Every Tuesday, 10:10-12:00, March 3 - June 30, 2020

Room 440, Astronomy-Mathematics Building, NTU

Outline

Lecturer: Prof. Chih-Hao Hsieh (謝志豪)

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

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.

REFERENCES

[1]. Hsieh, C.H., S.M. Glaser, A.J. Lucas, and G. Sugihara (2005) Distinguishing random environmental fluctuations from ecological catastrophes for the North Pacific Ocean. Nature. 435: 336-340.

[2]. Hsieh, C.H., C. Anderson, and G. Sugihara (2008) Extending nonlinear analysis to short ecological time series. American Naturalist. 171: 71-80.

[3]. Sugihara, G., R. May, H. Ye, C. H. Hsieh, E. Deyle, M. Fogarty, and S. Munch (2012) Detecting causality in complex ecosystems. Science 338: 496-500.

[4]. Deyle, E., M. F. Fogarty, C. H. Hsieh, L. Kaufman, A. MacCall, S. B. Munch, C. Perretti, H. Ye, and G. Sugihara (2013) Predicting climate effects on Pacific sardine. PNAS. 110: 6430-6435.

[5]. Ye, H., R. J. Beamish, S. M. Glaser, S. C. H. Grant, C. H. Hsieh, L. J. Richards, J. T. Schnute, and G. Sugihara (2015) Equation-free mechanistic ecosystem forecasting using empirical dynamic modeling. PNAS. 112: E1569–E1576.

[6]. Takens, F. 1981. Detecting strange attractors in turbulence. Pages 366-381 in D. A. Rand and L. S. Young, editors. Dynamic systems and turbulence. Springer-Verlag, New York.

[7]. Anderson, C. N. K., C. H. Hsieh, S. A. Sandin, R. Hewitt, A. Hollowed, J. Beddington, R. M. May, and G. Sugihara. 2008. Why fishing magnifies fluctuations in fish abundance. Nature 452:835-839.

[8]. Deyle, E. R., R. M. May, S. B. Munch, and G. Sugihara. 2016. Tracking and forecasting ecosystem interactions in real time. Proceedings of the Royal Society of London B: Biological Sciences 283.

[9]. Ushio, M., C. H. Hsieh, R. Masuda, E. Deyle, H. Ye, C. W. Chang, G. Sugihara, and M. Kondoh. 2018. Fluctuating interaction network and dynamic stability of natural fish community. Nature 554: 360-363.

[10]. Liu, H., M. J. Fogarty, S. M. Glaser, I. Altman, C. Hsieh, L. Kaufman, A. A. Rosenberg, and G. Sugihara. 2012. Nonlinear dynamic features and co-predictability of the Georges Bank fish community. Marine Ecology Progress Series.

Lecturer: Prof. Chao-Ping Hsu (許昭萍) & Dr. Ching-Cher Sanders Yan (顏清哲)

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:

1. General introduction: Dynamics in a cell and the importance of noises

2. The chemical master equation

3. The Langevin equation

4. Treatment for gene expression: effect of Bursts

5. Noise-filtering mechanism: Nonlinear versus Linear regulation

6. Noise-filtering mechanism: feed-forward motifs

Lecturer: Prof. Chih-Hung Chang (張志鴻)

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:

1. Period 3 and Sharkovskii’s theorem

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

3. Critical points and basins of attraction (2 weeks)

4. Introduction of kneading theory (2 weeks)

5. Fractals and iterated function systems