Email H. Bruin for further information for this course.

Easter Break is from 15 to 28 April

Pentacost is on June 8 & 9 (Vorlesungsfrei).

Exam (first opportunity) is planned on Wednesday June 26, 12:00-14:00, in HS02.

Exam (second opportunity) is planned on Friday July 19, 10:00-12:00, in HS03.

Exam (third opportunity) is planned on Thursday September 26, 10:00-12:00, in

Exam (fourth opportunity) is planned on Friday Jnauary 10, 10:00-12:00, in

Day | Time | Room | from | until | |
---|---|---|---|---|---|

Tuesday | 13:15--14:45 | HS02 | Lecture | 05.3.2019 | 26.06.2019 |

Wednesday | 13:15-14:45 | HS02 | Lecture/Proseminar | 7.3.2019 | 27.06.2019 |

Basic notions for continuous and discrete dynamical systems; flows, attractors, and chaos; stability of stationary points by linearization and by Lyapunov functions, Poincaré-Bendixson theory, bifurcations.

The course will be given in English

** Contents of the course **
(B = Exercises ,
S = Schmeiser's notes, T =
Teschl's book),
St = Strogatz lecture on youtube.

- Bifurcations (S§5. T§6.5. T§11.1.)
- Chaos (T§10.3)
- Cusp bifurcation (S§5.4, St)
- Existence, uniqueness, continuity of solutions of ODEs (S§1. T§2.2. T§2.4.)
- Euler-Lagrange equations (T§8.3. S§10 and B)
- Feigenbaum (bifurcation diagram) (S§6. T§11.1, St)
- Hamiltonian systems (S§10. T§8.3. St and B)
- Harmonic and other oscillators (S§8.3.)
- Homoclinic orbits and tangles (S Def 10. T§13.2)
- Hopf bifurcation (S§8.4. T p219-220)
- Inhomogeneous linear ODEs (S§2.1. T§3.2.)
- (In)stability of equilibria, sink, sources, saddles, centers (T§3.2.)
- (In)stability of fixed points (T§6.5, T§10.2.)
- Iteration of maps, cobweb diagrams (T§10.2.)
- Kepler problem = two-body problem (T§8.5. S§10 and B )
- Lagrangian = Lagrange function (T§8.3. S§10 and B )
- Legendre transform (T§8.3. S§10 and B )
- Limit cycles (S§8), see also Van der Pol equation and Poincaré-Bendixson.
- Linear ODEs (S§2. T§3.2. T§3.3 St)
- Linearization, the Hartman-Grobman theorem (S Thm 8. T§9.3.)
- Logistic differential equation (T§10.1.)
- Logistic (= quadratic) map (T§10.1. St)
- Lorenz system (S§9. T§8.2., St and St)
- Lotka-Volterra system (preditor-pray) T§7.1.
- Lyapunov functions (S§7. T§6.6. St))
- Orbits, omega-limit sets (T§6.3 T§8.1. S Def 4.)
- Pendulum (T§6.7. S§10 (Example 8) and B .)
- Phase portraits (T§3.2.)
- Poincaré maps (T§6.4. T§12.2)
- Poincaré-Bendixson Theorem (S§8.5. T§7.3, St)
- Sensitive dependence on initial conditions (T§10.3)
- Sharkovskiy's Theorem (T§11.2)
- Smale's horseshoe (T§13.1.)
- Stable/unstable manifolds (S Thm 7. T§9.2. T§12.3.)
- Structural stability
- Symbolic dynamics (T§11.5.)
- Strange attractors (T§11.6.)
- Van der Pol equations (S§8, T§7.2, St)

Sharkovkiy's Theorem

Day | Material | |
---|---|---|

Week 1 | ||

5/3/2019 | Introduction, 1-D ODEs, Stationary points and their stability Logistic ODEs vs. Logistic map | |

6/3/2019 | Two-dimensional linear ODEs | |

Week 2 | ||

12/3/2019 | No class (Rektorstag) Logistic ODEs vs. Logisitc map | |

13/3/2019 | Proseminar: Exercises 1-5 | |

Week 3 | ||

19/3/2019 | structural stability | |

20/3/2019 | Hartman-Grobman | |

Week 4 | ||

26/3/2019 | bifurctions (transcritical, saddlenode/fold,pitchfork) | |

27/3/2019 | Proseminar: Exercises 6-11 | |

Week 5 | ||

2/4/2019 | bifurcations (period doubling, cusp + hysteresis) | |

3/4/2019 | mathematical chaos, Lyapunov exponents, sensitive dependence | |

Week 6 | ||

9/4/2019 | | |

10/4/2019 | Proseminar: Exercises 6b,c, 10,11,12,14 | |

Week 7 | ||

30/4/2019 | Limit cykels, Van der Pol equation | |

1/5/2019 | No class (Labour Day) | |

Week 8 | ||

7/5/2019 | Poincaré-Bendixson Hopf bifurcation | |

8/5/2019 | Test 1 (material up to Exercise 12) Proseminar: Exercises 14,15,16,17 | |

Week 9 | ||

14/5/2019 | Lotka-Volterra system Lyapunov functions | |

15/5/2019 | Hamiltonian and Lagrangian dynamics Legendre transform | |

Week 10 | ||

21/5/2019 | Noether's Theorem | |

22/5/2019 | Proseminar: Exercises 16, 17, 18, 19, 20. | |

Week 11 | ||

28/5/2019 | Integrable systems n-body problem | |

29/5/2019 | ||

Week 12 | ||

4/6/2019 | Smale's horseshoe Homoclinic bifurcation | |

5/6/2019 | Proseminar: Exercises 7-11 | |

Week 13 | ||

11/6/2019 | No class Vorlesungsfrei | |

12/6/2019 | Circle homeomorphisms Sascha Troscheit replaces me | |

Week 14 | ||

18/6/2019 | Lorenz attractors | |

19/6/2019 |
Proseminar Test 2: all exercises up to 20 and
Exercises 7-11
New exercises: Exercises 21,22,26 | |

Week 14 | ||

25/6/2019 | | |

26/6/2019 | Exam (12:00--14:00 in HS02) |

- Kathy Alligood, Tim Sauer and James Yorke, Chaos, an Introduction to Dynamical Systems, Springer 1996 ISBN 0-387-94677-2
- Michael Brin and Garrett Stuck, Introduction to Dynamical Systems, Cambridge University Press 2002 ISBN 0-521-80841-3
- Robert Devaney, An Introduction to Cahotic Dynamical Systems, Benjamin and Cummings Publishing 1986, ISBN 0-8053-1601-9
- Clark Robinson, Dynamical Systems (Stability, Symbolic Dynamics, and Chaos), CRC Press 1995 ISBN 0-8493-8493-1.
- Steven Strogatz, Nonlinear dynamics and chaos, with applications to physics, biology and engineering, CRC Press, 2015, ISBN-13: 978-0813349107 or ISBN-10: 0813349109
- G. Teschl, Ordinary Differential Equations and Dynamical Systems ODE Classnotes (or in fact book).

Assessment of the course will be based on an written exam.

Assessment of the course proseminar will be based on two class tests (30 minues, worth 1/3) and
the performance in class (solving exercises at the board), worth 1/3.

- Class notes in pdf (written by Christian Schmeiser)
- Exercises for the Proseminar.
- An Applet for cobweb diagrams for the logistic family.
- An Applet for the bifurcation diagram for the logistic family. Also this one .
- One applet and another about period doubling bifurcations in the logistic family.
- Hand-out about structural stability
- Some bifurcation diagrams: Strogatz on the same topic.
- Some phase portraits for the Van der Pol oscillator:

and also some youtube videos here, here, and a lecture by Strogatz (Cornell University). - Some youtube videos on Huygens resonance here, here, here, and here, and if you still didn't have enough, also here.
- Some youtube videos on Smale and his horseshoe: here and here.
- Some youtube videos on Lorenz and the Lorenz attractor here, here, here, here and the Wikipedia page
- Some youtube videos on three-body choreographies, here, here, and here and here. Or this one.
- Some youtube videos on the Kepler problem (motion by the Earth and Sun), on gravitation, by Feynman (sort of), and general.
- Online Lecture by Anima Nagar

The limit-cykel of the Van der Pol equation for large ε has the shape of a canard (=duck), if you squint your eye just right... Note, however, that in this picutre you only have some transiet (=initial) orbits +

Smale's horseshoe | A homoclinic tangle | Another homoclinic tangle |

Updated May 2 2019