Nonlinear Dynamics 1: Geometry of Chaos

1. 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

1. 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.

1. 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.
2. 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.

1. 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.
2. Pinball wizard
   The dynamics that we have the best intuitive grasp on is the dynamics of billiards. For billiards, discrete time is altogether natural; a particle moving through a billiard suffers a sequence of instantaneous kicks, and executes simple motion in between.

1. Discrete symmetries of dynamics
   What is a symmetry of laws of motion? The families of symmetry-related full state space cycles are replaced by fewer and often much shorter "relative" cycles, and the notion of a prime periodic orbit is replaced by the notion of a "relative" periodic orbit, the shortest segment that 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-disk billiard, the simpler problem - symmetry reduction of 1d dynamics that is equivariant under a reflection, the most common symmetry in applications - seems to baffle everyone. So here is a step-by-step walk through to this simplest of all symmetry reductions.

1. Continuous symmetries of dynamics
   Symmetry reduction: If the symmetry is continuous, the interesting dynamics unfolds on a lower-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 orbit to pick out one representative for the whole family. For continuous spatial symmetries, this is achieved by slicing. And then there is dicing.

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

1. 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.
2. 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.

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