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Nonlinear Dynamics and Chaos

Instructor: Björn Hof, Burak Budanur

Teaching Assistant: Chaitanya Paranjape

Description

 

The course will begin by introducing continuous and discrete time dynamical systems, followed by fundamental concepts such as stability, Poincare sections, bifurcation theory, transition to chaos, and qualitative dynamics. Second half of the class will cover a selection of advanced topics and applications according to the interests of participants. Possible topics are spatiotemporal chaos, transition to turbulence, symmetries and symmetry reduction, operator-based methods, periodic orbit theory, and quantum chaos.

 


Requirements/Exams/Grading

 6 Homework assignments will be assigned weekly during first 6 weeks (10% each)

 1 term project (40%) : project report + presentation

 
 
Textbooks

 

    ChaosBook.org (Cvitanovic et al.)   

    Nonlinear Dynamics and Chaos (S. H. Strogatz)

 

 


Credits

 6


Course Calender (Tentative)

• Week 1 - Introduction to dynamical systems: Continuous and discrete time dynamical systems,
                   phase portraits, cobweb plots, numerical integration
• Week 2 - Time invariant sets: Equilibria, periodic orbits, linear stability, Poincare ́ sections
• Week 3 - Symmetries: Discrete and continuous symmetries, equivariance, invariant subspaces
• Week 4 - Bifurcations: Saddle-node, Pitchfork, Period-doubling, Hopf
• Week 5 - Transition to chaos: Period doubling and others
• Week 6 - Qualitative dynamics: Symbolic dynamics of unimodal maps
• Week 6 - 12 will cover a selection of advanced subjects.

 



 

 

 

 

Schedule (subject to change)

Date Time Location Topic
Tue, 10-Oct-2017 10:15 - 11:30 Mondi 1 Introduction/Historical overview
Tue, 10-Oct-2017 11:45 - 12:30 Mondi 1 No recitation
Thu, 12-Oct-2017 11:15 - 12:30 Seminar room Ground floor / Lab Bldg West Flows and maps
Tue, 17-Oct-2017 10:15 - 11:30 Seminar room Ground floor / Lab Bldg West Equilibria, fixed points, and linear stability
Tue, 17-Oct-2017 11:45 - 12:30 Seminar room Ground floor / Lab Bldg West Recitation: Numerical integration
Thu, 19-Oct-2017 11:15 - 12:30 Seminar room Ground floor / Lab Bldg West Periodic orbits, Floquet multipliers and vectors
Tue, 24-Oct-2017 10:15 - 11:30 Seminar room Ground floor / Lab Bldg West Poincaré sections and Poincaré maps
Tue, 24-Oct-2017 11:45 - 12:30 Seminar room Ground floor / Lab Bldg West Recitation
Tue, 31-Oct-2017 10:15 - 11:30 Seminar room Ground floor / Lab Bldg West Discrete symmetries
Tue, 31-Oct-2017 11:45 - 12:30 Seminar room Ground floor / Lab Bldg West Recitation: Newton method
Thu, 02-Nov-2017 11:15 - 12:30 Seminar room Ground floor / Lab Bldg West Continuous symmetries
Tue, 07-Nov-2017 10:15 - 11:30 Seminar room Ground floor / Lab Bldg West Bifurcations of equilibria
Tue, 07-Nov-2017 11:45 - 12:30 Seminar room Ground floor / Lab Bldg West Recitation: Computing Poincaré sections
Thu, 09-Nov-2017 11:15 - 12:30 Seminar room Ground floor / Lab Bldg West Hopf bifurcation
Tue, 14-Nov-2017 10:15 - 11:30 Mondi 2 Bifurcations of maps
Tue, 14-Nov-2017 11:45 - 12:30 Mondi 2 Recitation: Arnoldi iteration
Thu, 16-Nov-2017 11:15 - 12:30 Seminar room Ground floor / Lab Bldg West Period doubling I
Tue, 21-Nov-2017 10:15 - 11:30 Seminar room Ground floor / Lab Bldg West NO LECTURE
Tue, 21-Nov-2017 11:45 - 12:30 Seminar room Ground floor / Lab Bldg West NO RECITATION
Thu, 23-Nov-2017 11:15 - 12:30 Seminar room Ground floor / Lab Bldg West NO LECTURE
Tue, 28-Nov-2017 10:15 - 11:30 Mondi 2 Bifurcations and transition to chaos in fluid dynamics (Björn Hof)
Tue, 28-Nov-2017 11:45 - 12:30 Mondi 2 Recitation: GMRES
Thu, 30-Nov-2017 11:15 - 12:30 Mondi 2 Renormalization theory of universality I
Tue, 05-Dec-2017 10:15 - 11:30 Seminar room Ground floor / Lab Bldg West Renormalization theory of universality II
Tue, 05-Dec-2017 11:45 - 12:30 Seminar room Ground floor / Lab Bldg West No recitation
Thu, 07-Dec-2017 11:15 - 12:30 Seminar room Ground floor / Lab Bldg West NO LECTURE
Tue, 12-Dec-2017 10:15 - 11:30 Seminar room Ground floor / Lab Bldg West Other routes to chaos (Björn Hof)
Tue, 12-Dec-2017 11:45 - 12:30 Seminar room Ground floor / Lab Bldg West Newton-Krylov method
Thu, 14-Dec-2017 11:15 - 12:30 Seminar room Ground floor / Lab Bldg West Symbolic dynamics I: Unimodal maps
Tue, 19-Dec-2017 10:15 - 11:30 Seminar room Ground floor / Lab Bldg West Stable/Unstable manifolds and the homoclinic tangle
Tue, 19-Dec-2017 11:45 - 12:30 Seminar room Ground floor / Lab Bldg West Symbolic dynamics II: The horseshoe
Thu, 21-Dec-2017 12:00 - 13:15 Seminar room Ground floor / Lab Bldg West Subcritical transition to turbulence (Björn Hof)
Tue, 09-Jan-2018 10:15 - 11:30 Seminar room Ground floor / Lab Bldg West Subcritical transition to turbulence (Björn Hof)
Tue, 09-Jan-2018 11:45 - 12:30 Seminar room Ground floor / Recitation Transition to turbulence in plane Poiseuille flow (Chaitanya Paranjape)
Thu, 11-Jan-2017 11:15 - 12:30 Seminar room Ground floor / Lab Bldg West  
Tue, 16-Jan-2018 10:15 - 11:30 Seminar room Ground floor / Lab Bldg West Lagrangian chaos, guest lecturer: Francesco Romano (TU Wien)
Tue, 16-Jan-2018 11:45 - 12:30 Seminar room Ground floor / Recitation  
Thu, 18-Jan-2017 11:15 - 12:30 Seminar room Ground floor / Lab Bldg West Concluding remarks: What have we skipped?
Tue, 23-Jan-2017 12:00 - 13:15 Seminar room Ground floor / Lab Bldg West Student presentations I
Thu, 25-Jan-2017 11:15 - 12:30 Seminar room Ground floor / Lab Bldg West Student presentations II

Homework

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