| 週次 |
授課內容 |
| 第1週 |
Introduction: chaos, fractals, dynamics, and importance of being nonlinear.
|
| 第2週 |
One-dimensional flows (systems): geometric ways of thinking, fixed points and stability, structural change, and system potentials. |
| 第3週 |
Bifurcations: saddle-node bifurcation, transcritical bifurcation, supercritical and subcritical pitch-fork bifurcations, imperfect bifurcations and catastrophes.
|
| 第4週 |
Bifurcations: saddle-node bifurcation, transcritical bifurcation, supercritical and subcritical pitch-fork bifurcations, imperfect bifurcations and catastrophes. Phase-locking phenomenon in a nonuniform oscillator.
|
| 第5週 |
Linear 2-D systems: phase portraits, eigenvalues and eigenvectors, classification of stabilityies. |
| 第6週 |
Nonlinear 2-D systems: phase portraits, numerical integration and spectral analysis-FFT, fixed points and linearization, conservative systems, reversible systems.
|
| 第7週 |
Nonlinear 2-D systems: phase portraits, numerical integration and spectral analysis-FFT, fixed points and linearization, conservative systems, reversible system. |
| 第8週 |
Midterm exam: in class. |
| 第9週 |
Limit cycles: Liapunov functions, Poincaré-Bendixson theorem, Liénard systems, relaxation oscillators, weakly nonlinear oscillators.
|
| 第10週 |
Bifurcations revisited in 2-D systems: saddle-node, transcritical, and pitch-fork bifurcations, Hopf bifurcations, oscillating chemical-reactions, hysteresis, quasiperiodic oscillations, Poincaré maps.
|
| 第11週 |
Bifurcations revisited in 2-D systems: saddle-node, transcritical, and pitch-fork bifurcations, Hopf bifurcations, oscillating chemical-reactions, hysteresis, quasiperiodic oscillations, Poincaré maps. |
| 第12週 |
Duffing’s equation: forced double-well oscillator, a route from periodic oscillation to chaos.
|
| 第13週 |
Chaos on 1-D maps and strange attractor: logistic map, period-doubling, Liapunov exponent, orbit diagram and periodic widow, Lorenz map, signal-masking. |
| 第14週 |
Lorenz Equations and other chaotic systems: properties of strange attractor and route to chaos. |
| 第15週 |
Proposal and discussion for the final project that requires numerical simulations and analyses of a dynamical system selected from published papers or chapters of texts discussing chaotic/fractal behaviors.
|
| 第16週 |
Introduction to fractals: Cantor set, dimension of self-similar fractals, box dimensions, pointwise and correlation dimensions.
|
| 第17週 |
Oral presentation and written report of the final project.
|
| 第18週 |
Self-directed learning and discussion on final project for systems having chaotic/fractal behaviors.
|
|
教科書: Strogatz, S. H., Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry, and Engineering, 2nd edition, CRC Press, Boca Raton, FL, 2018.
參考書: Moon, F. C., Chaotic Vibrations: An Introduction for Applied Scientists and Engineers, Wiley, New York, 2004. |