Dynamical SystemsBack JP / EN

Numbering Code Year/Term U-ENG29 39080 LJ55U-ENG29 39080 LJ10 2021 ・ First semester 2 Lecture Japanese Thu.3 YAGASAKI KAZUYUKI (Graduate School of Informatics Professor) 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. (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 Introduction to Dynamical Systems,5-6times,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,4-5times,Bifurcations of equilibria and fixed points, center manifold reductions and normal forms are discussed. Chaos,4-5times,Horseshoe maps, homoclinic theorem and Melnikov's method are discussed. Evaluation depends mainly on marks of examination, but marks of exercises and homework are taken into account when needed. Calculus, Linear Algebra and Differential Equations Handouts K.T. Alligood, T. Sauer and J.A. Yorke, Chaos: An Introduction to Dynamical Systems，Springer isbn{}{9780387946771} M.W. Hirsch，S. Smale and R.L. Devaney,Differential Equations, Dynamical Systems, and an Introduction to Chaos isbn{}{9780123820105} J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer isbn{}{0387908196} J.D. Meiss, Differential Dynamical Systems, SIAM isbn{}{9780898716351} S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer isbn{}{0387001778}