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Advanced Nonlinear Systems_E
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| Instructor(s) |
Ryuji TOKUNAGA
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| E-Mail |
tokunaga(at)cs.tsukuba.ac.jp |
| URL |
http://www.chaos.cs.tsukuba.ac.jp/ND/index.html |
| Office hours |
Friday 6 |
| Cource# |
01CH101 |
| Area |
Information Mathematics and Modeling |
| Basic/Advanced |
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| Course style |
Lecture |
| Term |
秋AB |
| Period |
金5,6 |
| Room# |
3B301 |
| Keywords |
invariant set, stability, orbital instability, attractor, self-similarity, bifurcation |
| Prerequisites |
basic mathematical analysis, ordinary differential equation |
| relation degree program competence |
Knowledge Utilization Skills,Research Skills,Expert Knowledge |
| Goal |
|
| 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.
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| Textbook |
lecture slides (.ppt files) |
| References |
- 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)
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| Evaluation |
oral examination |
| TF / TA |
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| Misc. |
open in an even number year |