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1MA217 Homepage

A collection of links to various Dynamical Systems resources

Homework

Timetable





Course Info:

  • Course: Dynamical Systems 1MA217

  • Lecturer: Denis Gaidashev, Ång 14231, gaidash at math.uu.se

  • First Day Handout

    • Textbooks:

  • Primary text: M. Brin, G. Stuck, Introduction to Dynamical Systems, Cambridge University Press 2002.

  • Secondary text: M. W. Hirsch, S. Smale, R. L. Devaney, Differential Equations, Dynamical Systems and an Introduction to Chaos, Academic Press (Elsevier) 2004.

  • Bits and pieces from: A. Katok, B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press 1995.


    TAKEHOME, DUE JAN 1, 2017, FINAL VERSION, 9 problems


    Things on the current agenda:

    My Lecture Notes, based on BS and KH, a work in progress, updated 2016/11/28.

    Period three implies chaos, J. Yorke ans T.-Y. Li.

    A little bit about the exponential map and geodesics on Riemannian manifolds.

    A horseshoe and stable/unstable manifolds for Henon maps, an applet .

    O. Lanford's computer assisted proof of the Feigenbaum conjecture and the renormalization hyperbolicity .

    Universal properties of the maps of the interval, P. Collet, J.-P. Eckmann and O. Lanford .

    C. Liverani's notes on the stable-unstable manifold theorem .

    John Milnor's lectures on dynamics in one complex variable .

    A paper about the recurrence property of the Arnold cat map .

    Jaume Alonso's slides on the Siegel theorem .

    Iterations and the bifurcation diagram for the quadratic map

    Interactive cobweb plot for the quadratic map by Andre Burbanks

    Paper by J. Banks et al, "On Devaney's Difinition of Chaos".



    Course Objectives/Outcomes:

    1) Understand the fundamental concepts of the dynamical systems, specifically:

    • dynamics of the quadratic family,
    • conjugacies, specifically as relates to circle maps,
    • topological dynamics; recurence, mixing and transitivity,
    • symbolic dynamics, shifts
    • hyperbolicity and structural stability,
    • stable and unstable manifolds, homoclinic and heteroclinic intersections, horseshoes,
    • fundamental concepts of continuous dynamics, Poincare-Bendixson theorem, Poincare return maps, etc
    • fundamental concepts of Hamiltonian dynamics and Kolmogorov-Arnold-Moser theory,
    • fundamental concepts of ergodic theory,
    • some concepts from complex dynamics, such as Julia and Mandelbrot sets.
    2) Solve representative problems in the above-mentioned topics.

    3) Carry out numerical studies of dynamical systems.

    4) Understand and be able to explain/present some applications of the theory.


    Grading:

    There will be a takehome final, 100% of the final grade.


    Schedule

    Below: BS stands for Brin-Stuck, HSD for Hirsch, Smale and Devaney, KH for Katok Hasselblat.

    1. 2016-08-30
      • Introduction, course information.
      • What is "dynamics"?
      • The quadrtic family as the fundamental example of a chaotic dynamical system.

      Reading: 1.1 BS, 1.5 BS, 15.1 HSD, 15.3 HSD, 15,4 HSD


    2. 2016-09-01
      • The quadratic family continued.

      Reading: 1.1 BS, 1.5 BS, 15.1 HSD, 15.3 HSD, 15,4 HSD


    3. 2016-09-07
      • The quadratic family continued.
      • Expanding endomorphisms of the circle

      Reading: 1.3 BS, 1.4 BS. 1.9 a) and b) in KH


    4. 2016-09-08
      • Symbolic dynamics.
      • Topological dynamics: transitivity and mixing.

      Reading: 2.2 and 2.3 BS, Lemma 1.4.2 in KH


    5. 2016-09-14
      • Topological dynamics continued: transitivity and mixing.
      • Discussion of the Devaney's definition of chaos.
      • Topological dynamics continued: limit sets, reccurent sets.

      Reading: Paper by J. Banks under "Current Agenda", 2.1 BS, 3.3 KH, 10.1 HSD


    6. 2016-09-14
      • Topological dynamics continued: transitivity and mixing.
      • Discussion of the Devaney's definition of chaos.
      • Topological dynamics continued: limit sets, reccurent sets.

      Reading: Paper by J. Banks under "Current Agenda", 2.1 BS, 3.3 KH, 10.1 HSD


    7. 2016-09-16
      • Irrational rotations of the circle.
      • Topological dynamics continued: topological entropy.

      Reading: 1.3 BS, 3.3 KH, 2.5 BS, 2.6 BS, 1.9 c) and d) in KH


    8. 2016-09-20
      • Topological dynamics continued: topological entropy.

      Reading: 2.5 BS, 2.6 BS, 1.9 c) and d) in KH


    9. 2016-09-23
      • Subshifts.
      • Perron-Frobenius Theorem.

      Reading: 3.1 BS, 3.2, 3.3 BS, 1.9 in KH


    10. 2016-09-26
      • Perron-Frobenius Theorem.
      • Circle homeomorphisms.

      Reading: 3.3, 7.1-7.2 BS


    11. 2016-09-30
      • Circle homeo- and diffeomorphisms.

      Reading: 7.1-7.2 BS


    12. 2016-10-05
      • Circle homeomorphismsm, rotation number.
      • Denjoy's example of a non-transitive circle diffeomorphism

      Reading: 7.2 BS.


    13. 2012-10-11
      • Poincare's classification of irrational circle homeos, possible limit sets for an irrational circle homeo.

      Reading: 7.1-7.2 BS, 11.1 KH.


    14. 2012-10-14
      • Ordered families of orientation preserving homeomorphisms, monotonicity of the rotation number.

      Reading: my notes, 11.1 KH.


    15. 2012-10-24
      • Devils staircase, Arnold's tongues
      • Rational rotations
      • Circle diffeos, Denjoy's theorem

      Reading: my notes, 11.1 KH, 11.2 KH.


    16. 2016-10-31
      • Introduction to Kolmogorov-Arnold-Moser theory.

      Reading: Jaume Alonso's slides on the Siegel theorem in Current Agenda, a full proof in Carleson-Gamelin, 12.3 KH, 12.5 KH, Another proof of a version of the theorem, page 21 here


    17. 2012-11-3
      • Sharkovsky Theorem.

      Reading: 7.3 BS.


    18. 2012-11-08
      • Sharkovsky theorem

      Reading: 7.3 BS.


    19. 2012-11-15
      • Period doubling and renormalization

      Reading: my lecture notes


    20. 2012-11-17
      • The linearization problem and the Hartman-Grobman theorem
      • The Hadamard-Perron Theorem.

      Reading: my lecture notes, Liverani's notes in Current Agenda


    21. 2012-11-28
      • The Hadamard-Perron Theorem.
      • Hyperbolic sets and shadowing.

      Reading: my lecture notes, Liverani's notes in Current Agenda


    22. 2012-11-30
      • Hyperbolic sets and shadowing.
      • The horseshoe.

      Reading: my lecture notes


    23. 2012-12-7
      • Persistence of hyperbolic sets.

      Reading: my lecture notes


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