PDIFF piecewise-smooth manifolds

From: ctm@bosco berkeley edu (C. T. McMullen)
Newsgroups: sci math research
Subject: Re: PL and DIFF manifolds: a question
Date: 21 Aug 1997 18:26:12 GMT
> It is often stated
>that DIFF manifolds are a particular case of PL manifolds, but this fact
>is not obvious. How does one go about proving it?
A DIFF manifold carries a canonical PL structure, which can
be obtained by
1) introducing a Riemannian metric, then
2) choosing a fine triangulation whose 1-skeleton is piecewise
geodesic, with triangles of bounded geometry; and
3) using barycentric coordinates (relative to the given metric)
to get new, nearby simplices, each endowed with a
linear structure.
It is not quite true that a DIFF manifold *is* a PL manifold;
one should introduce the intermediate category of
PDIFF, piecewise-smooth manifolds, which clearly contains DIFF
manifolds. A clear discussion appears in the book by
Thurston, “Three-Dimensional Geometry and Topology”, PUP, 1997.
-C. McMullen
Piecewise smooth dynamical systems
Knew you must exist.
“(a) A representative grazing orbit and (b) a typical bifurcation diagram where the grazing shown in panel (a) causes the sudden transition from a periodic to a chaotic attractor at μ≈−0.315 . Here μ represents the position of the constraint in phase space.”
Arnold tongues in piecewise-smooth, continuous maps.
Oscillators subject to external forcing may be entrained to a rational multiple of the forcing frequency. Regions of parameter space for which this occurs are known as Arnold tongues. The figure below shows Arnold tongues for a piecewise-linear, continuous map (different colours correspond to different periods). We have a good understanding of the distinctive chain structure of the tongues but it remains to properly explain many other features, such as the small sequence visible in the lower right and the creation of chaotic solutions.
IF-THEN. Choices. Impacts. Cases. Branches. Decisions. This-or-That showing both. All from finding about “piecewise”.
“The results are illustrated with impact oscillators, relay control, automated balancing control, predator-prey systems, ocean circulation, and the McKean and Wilson-Cowan neuron models.”

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