Taiwan Mathematics School: Dynamics in Network Systems
Every Tuesday, 10:10-12:00
R440, Astronomy-Mathematics Building, NTU
Speaker(s):
Chih-hao Hsieh (Institute of Oceanography, National Taiwan University)
Chao-Ping Hsu (Academia Sinica)
Chih-Hung Chang (National University of Kaohsiung)
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.
二、課程之大綱與講者：
Lecturer: Prof. Chih-Hao Hsieh (謝志豪)
Date: Mar. 17, 24, 2020 (2 weeks in total)
Institute of Oceanography, National Taiwan University
Title: Time series analysis for nonlinear dynamical systems
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.
Lecturer: Prof. Chao-Ping Hsu (許昭萍)
Institute of Chemistry, Academia Sinica
Date: May 12 -- Jun. 16, 2020 (6 weeks in total)
Title: Noise effect on network system
Abstract:
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.
Topic:
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. 7, 14, 21, 28, May 5, Jun. 23 and 30, 2020 (10 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.
Topic:
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)
