 2 

Contact: murphyyu@ncts.ntu.edu.tw

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