Email H. Bruin for further information for this course.

First class on March 5

March 12 is canceled due to Rector's Day.

Easter Break is from March 26 to April 6.

Whit Monday is on May 21.

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

Monday | 9:45--11:15 | HS02 | Lecture | 05.3.2018 | 25.06.2018 |

Wednesday | 13:15-14:00 | HS02 | Lecture | 7.3.2018 | 27.06.2018 |

Wednesday | 14:15-15:00 | HS02 Proseminar Henna Koivusalo | 7.3.2018 | 27.06.2018 |

Day | Exercises |
---|---|

March 21 | 1-5 |

April 18 | 6-10 |

May 2 | 9,10,12,13,11 |

May 16 | 22,16,17,18 |

May 30 | 23,20,21,25 |

June 6 | 26,27,28 |

June 20 | 29,32,31 |

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 **(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)
- Feigenbaum (bifurcation diagram) (S§6. T§11.1, St)
- Hamiltonian systems (S§10. T§8.3. St)
- 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)
- Lagrangian = Lagrange function (T§8.3. S§10)
- Legendre transform (T§8.3. S§10)
- 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)
- 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).)
- 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)

- 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
- ODE Classnotes (or in fact book) of Prof. G. Teschl.

Will be based on an oral exam (in English by default, aber auf Deutsch ist auch möglich).

Exam Material:

- Linear ODEs and phase portraits
- Existence, uniqueness and continuity of solutions of ODEs
- Iteration of maps on the line (cobweb diagrams), circle and plane
- Types of orbits (fixpoint/equilibria, periodic, saddle/sink/source/center) and their stability (hyperbolicity)
- Stable and unstable manifold and spaces - Hartman-Grobman Theorem
- Bifurcations (saddle-node/font, transcritical, pitchfork, period doubling, Hopf, cusp)
- Poincaré-Bendixson Theorem, alpha/omega-limit sets and limit cycles
- Lyapunov functions
- Definitions of chaos, sensitive dependence on initial conditions.
- Symbolic dynamics
- Smale's horseshoe
- Hamiltonian systems and first integrals
- Examples: logistic maps, logistic differential equations, Van der Pol equations, Lorenz equations, harmonic oscillator and pendulum

- 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.
- Hand-out about structural stability
- Hand-out about symbolic itineraries
- 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, and here and the Wikipedia page
- Written out class notes of June 18 in pdf.
- 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

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

Updated April 2018