Sommersemester 2020

Seminar (Hi)stories of Dynamics

Prof. Dr. Bernold Fiedler, Alejandro López Nieto

Schedule

Wednesdays 10:00-12:00, starting 27.05.20 via Zoom, Meeting ID: 898 1838 5601.

The distribution of topics will take place on Thursday April 23 12:00-, right after the second lecture of Dynamical Systems I.

Format

Due to the current situation, the course will be completely online, we will be using Zoom for our appointments.
If you wish to make sure you don't miss anything, just let Alejandro know that you are interested. The specific format of the presentations will be discussed in further detail during the preliminary session on April 23, but it will be possible present in either German or English, completely up to you.

Description

Through this seminar we'd like to explore a few pearls of dynamics that complement the contents of the lecture Dynamical Systems I. Most of the topics that we offer will deal with discrete dynamics. However, for obvious reasons, we also feel the need to dive in the direction of models of epidemics and the study of infectious diseases as a dynamical system.

Themen

Bifurcating into chaos: Period doubling and the logistic map

• The logistic map and period doubling (phenomenology and numerics)
• Conjugation to the tent-map for suitable parameters
• Connection to continued fractions
• References: [1], [7], Collet and Eckmann, van Strien

More space, more disorder: the Hénon attractor

• Approximation of the logistic flow through Euler and central differences
• Hénon map as an interpolation of both approximations
• Period doubling in the Hénon map
• Numerics
• References: [6]

Anosov cat map and the butterfly effect

• The cat map
• Periodic orbits, dense orbits
• References: [3] Introduction, [2] 3.13

Fractals and fractional dimensions

• What's a fractal? Examples
• Hausdorff measure, Hausdorff dimension and examples
• Comparison to box counting dimension
• References: [5]

Self-similar fractals

• Examples: the Cantor set, Sierpinski triangle, Koch curve
• Iterated function systems
• References: [5]

Michael Crichton: the monster of Jurassic Park

• Construction and properties of the dragon curve
• References: [4]

Compartmental disease models: The numbers behind social distancing.

• Compartmental disease models: SIR vs SIS vs SIRS
• Global stability: Lyapunov functions and the basic reproduction number
• References: [8], [9]

References

• [1] Kathleen T Alligood, Tim D Sauer, and James A Yorke. Chaos. Springer, 1996.
• [2] Vladimir Igorevich Arnold. Geometrical methods in the theory of ordinary differential equations. Vol. 250. Springer, 1983.
• [3] Ehrhard Behrends and Bernold Fiedler. “Periods of discretized linear Anosov maps”. In: Ergodic Theory and Dynamical Systems 18.2 (1998), pp. 331–341.
• [4] Michael Crichton. Jurassic park: A novel. Vol. 1. Ballantine Books, 1990.
• [5] Kenneth Falconer. Fractal geometry: mathematical foundations and applications. John Wiley & Sons, 2004.
• [6] Jack K Hale and Hüseyin Koçak. Dynamics and bifurcations. Vol. 3. Springer Science & Business Media, 2012.
• [7] Steven H Strogatz. Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. Hachette UK, 2001.
• [8] A Lajmanovic and J A Yorke, "A deterministic model for gonorrhea in a nonhomogeneous population", Mathematical Biosciences (1976).
• [9] A Korobeinikov and G C Wake, "Lyapunov Functions and Global Stability for SIR, SIRS, and SIS Epidemiological Models", Applied Mathematics Letters (2002).