135 research outputs found
Weak index pairs and the Conley index for discrete multivalued dynamical systems
Motivated by the problem of reconstructing dynamics from samples we revisit
the Conley index theory for discrete multivalued dynamical systems. We
introduce a new, less restrictive definition of the isolating neighbourhood. It
turns out that then the main tool for the construction of the index, i.e. the
index pair, is no longer useful. In order to overcome this obstacle we use the
concept of weak index pairs
Discretization strategies for computing Conley indices and Morse decompositions of flows
Conley indices and Morse decompositions of flows can be found by using
algorithms which rigorously analyze discrete dynamical systems. This usually
involves integrating a time discretization of the flow using interval
arithmetic. We compare the old idea of fixing a time step as a parameters to a
time step continuously varying in phase space. We present an example where this
second strategy necessarily yields better numerical outputs and prove that our
outputs yield a valid Morse decomposition of the given flow
Conley-Morse-Forman theory for combinatorial multivector fields on Lefschetz complexes
We introduce combinatorial multivector fields, associate with them multivalued dynamics and study their topological features. Our combinatorial multivector fields generalize combinatorial vector fields of Forman. We define isolated invariant sets, Conley index, attractors, repellers and Morse decompositions. We provide a topological characterization of attractors and repellers and prove Morse inequalities. The generalization aims at algorithmic analysis of dynamical systems through combinatorialization of flows given by differential equations and through sampling dynamics in physical and numerical experiments. We provide a prototype algorithm for such applications
\v{C}ech-Delaunay gradient flow and homology inference for self-maps
We call a continuous self-map that reveals itself through a discrete set of
point-value pairs a sampled dynamical system. Capturing the available
information with chain maps on Delaunay complexes, we use persistent homology
to quantify the evidence of recurrent behavior. We establish a sampling theorem
to recover the eigenspace of the endomorphism on homology induced by the
self-map. Using a combinatorial gradient flow arising from the discrete Morse
theory for \v{C}ech and Delaunay complexes, we construct a chain map to
transform the problem from the natural but expensive \v{C}ech complexes to the
computationally efficient Delaunay triangulations. The fast chain map algorithm
has applications beyond dynamical systems.Comment: 22 pages, 8 figure
CAPD::RedHom - Reduction heuristics for homology algorithms
We present an efficient software package for computing homology of sets, maps and filtrations represented as cubical, simplicial and regular CW complexes. The core homology computation is based on classical Smith diagonalization, but the efficiency of our approach comes from applying several geometric and algebraic reduction techniques combined with smart implementation
Creating Semiflows on Simplicial Complexes from Combinatorial Vector Fields
Combinatorial vector fields on simplicial complexes as introduced by Robin
Forman have found numerous and varied applications in recent years. Yet, their
relationship to classical dynamical systems has been less clear. In recent work
it was shown that for every combinatorial vector field on a finite simplicial
complex one can construct a multivalued discrete-time dynamical system on the
underlying polytope X which exhibits the same dynamics as the combinatorial
flow in the sense of Conley index theory. However, Forman's original
description of combinatorial flows appears to have been motivated more directly
by the concept of flows, i.e., continuous-time dynamical systems. In this
paper, it is shown that one can construct a semiflow on X which exhibits the
same dynamics as the underlying combinatorial vector field. The equivalence of
the dynamical behavior is established in the sense of Conley-Morse graphs and
uses a tiling of the topological space X which makes it possible to directly
construct isolating blocks for all involved isolated invariant sets based
purely on the combinatorial information.Comment: 57 pages, 12 figure
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