13 research outputs found
Symplectomorphisms and discrete braid invariants
Area and orientation preserving diffeomorphisms of the standard 2-disc,
referred to as symplectomorphisms of , allow decompositions in
terms of positive twist diffeomorphisms. Using the latter decomposition we
utilize the Conley index theory of discrete braid classes as introduced in
[Ghrist et al., C. R. Acad. Sci. Paris S\'er. I Math., 331(11), 2000, Invent.
Math., 152(2), 2003] in order to obtain a Morse type forcing theory of periodic
points: a priori information about periodic points determines a mapping class
which may force additional periodic points.Comment: 31 pages, in print in Journal of Fixed Point Theory and Application
Existence of periodic solutions of the FitzHugh-Nagumo equations for an explicit range of the small parameter
The FitzHugh-Nagumo model describing propagation of nerve impulses in axon is
given by fast-slow reaction-diffusion equations, with dependence on a parameter
representing the ratio of time scales. It is well known that for all
sufficiently small the system possesses a periodic traveling wave.
With aid of computer-assisted rigorous computations, we prove the existence of
this periodic orbit in the traveling wave equation for an explicit range
. Our approach is based on a novel method of
combination of topological techniques of covering relations and isolating
segments, for which we provide a self-contained theory. We show that the range
of existence is wide enough, so the upper bound can be reached by standard
validated continuation procedures. In particular, for the range we perform a rigorous continuation based on
covering relations and not specifically tailored to the fast-slow setting.
Moreover, we confirm that for the classical interval
Newton-Moore method applied to a sequence of Poincar\'e maps already succeeds.
Techniques described in this paper can be adapted to other fast-slow systems of
similar structure
Rigorous numerics for PDEs with indefinite tail: existence of a periodic solution of the Boussinesq equation with time-dependent forcing
We consider the Boussinesq PDE perturbed by a time-dependent forcing. Even
though there is no smoothing effect for arbitrary smooth initial data, we are
able to apply the method of self-consistent bounds to deduce the existence of
smooth classical periodic solutions in the vicinity of 0. The proof is
non-perturbative and relies on construction of periodic isolating segments in
the Galerkin projections
Periodic orbits of the FitzHugh-Nagumo equations - a computer assisted proof
Non UBCUnreviewedAuthor affiliation: Jagiellonian UniversityGraduat
Metody indeksu Conleya na przestrzeniach warkoczy w niskowymiarowych układach dynamicznych
Proponujemy metodę zdefiniowania indeksu Conleya dla przestrzeni warkoczy dopuszczających negatywne przecięcia. Poprzez rozkład dyfeomorfizmów (względnie symplektomorfizmów) płaszczyzny o zwartym nośniku na dyfeomorfizmy (symplektomorfizmy) z własnością pozytywnego twistu sprowadzamy sytuację do przypadku wyłącznie z pozytywnymi przecięciami. Omawiamy metody obliczania indeksu oraz możliwe zastosowania.The purpose of this thesis is to propose a framework for developing a forcingtheory on spaces of arbitrary braids. We establish a method of decomposingcompactly supported diffeomorphisms of the plane to positive twist mappingsand relate them to the recently developed machinery of Conley indexfor positive braid diagrams. Area preserving case is also covered. We discusspossible dynamical implications in study of non-Lagrangian systemsvia Poincaré maps
Safe Multi-agent Learning via Trapping Regions
One of the main challenges of multi-agent learning lies in establishing
convergence of the algorithms, as, in general, a collection of individual,
self-serving agents is not guaranteed to converge with their joint policy, when
learning concurrently. This is in stark contrast to most single-agent
environments, and sets a prohibitive barrier for deployment in practical
applications, as it induces uncertainty in long term behavior of the system. In
this work, we apply the concept of trapping regions, known from qualitative
theory of dynamical systems, to create safety sets in the joint strategy space
for decentralized learning. We propose a binary partitioning algorithm for
verification that candidate sets form trapping regions in systems with known
learning dynamics, and a heuristic sampling algorithm for scenarios where
learning dynamics are not known. We demonstrate the applications to a
regularized version of Dirac Generative Adversarial Network, a
four-intersection traffic control scenario run in a state of the art
open-source microscopic traffic simulator SUMO, and a mathematical model of
economic competition