Nonlinear dynamics: Chaos, and what to do about it?
开始时间: 04/22/2022
持续时间: 8 weeks
课程主页: https://www.coursera.org/course/chaosbook
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The theory developed here (that you will not find in any other course :) has much in common with (and complements)
statistical mechanics and field theory courses; partition functions and
transfer operators are applied to computation of observables and
spectra of chaotic systems.
Nonlinear dynamics I: Geometry of chaos (this course)
- Topology of flows - how to enumerate orbits, Smale horseshoes
- Dynamics, quantitative - periodic orbits, local stability
- Role of symmetries in dynamics
Nonlinear dynamics II: Chaos rules (second course)
- Transfer operators - statistical distributions in dynamics
- Spectroscopy of chaotic systems
- dynamical zeta functions
- Dynamical theory of turbulence
The course is in part an advanced seminar in nonlinear dynamics, aimed at
PhD students, postdoctoral fellows and advanced undergraduates in
physics, mathematics, chemistry and engineering.
课程大纲
- Trajectories
Read quickly all of Chapter 1 - do not worry if there are stretches
that you do not understand yet.
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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.
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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.
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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.
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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.
-
Pinball wizzard
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.
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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
-
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.
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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.
-
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.
-
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 paramete
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.
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Qualitative dynamics, for pedestrians
Qualitative properties of
a flow partition the state space in a topologically invariant way.
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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.
-
Finding cycles
Why nobody understands anybody? The bane of night fishing - plus
how to find all possible orbits by (gasp!) thinking.
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Finding cycles; long cycles, continuous time cycles
Multi-shotting; d-dimensional flows; continuous-time flows.
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