Topics in Dynamical Systems and Chaos A (22/23)
Faculteit  Science and Engineering 
Jaar  2022/23 
Vakcode  WMMA03105 
Vaknaam  Topics in Dynamical Systems and Chaos A (22/23) 
Niveau(s)  master 
Voertaal  Engels 
Periode  semester I a 
ECTS  5 
Rooster  rooster.rug.nl 
Uitgebreide vaknaam  Topics in Dynamical Systems and Chaos A (tweejaarlijks 22/23)  
Leerdoelen  At the end of the course, the student is able to: 1. apply the key concepts "Poincare map", "suspension flow", "orbit", and "invariant set" to concrete examples of dynamical systems (such as rotations of the circle, endomorphisms on the torus, and autonomous differential equations). 2. formulate definitions and prove the key theorems related to concepts of topological dynamical systems, such as "omega limit sets", "alpha limit sets", "topological recurrence", and "topological entropy". In addition, the student is able to apply these concepts to concrete examples of dynamical systems. 3. formulate definitions and prove the key theorems related to concepts of lowdimensional dynamical systems, such as "lift", "rotation number", and "Sharkovsky's theorem". In addition, the student is able to apply these concepts to concrete examples of dynamical systems. 4. formulate definitions and prove the key theorems related to basic concepts of hyperbolic dynamics, such as "hyperbolic sets", "stable and unstable spaces", "invariant cones". In addition, the student is able to apply these concepts to concrete examples of dynamical systems such as Smale's horseshoe map. 5. formulate definitions and prove the key theorems related to advanced concepts of hyperbolic dynamics, such as "GrobmanHartman theorem", "HadamardPerron theorem", "stable and unstable manifolds". In addition, the student is able to apply these concepts to concrete examples of dynamical systems. 6. formulate definitions and prove the key theorems related to concepts of symbolic dynamics, such as "shift maps", "coding maps", "topological Markov chains", and "zeta functions". In addition, the student is able to apply these concepts to concrete examples of dynamical systems. 

Omschrijving  Dynamical systems theory concerns the question of how deterministic systems evolve in time. This first of all concerns the longterm behaviour of systems which includes stationary, periodic, multiperiodic, and chaotic dynamics, but also transient behaviour is of interest. Moreover bifurcations or transitions between asymptotic states  in particular transitions between regular and chaotic motions  under variation of parameters are of great importance. Special stationary solutions or more generally invariant manifolds can also form the organizing centers for the dynamics in the state space of a dynamical system. Applications of dynamical systems theory range from molecular dynamics to celestial mechanics and extend to the life sciences, climate science, and many other fields. This course aims to provide a solid mathematical introduction to the field of dynamical systems. Topics covered in this course include: topological dynamics, lowdimensional systems (such as flows in the plane or circle maps), hyperbolic dynamics, and symbolic dynamics. 

Uren per week  
Onderwijsvorm  Hoorcollege (LC)  
Toetsvorm 
Mondeling tentamen (OR), Opdracht (AST)
(Final Grade = 0.2 OE + 0.8 TH, where OE is the mark for the oral exam and TH is the mark of the written assignment (take home exam). The student passes when the Final Grade, rounded to the nearest multiple of 0.5, is larger than or equal to 6. In case of a failing grade, the student can redo the oral exam and written assignment one more time. The same formula applies to the resit exam.) 

Vaksoort  master  
Coördinator  dr. A.E. Sterk  
Docent(en)  dr. A.E. Sterk  
Verplichte literatuur 


Entreevoorwaarden  The course assumes knowledge from: Analysis, Linear Algebra 1, Linear Algebra 2, Metric and Topological Spaces, Multivariable Analysis, Analysis on Manifolds  
Opmerkingen  
Opgenomen in 
