Numbering Code	U-ENG29 39080 LJ55 U-ENG29 39080 LJ10	Year/Term	2022 ・ First semester
Number of Credits	2	Course Type	Lecture
Target Year		Target Student
Language	Japanese	Day/Period	Tue.3
Instructor name	YAGASAKI KAZUYUKI (Graduate School of Informatics Professor)
Outline and Purpose of the Course	Dynamical systems represent general mathematical models such as differential equations for time-dependent phenomena and a mathematical field having originated in the work of the greatest mathematician in 19th century, Poincare. Dynamical systems theory provides tools to treat nonlinear phenomena such as bifurcations and chaos, and its application range is very wide since there are numerous time-dependent phenomena in natural and social sciences. This course provides fundamentals of dynamical systems theory with a special focus on differential equations.
Course Goals	(1) To understand dynamics of differential equations and maps near neighborhoods of equilibria and fixed points (2) To understand mechanisms for nonlinear phenomena such as bifurcations and chaos (3) To master fundamental techniques for dynamical systems
Schedule and Contents	Examples of Dynamical Systems, 1 time: Classical dynamical systems given by differential equations and maps are considered and nonlinear phenomena occurred in numerical simulations of these systems are reviewed. Introduction to Dynamical Systems, 5-6 times: Fundamentals of differential equations are reviewed and elementary concepts such as Poincare maps, stability, dynamics of linear systems and invariant manifolds are explained. Local Bifurcations, 4-5 times: Bifurcations of equilibria and fixed points, center manifold reductions and normal forms are discussed. Chaos, 3-4 times: Horseshoe maps, homoclinic theorem and Melnikov's method are discussed. Summary and learning achievement evaluation, 1 time: A summary and supplements of this course are given and the learning achievement of students is evaluated.
Evaluation Methods and Policy	Evaluation depends mainly on marks of mid-term examinations (20%) and final one (80%).
Course Requirements	Calculus, Linear Algebra and Differential Equations
Study outside of Class (preparation and review)	Prepare and review the lectures and solving the problems given on KULASIS or PANDA to understand the contents of the textbook and lectures.
Textbooks	Textbooks/References	Handouts
References, etc.	Chaos: An Introduction to Dynamical Systems, K.T. Alligood, T. Sauer and J.A. Yorke, (Springer), ISBN:9783642592812 Differential Equations, Dynamical Systems, and an Introduction to Chaos , M.W. Hirsch，S. Smale and R.L. Devaney, (Elsevier, 2013), ISBN:9780123820105 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, J. Guckenheimer and P. Holmes, (Springer, 1983), ISBN:0387908196 Differential Dynamical Systems, J.D. Meiss, (SIAM, 2017), ISBN:9780898716351 Introduction to Applied Nonlinear Dynamical Systems and Chaos, S. Wiggins, (Springer, 2003), ISBN:0387001778
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