Syllabus

Two lectures per week, one homework

For bird's eye overview, read thesummary of part I of ChaosBook.

Week 1: Flows and maps

  1. Trajectories
    2. Read quickly all of Chapter 1 - do not worry if there are stretchesthat you do not understand yet. Study all of Chapter 2
  2. Flow visualized as an iterated mapping
    Discrete time dynamical systems arise naturally byrecording the coordinates of the flow when a special event happens:the Poincaré section method, key insight for much that is tofollow.


Week 2: Linear stability

  1. There goes the neighborhood
    So farwe have concentrated on description of the trajectoryof a single initial point.Our next task is to define and determine the size of aneighborhood, and describe the local geometry ofthe neighborhood by studying the linearized flow.What matters are the expanding directions.
  2. Cycle stability
    If a flow issmooth, in a sufficiently small neighborhood it is essentiallylinear. Hence in this lecture, which might seem an embarrassment(what is a lecture on linear flows doing in a book on nonlinear dynamics?), offers a firm stepping stone on the way tounderstanding nonlinear flows. Linear charts are the key tool ofdifferential geometry, general relativity, etc, so we are in goodcompany.


Week 3: Linear stability

  1. Stability exponents are invariants of dynamics
    We prove that (1) Floquet multipliers are the same everywherealong a cycle, and (b) that they are invariant under any smoothcoordinate transformation.
  2. Pinball wizard
    The dynamicsthat we have the best intuitive grasp onis the dynamics of billiards.For billiards, discrete time is altogether natural;a particle moving through a billiardsuffers a sequence of instantaneous kicks,and executes simple motion in between.


Week 4: World in a mirror

  1. Discrete symmetries of dynamics
    What is a symmetry of laws of motion?The families of symmetry-related full state space cyclesare replaced by fewer and often much shorter"relative" cycles, andthe notion of a prime periodic orbitis replaced by the notion ofa "relative" periodic orbit, the shortest segmentthat tiles the cycle under the action of the group.Discrete symmetries: a review of the theory of finite groups
  2. Discrete symmetry reduction of dynamics to a fundamental domain
    While everyone can visualize the fundamental domain for a 3-diskbilliard, the simpler problem - symmetry reduction of 1d dynamicsthat is equivariant under a reflection, the most common symmetry inapplications - seems to baffle everyone. So here is a step-by-stepwalk through to this simplest of all symmetry reductions.


Week 5: Relativity for cyclists

  1. Continuous symmetries of dynamics
    Symmetry reduction:If the symmetry is continuous, the interesting dynamics unfolds on alower-dimensional "quotiented" system, with"ignorable" coordinates eliminated (but not forgotten). Hilbert's invariant polynomials. Cartan's moving frames.
  2. Got a continuous symmetry? Freedom and its challenges
    Whenever you have a continuous symmetry, you need to cut the orbitto pick out one representative for the whole family. For continuousspatial symmetries, this is achieved by slicing. And then there isdicing.


Week 6: Charting the state space

  1. Slice and dice
    Symmetry reduction is the identification of a unique point on agroup orbit as the representative of this equivalence class. Thus,if the symmetry is continuous, the interesting dynamics unfolds ona lower-dimensional `quotiented', or `reduced' state space M/G. Inthe method of slices the symmetry reduction is achieved by cuttingthe group orbits with a set of hyperplanes, one for each continuousgroup parameter  Moving frames give us a great deal offreedom - we discuss how to choose a frame The most natural of allmoving frames: the comoving frame, the frame for space cowboys.
  2. Qualitative dynamics, for pedestrians
    Qualitative properties ofa flow partition the state space in a topologically invariant way.


Week 7: Stretch, fold, prune

  1. The spatial ordering of trajectories from the time ordered itineraries
    Qualitative dynamics: (1) temporal ordering, or itinerary withwhich a trajectory visits state space regions and (2) the spatial orderingbetween trajectory points, the key to determining the admissibilityof an orbit with a prescribed itinerary. Kneading theory.
  2. Qualitative dynamics, for cyclists
    Dynamical partitioning of a plane.Stable/unstable invariant manifolds, and how they partition thestate space in intrinsic, topologically invariant manner. Henon mapis the simplest example.


Week 8: Fixed points, and how to get them

  1. Finding cycles
    Why nobody understands anybody? The bane of night fishing - plushow to find all possible orbits by (gasp!) thinking.
  2. Finding cycles; long cycles, continuous time cycles
    Multi-shooting; d-dimensional flows; continuous-time flows.