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Taiwan Mathematics School: Dynamics in Network Systems
Every Tuesday, 10:10-12:00, March 3 - June 30, 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)

Date: Mar. 17, 2020 and Mar. 24, 2020 (2 weeks in total)

Institute of Oceanography, National Taiwan University

Title: Time series analysis for nonlinear dynamical systems


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.

Lecturer: Prof. Atsushi Mochizuki  

Institute for Frontier Life and Medical Sciences, Kyoto University, Japan

Date: Apr. 7, 2020 and Apr. 14, 2020 (2 weeks in total)

Title: Understanding dynamics of complex biological systems from network structure


Many biological systems have been identified as complex network systems consisting of a large number of biomolecules and interactions between them. Dynamics of molecular activities based on such networks are considered to be the origin of biological functions. However, it has been a very difficult problem to understand dynamics of complex systems from information of networks. In this lecture, I will introduce some novel mathematical theories, by which important aspects of dynamical properties of systems are determined from information of the network structures, alone.

The first theory named Linkage Logic mathematically assures that i) any long-term dynamical behavior of a whole system can be identified/controlled by a subset of molecules in the system, and that ii) the subset is determined from the network topology alone as a feedback vertex set (FVS) of the network. We apply this theory to the gene regulatory network for cell differentiation of ascidian embryo. We show that the dynamical system was successfully controlled by a small set of genes identified as a FVS from the network topology. 
In the second theory named Structural Sensitivity Analysis we show that i) sensitivity responses of a steady state of a reaction system to perturbations of reaction rates are determined from the reaction network, alone. We found that ii) nonzero responses are localized in a finite region in a network, and that the extent of nonzero responses are governed by an index which is an analogue of Euler characteristics. We apply our method to the metabolic network of the E. coli including more than 45 reactions. We demonstrate how these theories are practically useful to understand behaviors of complex biological systems. 

Further Reading

[1] Mochizuki, A., Fiedler, B. et al. (2013) J. Theor. Biol., 335, 130-146.

[2] Kobayashi., et al. (2018) iScience 4, 281–293

[3] Okada T. and Mochizuki A. (2016) Phys. Rev. Lett. 117, 048101.

[4] Okada T., Tsai JC, Mochizuki A. (2018) Phys. Rev. E 98, 012417.

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

Date: May 12 -- Jun. 16, 2020 (6 weeks in total)

Title: Noise effect on network system


Recently there is a growing interest in the effect of noise in cell biology. The ubiquity of noise motivate the fundamental questions such as (1) how does noise at the cellular level translates into the robust behavior at the

macroscopic level, and (2) can noise be exploited to enhance the performance of cellular function? These questions suggest that mathematical biologists and applied mathematicians need to have some background in stochastic process and its application in biological systems. In this course, we will introduce the theories in stochastic process and give its applications in some realistic biological systems. We will start from the tutorial level and then gives an overview of current state-of-the-art approaches for the application of stochastic process in cell biology. The goals of this course are to help the students with the following capabilities,

(1) basic mathematical and computational tools for describing stochastic dynamics, mainly for describing processes in a cell.

(2) building models for biological processes in a cell.


  • General introduction: Dynamics in a cell and the importance of noises
  • The chemical master equation
  • The Langevin equation
  • Treatment for gene expression: effect of Bursts
  • Noise-filtering mechanism: Nonlinear versus Linear regulation
  • Noise-filtering mechanism: feed-forward motifs

Lecturer: Prof. Chih-Hung Chang (chchang@nuk.edu.tw)

Department of Applied Mathematics, National University of Kaohsiung

Date: Mar. 3, 10, 31, Apr. 21, 28, May 5, Jun. 23 and 30, 2020 (8 weeks in total)

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.


Ø   Period 3 and Sharkovskii’s theorem

Ø   Period 3 window and subshift of finite type (2 weeks)

Ø   Critical points and basins of attraction (2 weeks)

Ø   Introduction of kneading theory (2 weeks)

Ø   Fractals and iterated function systems

三、 課程詳細時間地點以及方式:

Every Tuesday 10:10-12:00

  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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