University of Tsukuba | Grad. Scho. Syst. and Info. Eng. | Dept. Comp. Sci. | List of Courses
Advanced Nonlinear Systems_E
E-Mail tokunaga(at)
Office hours Friday 6
Cource# 01CH101,01CJ214
Area Information Mathematics and Modeling
Course style Lecture
Keywords invariant set, stability, orbital instability, attractor, self-similarity, bifurcation
Prerequisites basic mathematical analysis, ordinary differential equation
Outline Lectures on nonlinear phenomena generated by low-dimensional diffrential and difference dynamical systems, Chaos, Fractals, Bifurcations.
Course plan
1. chaos generated by 1-dimensional mapping
reductionism and deterministic dynamical system, countable set and uncountable set, periodic orbit and non-periodic orbit, dense orbit.
2. invariant sets generated by 1-dimensional mapping
rigid rotation and quasi-periodic 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, saddle-node bifurcation, period doubling.
7. global bifurcation
global bifurcations generated by unimodal mapping, global bifurcations generated by circle map, homo-clinic bifurcations.
Textbook lecture slides (.ppt files)
J.Guckenheimer and P.J.Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Springer-Verlag, New York (1983)
R.L. Devaney, An introduction to chaotic dynamical systems, Addison-Wesley, 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, Springer-Verlag, New York (1993)
Evaluation oral examination
Misc. open in an even number year