Meanders describe Jordan curves which cross a horizontal axis transversely,
at N points. They also relate to pairs of parenthesis stackings
(Dyck words), folded strips of stamps, to traces in Temperley-Lieb
algebras, certain Cartesian billiards, the shooting approach to nonlinear
second order boundary value problems in ordinary differential
equations (ODEs), and the geometry of global attractors of scalar partial
differential equations of parabolic type (PDEs). But, even their
asymptotic combinatorics remains open.
We illustrate how the PDE global attractors of ODE meanders turn
out to be regular Thom-Smale cell complexes, with an additional sign
Many examples will accompany our approach. We happily exploit
contributions by Pablo Castaneda, Anna Karnauhova, Phillipo Lappicy,
Alejandro López Nieto, Carlos Rocha, Matthias Wolfrum, Peter
Zograf, and many others. Theoretical flatworm chopping and grafting
by Angela Stevens et al., however, presents a surprise.