Syllabus
Four in-class lectures, two problem-solving sessions
For bird's eye overview, read the summary of part I of ChaosBook.Week 1: Flows and maps
- Trajectories Read quickly all of Chapter 1 - do not worry if there are stretches that you do not understand yet. Study all of Chapter 2
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Flow visualized as an iterated mapping
Discrete time dynamical systems arise naturally by recording the coordinates of the flow when a special event happens: the Poincare section method, key insight for much that is to follow.
Week 2: Linear stability
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There goes the neighborhood
So far we have concentrated on description of the trajectory of a single initial point. Our next task is to define and determine the size of a neighborhood, and describe the local geometry of the neighborhood by studying the linearized flow. What matters are the expanding directions. -
Cycle stability
If a flow is smooth, in a sufficiently small neighborhood it is essentially linear. 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 to understanding nonlinear flows. Linear charts are the key tool of differential geometry, general relativity, etc, so we are in good company.
Week 3: Linear stability
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Stability exponents are invariants of dynamics
We prove that (1) Floquet multipliers are the same everywhere along a cycle, and (b) that they are invariant under any smooth coordinate transformation.
Week 6: Charting the state space
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Qualitative dynamics, for pedestrians
Qualitative properties of a flow partition the state space in a topologically invariant way.
Week 7: Stretch, fold, prune
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The spatial ordering of trajectories from the time ordered itineraries
Qualitative dynamics: (1) temporal ordering, or itinerary with which a trajectory visits state space regions and (2) the spatial ordering between trajectory points, the key to determining the admissibility of an orbit with a prescribed itinerary. Kneading theory. -
Qualitative dynamics, for cyclists
Dynamical partitioning of a plane. Stable/unstable invariant manifolds, and how they partition the state space in intrinsic, topologically invariant manner. Henon map is the simplest example.
Week 8: Fixed points, and how to get them
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Finding cycles
Why nobody understands anybody? The bane of night fishing - plus how to find all possible orbits by (gasp!) thinking. -
Finding cycles; long cycles, continuous time cycles
Multi-shooting; d-dimensional flows; continuous-time flows.