29 research outputs found
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
Continuous and piecewise affine Lyapunov functions using the Yoshizawa construction
6 p.International audienceWe present a novel numerical technique for the computation of a Lyapunov function for nonlinear systems with an asymptotically stable equilibrium point. Our proposed approach constructs a continuous piecewise affine (CPA) function given a suitable partition of the state space, called a triangulation, and values at the vertices of the triangulation. The vertex values are obtained from a Lyapunov function in a classical converse Lyapunov theorem and verification that the obtained CPA function is a Lyapunov function is shown to be equivalent to verification of several simple inequalities. Furthermore, by refining the triangulation, we show that it is always possible to construct a CPA Lyapunov function. Numerical examples are presented demonstrating the effectiveness of the proposed method
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Analysing dynamical systems towards computing complete Lyapunov functions
Ordinary differential equations arise in a variety of applications, including e.g. climate systems, and can exhibit complicated dynamical behaviour. Complete Lyapunov functions can capture this behaviour by dividing the phase space into the chain-recurrent set, determining the long-time behaviour, and the transient part, where solutions pass through. In this paper, we present an algorithm to construct complete Lyapunov functions. It is based on mesh-free numerical approximation and uses the failure of convergence in certain areas to determine the chain-recurrent set. The algorithm is applied to three examples and is able to determine attractors and repellers, including periodic orbits and homoclinic orbits
Simplicial Multivalued Maps and the Witness Complex for Dynamical Analysis of Time Series
Topology based analysis of time-series data from dynamical systems is
powerful: it potentially allows for computer-based proofs of the existence of
various classes of regular and chaotic invariant sets for high-dimensional
dynamics. Standard methods are based on a cubical discretization of the
dynamics and use the time series to construct an outer approximation of the
underlying dynamical system. The resulting multivalued map can be used to
compute the Conley index of isolated invariant sets of cubes. In this paper we
introduce a discretization that uses instead a simplicial complex constructed
from a witness-landmark relationship. The goal is to obtain a natural
discretization that is more tightly connected with the invariant density of the
time series itself. The time-ordering of the data also directly leads to a map
on this simplicial complex that we call the witness map. We obtain conditions
under which this witness map gives an outer approximation of the dynamics, and
thus can be used to compute the Conley index of isolated invariant sets. The
method is illustrated by a simple example using data from the classical H\'enon
map.Comment: laTeX, 9 figures, 32 page
Finite Resolution Dynamics
We develop a new mathematical model for describing a dynamical system at
limited resolution (or finite scale), and we give precise meaning to the notion
of a dynamical system having some property at all resolutions coarser than a
given number. Open covers are used to approximate the topology of the phase
space in a finite way, and the dynamical system is represented by means of a
combinatorial multivalued map. We formulate notions of transitivity and mixing
in the finite resolution setting in a computable and consistent way. Moreover,
we formulate equivalent conditions for these properties in terms of graphs, and
provide effective algorithms for their verification. As an application we show
that the Henon attractor is mixing at all resolutions coarser than 10^-5.Comment: 25 pages. Final version. To appear in Foundations of Computational
Mathematic
Chain recurrence rates and topological entropy
We investigate the properties of chain recurrent, chain transitive, and chain
mixing maps (generalizations of the well-known notions of non-wandering,
topologically transitive, and topologically mixing maps). We describe the
structure of chain transitive maps. These notions of recurrence are defined
using \ep-chains, and the minimal lengths of these \ep-chains give a way to
measure recurrence time (chain recurrence and chain mixing times). We give
upper and lower bounds for these recurrence times and relate the chain mixing
time to topological entropy