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1MA444 Homepage

A collection of links to various Dynamical Systems resources

Maple worksheets

Timetable






Course Info:

  • Course: Applied Dynamical Systems 1MA444
  • 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.

  • Secondary text: Differential Equations, Dynamical Systems, and an Introduction to Chaos , Morris W. Hirsch, Stephen Smale, Robert L. Devaney Academic Press.



    Things on the current agenda:

    HOMEWORK 2, DUE 3/12: Homework 2


    PROJECTS:

    • Synchronization in chaos (9.6.1, 9.6.2, 9.6.3 in Strogatz)

    • Strange attractor in Rossler system (12.3.1, 12.3.2, 12.3.3 in Strogatz)

    • Quasiperiodic motion (8.6.2, 8.6.3, 8.6.5, 8.6.6 in Strogatz)

    • Correlation dimension of the Lorenz attractor (11.5.1 in Strogatz)

    • Method of averaging (7.6.25, 7.6.26 in Strogatz)

    • Chemical reaction networks (Chapter 2 in Mathematical Modelling in Systems Biology: An Introduction by Brian Ingalls, et al. Exercise 2.4.7 )

    • Metabolic networks ( Chapter 5 in Mathematical Modelling in Systems Biology: An Introduction by Brian Ingalls, et al. Exercise 5.6.2 )



    Example of a program that finds an attracting cycle in the quadratic family, in SageMath, 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.
    2) Understand and be able to explain/present some applications of the theory.

    3) Outline the mathematical methods and techniques used to analyse these models and understand in what situations these methods can be applied.

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

    5) Carry out numerical studies of dynamical systems.



    Grading:

    Two homeworks, 66%. A project 34%



    Schedule

    1. 2019-09-03
      • 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. 2019-09-04
      • The quadratic family continued. Periodic and stochastic behaviour.
      • Functions vs maps.
      • Periodic orbits and their stability.

      Reading: Ch 10

    3. 2019-09-11
      • The bifurcation cascade.
      • Intermittency and period 3.
      • What is chaos?
      • Fractals
      • Box counting dimension

      Reading: Ch 10 and p 323, Ch 11

    4. 2019-09-16
      • Fractals.
      • Cardinality and Cantor set.

      Reading: Ch 11

    5. 2019-09-19
      • Iterated Function System.
      • Hausdorff dimension.

      Reading: Ch 11

    6. 2019-09-26
      • Introduction to ODE's.
      • One-dimensional flows.
      • Bifurcation theory.

      Reading: Ch 3

    7. 2019-09-30
      • Bifurcation theory.
      • Stable/unstable manifold theorem.
      • Pendulum

      Reading: Ch 5. Ch 6.0-6.4. Ch 3,4 in Hirsch, Smale, Devaney, Ch 6.7

    8. 2019-10-04
      • Stability of equilibria. Lyapunov function.
      • Lyapunov function for damped pendulum and Lorenz system.

      Reading: Ch 7.2. Ch 9.2 in Hirsch, Smale, Devaney, Ch 7.5

    9. 2019-10-07
      • Limit sets and cycles.
      • Poincare-Bendixson theorem

      Reading: Ch 7.

    10. 2019-10-10
      • Poincare-Bendixson theorem
      • Oscilating reactions, Selkov system and Hopf bifurcations
      • 2D bifurcations.

      Reading: Ch 7 (esp 7.3), Ch 8.2, 8.3. Read about bifurcations here

    11. 2019-10-15
      • More about the Selkov system
      Pendulum

      Reading: 7.3, 6.5

    12. 2019-10-17
      • Pendulum
      • Damped pendulum

      Reading: 6.5, 6.7

    13. 2019-10-21
      • Lorenz equations and chaos

      Reading: Chapter 9

    14. 2019-10-22
      • Lorenz equations and chaos

      Reading: Chapter 9

    15. 2019-11-04
      • Lorenz equations and chaos

      Reading: Chapter 9

    16. 2019-11-15
      • Henon map and horseshoe
      • Horseshoe as a fundamental example of chaos

      Reading: Chapter 12, exercise 12.1.7

    17. 2019-11-19
      • Henon map and strange attractors

      Reading: Chapter 12

    18. 2019-11-21
      • Homoclinic bifurcations, homoclinic tangle.
      • Rossler attractor

      Reading: 12.0-12.3. A mostly historic review of the discovery of homoclinic penomena by H. Poincare here

    19. 2019-11-25
      • Standard map,
      • Tokomak.
      • Siegel linearization theorem

      Reading: 8.6, also, a much more detailed but accessible proof of the Siegel theorem (simplest KAM theorem) can be found in L. Carleson, Th. Gamelin "Complex Dynamics" in Chapter "Irrationally neutral fixed points".

    20. 2019-11-27
      • Quasiperiodic motion, KAM theory and the standard map
      • N-body problem
      • Kepler two-body problem

      Reading: 8.6, here, here

    21. 2019-12-03
      • N-body problem
      • Euler and Lagranfe solutions
      • Restricted three body problem

      Reading: here

    22. 2019-12-4
      • Michaelis-Menten kinetics;

      Reading: here and a presentation here

    23. 2019-12-10
      • Biological oscillators and other models

      Reading: here

    24. 2019-12-12
      • Models of enzyme kinetics

      Reading:

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