Nonlinear Dynamics & Chaos I151-0532-00 (FS2015)

Dr. Daniel Karraschkarrasch@imes.mavt.ethz.chETH Zrich, LEE M 201

044 632 77 55www.zfm.ethz.ch/karrasch

mailto:karrasch@imes.mavt.ethz.chhttp://www.zfm.ethz.ch/~karrasch

Course outline

1 Basic facts about nonlinear systems: Existence, uniqueness,dependence on initial data

2 Near equilibrium dynamics: Linear and Lyapunov stability3 Bifurcations of equilibria: Center manifolds, normal forms, elementary

bifurcations4 Nonlinear dynamical systems on the plane: Phase plane techniques,

limit sets, limit cycles5 Time-periodic dynamical systems: Floquet theory, Poincare maps,

averaging methods, resonance6 (Possibly) Chaotic dynamics: Homoclinic dynamics, Melnikovs

method, attractors, Lyapunov exponents

Logistics

Course language : English.Prerequisites: Analysis, linear algebra and a basic course in differentialequations.Exam: a written, two-hour exam in English.Homework assignments:

one assignment roughly every other week,one randomly chosen exercise from each assignment will be graded,hints to solutions (possibly full solutions) to the assignments will beposted online after the submission deadline.

Grade policy: 25% from homework, 75% from exam.Class times and location: Wed 1012 and Thu 1617 HG D7.2.Office hours: Mon & Fri, 1213, LEE M 207.Lecture notes will be posted online (these are not intended as areplacement for your own notes!)

Motivation

Complex systems can show simple behavior (after 605 trials).Movie

Motivation

Simple systems can show complex behavior (right away)

A model for atmospheric convection (Lorentz [1963]):dxdt

= (y x),

dydt

= x( z) y , (x , y , z): amplitudes of velocity modes

dzdt

= xy z .

Objectives

1 Learn methods to analyze nonlinear dynamics without solving theunderlying differential equations

2 Develop intuition for geometry of nonlinear systems through numericalassignments

3 Learn a few important techniques from applied mathematics4 Analyze applications from various areas of engineering and applied

science

Recommended books

1 J. Guckenheimer & P. Holmes, Nonlinear Oscillations, DynamicalSystems, and Bifurcations of Vector Fields, Springer, 1983.

2 F. Verhulst, Nonlinear Differential Equations and Dynamical Systems,Springer, 1996.

3 V. I. Arnold, Ordinary Differential Equations, Springer, 1992.4 S. Strogatz, Nonlinear Dynamics and Chaos, Addison-Wesley, 1994.5 Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer,

2004.

http://link.springer.com/book/10.1007%2F978-1-4612-1140-2http://link.springer.com/book/10.1007%2F978-1-4612-1140-2http://link.springer.com/book/10.1007%2F978-3-642-61453-8http://dx.doi.org/10.1007/978-1-4757-3978-7