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1MA151 Homepage

A collection of links to various Dynamical Systems resources

Maple worksheets

Timetable





Course Info:

  • Course: Applied Dynamical Systems 1MA151

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

  • First Day Handout

    • Textbooks:

  • Primary text: Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering (Studies in nonlinearity), Steven H. Strogatz, Publisher: Westview Press 2001-01-19, 512 Pages, ISBN: 0738204536


    Things on the current agenda:


    Example of a program that finds an attracting cycle in the quadratic family, in SageMath, here

    TAKEHOME, all problems here

    LORENZ SYSTEM

    An applet, and some applete in the wiki page

    The Lorinz attractor folding onto itself

    CONSERVATIVE MAPS, HORSESHOES AND CHAOS

    Various applets here

    My notes about hyperbolic sets and the horseshoe

    QUADRATIC MAP

    An exposition about the dynamics of the quadratic map

    Iterations and the bifurcation diagram for the quadratic map

    Another iteration engine for the quadratic map

    Intermitency in the quadratic map

    Period doubling cascade

    Interactive cobweb plot for the quadratic map by Andre Burbanks

    CHAOS

    Double pendulum applet

    Equations of motion for a double pendulumt

    Paper by J. Banks et al, "On Devaney's Difinition of Chaos". A proof of full Sharkovsky theorem, a paper in which Sharkovsky's theorem was rediscovered.

    FRACTALS

    Serpinski Gasket.

    An IFS applet.

    IFS,explanation.

    Fractal image compression.

    A Barnsley fern, explanation.

    A Barnsley fern, applet.

    Some fractal applets here and many more here

    Box counting dimension of coastline, applet

    Covering coastline of Norway, applet

    Examples of fractals in nature here and here

    Devil's staircase for the Cantor set here , here and here



    Course Objectives/Outcomes:

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

    • dynamics of the quadratic family,
    • topological dynamics; recurrence, mixing and transitivity,
    • stable and unstable manifolds, homoclinic and heteroclinic intersections, horseshoes,
    • fractals,
    • one and two dimensional flows, phase space, limit cycles and Poincar\'e-Bendixson Theorem,
    • bifurcations in flows and maps.
    4) Understand and be able to explain/present some applications of the theory.

    3) Solve representative problems in the above-mentioned topics.

    4) Carry out numerical studies of dynamical systems.


    Grading:

    A take home final, 100%.


    Schedule

    1. 2015-10-26
      • Introduction, course information.
      • What is "dynamics"?
      • Continous and descrete dynamical systems.
      • The quadrtic family as the fundamental example of a chaotic dynamical system.

      Reading: Ch 10 in Strogatz


    2. 2015-10-28
      • The quadratic family continued. Periodic and stochastic behaviour.
      • Functions vs maps.
      • Periodic orbits and their stability.

      Reading: Ch 10

    3. 2015-11-3
      • The bifurcation cascade.
      • Renormalization

      Reading: Ch 10

    4. 2015-11-4
      • Intermittency and period 3.
      • What is chaos?

      Reading: Ch 10 and p 323

    5. 2015-11-6
      • Fractals.
      • Cardinality and Cantor set.

      Reading: Ch 11

    6. Stable/unstable manifold theorem
    7. 2015-11-9
      • Fractals.
      • Iterated Function System.
      • Fractal dimensions.

      Reading: Ch 11

    8. 2015-11-10
      • Hausdorff dimension.
      • Rotation and devil's staircase.
      • An exmaple of a "dynamical Cantor set": the Feigenbaum attractor.

      Reading: Ch 11

    9. 2015-11-12
      • An exmaple of a "dynamical Cantor set": the Feigenbaum attractor.
      • Bifurcation theory

      Reading: "An exposition about the dynamics of the quadratic map" in the "current agenda". Ch 3 in Strogatz.

    10. 2015-11-13
      • Bifurcation theory.

      Reading: Ch 3

    11. 2015-11-13
      • Introduction to ODE's.
      • Stable/unstable manifold theorem.

      Reading: Ch 5. Ch 6.0-6.3. Ch 3,4 in Hirsch, Smale, Devaney,

    12. 2015-11-24
      • Stable/unstable manifold theorem
      • Stability of equilibria. Lyapunov function.
      • Ex: Pendulum

      Reading: Ch 7.2. Ch 9.2 in Hirsch, Smale, Devaney. Ch 6.7

    13. 2015-11-25
      • Lyapunov function for damped pendulum and Lorenz system.
      • Limit sets and cycles.
      • Poincare-Bendixson theorem
      • Oscilating reactions, Selkov system and Hopf bifurcations

      Reading: Ch 7.

    14. 2015-11-26
      • Poincare-Bendixson theorem
      • Oscilating reactions, Selkov system and Hopf bifurcations
      • 2D bifurcations.

      Reading: Ch 7. Ch 8.2, 8.3. Read about bifurcations here

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