Wintersemester 2020/2021
Seminar Geschichte(n) der Dynamik 2
Prof. Dr. Bernold Fiedler,
Isabelle Schneider
Schedule
 Wednesdays 10.0012.00
No seminar on Wednesday 04.11.20! The distribution of topics will be on Thursday 05.11.20, at 12:00, after the lecture on Dynamical Systems 2.
Beschreibung
In parallel to the lecture course "Dynamical Systems 2", we want to explore some exciting
methods, examples and applications of dynamical systems, such as curiosities amongst
discrete systems, fractals, oscillation patterns, and systems with time delays.
We will chose a selection of the following topics, depending on the interests of the participants.
Themen
1 Discrete dynamical systems
1.1 Computer Simulations
 logistic map
 Hénon map
 Lorenz attractor
 Literature: [ASY96]
1.2 Kneading
 iterates of unimodal maps
 itineraries and kneading sequences
 kneading matrix and kneading determinant, classification of dynamics
 Literature: [CE09]
1.3 The HénonAttraktor
 approximation of the logistic flow through Euler und central differences
 Hénon map as interpolation of both approximations
 period doubling in the Hénon map
 Computer experiments
 Literature: [HK12]
2. Fractals
2.1 Fractals and noninteger dimensions
 What are fractals? Examples
 Hausdorff measure
 Hausdorff dimension with examples
 if time allows: compare to boxcounting dimension
 Literature: [Fal04], notes
2.2 Selfsimilar fractals
 Examples: Cantor set, Sierpinski triangle, Koch curve
 Iterated function systems
 Literature: [Fal04], notes
2.3 Michael Crichton: the monster of Jurassic Park
 Construction of the dragon curve
 properties of the dragon curve
 Literature: [Cri90], exercise problem and notes
3 Oscillations
3.1 Inverse Pendulum
 Poincaré map, stability, swing, forced inverse pendulum
 If time allows: averaging, resonance
 Literature: [Arn92], [MW66]
3.2 The Kepler problem
 duality between Hooke’s law and the gravitational law
 Proof that orbits are elliptic
 Literature: [Arn90]
3.3 Hopf bifurcation
 The StuartLandau oscillator
 existence of bifurcating periodic solutions
 if time allows: stability
 Literature: [Kuz13]
s
3.4 Romeo and Julia: Turing instability in love
 Stability / instability in coupled iterated systems
 Literature: [Fie10]
3.5 How the zebra got its stripes
 Reactiondiffusion equations
 Patterns in the Schnakenberg model
 Literature: [Mur88], notes
4 Delay equations
4.1 Introduction and methods of steps
 Examples for delay equations
 What is a solution? In which space does it live?
 method of steps, continuity of solutions
 Literature: [Hal77], [HL93], [Smi11], [Dri77]
4.2 Small solutions and solving backwards
 What is a small solution? Why can it occur?
 explicit examples
 Nonuniqueness of backwards solutions
 Literature: [Hal77], [HL93], [Smi11], [Dri77]
4.3 Stability and characteristic equations
 stability of solutions
 exponential ansatz and characteristic equations
 Literature: [Hal77; HL93; Smi11]
4.4 Nicholson’s blowflies equation
 Modeling of Nicholson’s blowflies equation
 Stability of nontrivial steady state
 Existence of periodic solutions
 Literature: [BBI10; GBN80]
References
 [Arn90] Vladimir I Arnold. Huygens and Barrow, Newton and Hooke: pioneers in mathematical
analysis and catastrophe theory from evolvents to quasicrystals. Springer
Science & Business Media, 1990.
 [Arn92] Vladimir I. Arnold. Ordinary Differential Equations. Springer, 1992.
 [ASY96] Kathleen T Alligood, Tim D Sauer, and James A Yorke. Chaos. Springer, 1996.
 [BBI10] L Berezansky, Elena Braverman, and L Idels. “Nicholson’s blowflies differential
equations revisited: main results and open problems”. In: Applied Mathematical
Modelling 34.6 (2010), pp. 1405–1417.
 [CE09] Pierre Collet and JP Eckmann. Iterated maps on the interval as dynamical systems.
Springer Science & Business Media, 2009.
 [Cri90] Michael Crichton. Jurassic park: A novel. Vol. 1. Ballantine Books, 1990..
 [Dri77] Rodney David Driver. Ordinary and delay differential equations. Vol. 20. Springer
Science & Business Media, 1977.
 [Fal04] Kenneth Falconer. Fractal geometry: mathematical foundations and applications.
John Wiley & Sons, 2004.
 [Fie10] Bernold Fiedler. “Romeo and Juliet, Spontaneous Pattern Formation, and Turing’s
Instability”. In: Mathematics Everywhere (2010), p. 53.
 [GBN80] WSC Gurney, SP Blythe, and RM Nisbet. “Nicholson’s blowflies revisited”. In:
Nature 287.5777 (1980), pp. 17–21.
 [Hal77] Jack K Hale. Theory of functional differential equations. Vol. 3. Springer, 1977.
 [HK12] Jack K Hale and Hüseyin Koçak. Dynamics and bifurcations. Vol. 3. Springer
Science & Business Media, 2012.
 [HL93] Jack K Hale and Sjoerd M Verduyn Lunel. Introduction to functional differential
equations. Vol. 99. Springer Science & Business Media, 1993.
 [Kuz13] Yuri A Kuznetsov. Elements of applied bifurcation theory. Vol. 112. Springer
Science & Business Media, 2013.
 [Mur88] James D Murray. “How the leopard gets its spots”. In: Scientific American 258.3
(1988), pp. 80–87.
 [MW66] Wilhelm Magnus and Stanley Winkler. Hill’s equation. Interscience, 1966.
 [Smi11] Hal L Smith. An introduction to delay differential equations with applications to
the life sciences. Vol. 57. Springer New York, 2011.
Further literature will either be provided on the webpage or communicated directly to the
speakers.
