Advanced Nonlinear Systems_E

Instructor(s) 
Ryuji TOKUNAGA

EMail 
tokunaga(at)cs.tsukuba.ac.jp 
URL 
http://www.chaos.cs.tsukuba.ac.jp/ND/index.html 
Office hours 
Friday 6 
Cource# 
01CH101,01CJ214 
Area 
Information Mathematics and Modeling 
Basic/Advanced 

Course style 
Lecture 
Term 

Period 

Room# 

Keywords 
invariant set, stability, orbital instability, attractor, selfsimilarity, bifurcation 
Prerequisites 
basic mathematical analysis, ordinary differential equation 
Goal 

Outline 
Lectures on nonlinear phenomena generated by lowdimensional diffrential and difference dynamical systems, Chaos, Fractals, Bifurcations. 
Course plan 
 1. chaos generated by 1dimensional mapping
 reductionism and deterministic dynamical system, countable set and uncountable set, periodic orbit and nonperiodic orbit, dense orbit.
 2. invariant sets generated by 1dimensional mapping
 rigid rotation and quasiperiodic orbit, compact invariant set,Cantor set and similarity dimension.
 3. chaotic invariant sets
 fixed point theorem and asymptotically stability, tangent mapping and stability of fixed point, stable and unstable manifolds, definition of chaotic invariant sets.
 4. attractors
 attractor and trapping region, Lyapunov spectrum, horseshoe map and hyperbolic invariant set, Henon map
Lozi map and generalized hyperbolicity.
 5. nonlinear phenomena generated by ordinary differential equation
 electrical and kinetic dynamical systems, attractors generated by vector space, Poincare map and difference dynamical systems.
 6. local bifurcations
 diversity of fixed points, saddlenode bifurcation, period doubling.
 7. global bifurcation
 global bifurcations generated by unimodal mapping, global bifurcations generated by circle map, homoclinic bifurcations.

Textbook 
lecture slides (.ppt files) 
References 
 J.Guckenheimer and P.J.Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, SpringerVerlag, New York (1983)
 R.L. Devaney, An introduction to chaotic dynamical systems, AddisonWesley, New York (1989)
 M.F. Barnsley, Fractals everywhere, Academic Press Professional, (1988)
 T.Matsumoto, M.Komuro, H.Kokubu and R.Tokunaga, Bifurcations: sights, sounds, and mathematics, SpringerVerlag, New York (1993)

Evaluation 
oral examination 
TF / TA 

Misc. 
open in an even number year 