Recent development in rigorous computational methods in dynamical systems

Arai, Zin; Kokubu, Hiroshi; Pilarczyk, Paweł

doi:10.1007/bf03186541

Cited by 13 publications

(6 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The past decade has seen structural advances in the techniques for the rigorous computer-aided study of systems of ordinary differential equations (ODEs), see e.g. [3,1,9] (and the references therein), as well as the survey paper [2]. Nevertheless, relatively little attention has gone to the study of solutions on unbounded domains for non-autonomous ODEs.…”

mentioning

confidence: 99%

Rigorous Computation of a Radially Symmetric Localized Solution in a Ginzburg--Landau Problem

Berg¹,

Groothedde²,

Williams³

2015

SIAM J. Appl. Dyn. Syst.

View full text Add to dashboard Cite

We present a rigorous numerical method for proving the existence of a localised radially symmetric solution for a Ginzburg-Landau type equation. This has a direct application to the problem of finding spots in the Swift-Hohenberg equation. The method is more generally applicable to finding radially symmetric solutions of stationary PDEs on the entire space. One can rewrite such a problem in the form of a singular ODE. We transform this ODE to a finite domain and use a Green's function approach to formulate an appropriate integral equation. We then construct a mapping whose fixed points coincide with solutions to the ODE, and show via computeraided analytic estimates that the mapping is contracting on a small neighbourhood of a numerically determined approximate solution.

show abstract

mentioning

confidence: 99%

Rigorous Computation of a Radially Symmetric Localized Solution in a Ginzburg--Landau Problem

Berg¹,

Groothedde²,

Williams³

2015

SIAM J. Appl. Dyn. Syst.

View full text Add to dashboard Cite

show abstract

“…But there is no attractor-repeller pair for ϕ in which both invariant sets are non-empty. 1 The example shows that the assumption that M p be invariant also for the flow is necessary in Lemma 3.19. Such a function τ can always be constructed when the flow has a limit cycle L. One just has to replace 2π by the period of the limit cycle and extend τ from L to the whole phase space Y using Tietze's extension theorem.…”

Section: Shift Equivalencesmentioning

confidence: 97%

“…While the numerical approximation of ordinary differential equations has a long history, the systematic study of how to compute time invariant structures began in the 1980s. Rigorous computations of these structures is an even more recent phenomenon (see [1,8] and references therein). These latter efforts can be roughly divided into two approaches: direct computation of invariant sets, e.g., periodic orbits, heteroclinic and homoclinic orbits, invariant manifolds, and a more indirect approach based on identification of isolating neighborhoods.…”

Section: Introductionmentioning

confidence: 99%

Discretization strategies for computing Conley indices and Morse decompositions of flows

Mischaikow¹,

Mrożek

Weilandt

2016

JCD

View full text Add to dashboard Cite

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.K.

show abstract

“…To compute the topological invariants such as the number of distinct objects in an image, the algorithms based on topology, a major area of mathematics [8]- [10], can be more effective than ad hoc algorithms. Fortunately, recent advances of the field of computational topology made it possible to compute topological invariants in an accessible way [11]- [18]. However, the inventive algorithms of computational topology mostly aimed at off-line computation on serial [19]- [22] or parallel system computers [23]- [25].…”

Section: Introductionmentioning

confidence: 99%

Real-Time Computing of Touch Topology via Poincare–Hopf Index

Miura

Nakada

2015

IEEE Access

View full text Add to dashboard Cite

While visual or tactile image data have been conventionally processed via filters or perceptronlike learning machines, the recent advances of computational topology may make it possible to successfully extract the global features from the local pixelwise data. In fact, some inventive algorithms have succeeded in computing the topological invariants, such as the number of objects or holes and irrespective of the shapes and positions of the touches. However, they are mostly offline algorithms aiming at big data. A real-time algorithm for computing topology is also needed for interactive applications such as touch sensors. Here, we propose a fast algorithm to compute the Euler characteristics of touch shapes by using the Poincare-Hopf index for each pixel. We demonstrate that our simple algorithm, implemented solely as logical operations in Arduino, correctly returns and updates the topological invariants of touches in real time.

show abstract

Recent development in rigorous computational methods in dynamical systems

Cited by 13 publications

References 37 publications

Rigorous Computation of a Radially Symmetric Localized Solution in a Ginzburg--Landau Problem

Rigorous Computation of a Radially Symmetric Localized Solution in a Ginzburg--Landau Problem

Discretization strategies for computing Conley indices and Morse decompositions of flows

Real-Time Computing of Touch Topology via Poincare–Hopf Index

Contact Info

Product

Resources

About