A Survey of Methods for Computing (Un)stable Manifolds of Vector Fields

Krauskopf, Bernd; Osinga, Hinke M.; Doedel, Eusebius J.; Henderson, Michael E.; Guckenheimer, John; Vladimirsky, Alexander; Dellnitz, Michael; Junge, Oliver

doi:10.1142/s0218127405012533

Cited by 232 publications

(156 citation statements)

References 33 publications

Supporting

Mentioning

153

Contrasting

Unclassified

Order By: Relevance

“…While this method yields satisfactory results for the problem at hand, we mention that more advanced and accurate continuation methods are available for the computation of the manifold, for an overview of such methods see [74].…”

mentioning

confidence: 99%

Basins of attraction for chimera states

Martens

Panaggio

Abrams³

2016

New J. Phys.

View full text Add to dashboard Cite

Chimera states-curious symmetry-broken states in systems of identical coupled oscillatorstypically occur only for certain initial conditions. Here we analyze their basins of attraction in a simple system comprised of two populations. Using perturbative analysis and numerical simulation we evaluate asymptotic states and associated destination maps, and demonstrate that basins form a complex twisting structure in phase space. Understanding the basins' precise nature may help in the development of control methods to switch between chimera patterns, with possible technological and neural system applications.

show abstract

mentioning

confidence: 99%

Basins of attraction for chimera states

Martens

Panaggio

Abrams³

2016

New J. Phys.

View full text Add to dashboard Cite

show abstract

“…Many approaches have been formulated [Krauskopf & Osinga, 1999;Henderson, 2003;Dellnitz & Hohmann, 1997;Doedel et al, 1991aDoedel et al, , 1991bGuckenheimer & Vladimirsky, 2004]. A good review article summarizing a variety of methods is [Krauskopf et al, 2005]. Each of these methods was used to compute the "Lorenz Manifold," the two-dimensional stable manifold of the origin in the Lorenz system, which is, in effect, a three-dimensional problem.…”

Section: Methodsmentioning

confidence: 99%

“…Numerically, these manifolds are found by tracking solutions of System (4) backwards in time and triangulating to approximate the complete separatrix. A review of techniques that other authors have employed is available [Krauskopf et al, 2005].…”

Section: -3mentioning

confidence: 99%

Geometry of Basins of Attraction and Heteroclinic Connections in Coupled Bistable Systems

Lyons

Mahaffy

Wang

et al. 2014

Int. J. Bifurcation Chaos

View full text Add to dashboard Cite

The fundamental principle of bistability is widely used across various disciplines, including biology, chemistry, mechanics, physics, electronics and materials science. As the need for more powerful, efficient and sensitive complex-engineered systems grow, networks of coupled bistable systems have gained significant attention in recent years. Modeling and analysis of such higher-dimensional systems is usually focused on finding conditions for the existence and stability of typical invariant sets, i.e. steady states, periodic solutions and chaotic sets. High-dimensionality leads to complex patterns of collective behavior. Which type of behavior is exhibited by a network depends greatly on the initial conditions. Thus, it is also important to study the geometric structure and evolution of the basins of attraction of such patterns. In this manuscript, a complete study of the basins of attraction of a ring of bistable systems, coupled unidirectionally, is presented. 3D visualizations are included to aid the discussion of the changes in the basins of attraction as the coupling parameter varies. The results are broad enough that they can be applied to a wide range of systems with similar coupling topologies.

show abstract

“…(II.4) If kr 4 < a T < 0, there are another two interior steady states where 27) and the remaining the others are the same as in (23) and (24) by replacing the subscript 1 by 4 and 5.…”

Section: Propositionmentioning

confidence: 99%

On the long run sustainability of small jurisdictions

2020

View full text Add to dashboard Cite

In this paper, we analyse the long run sustainability of a small jurisdiction open to capital flows. Policy instruments used to reach an efficient steady state (when possible) are capital taxes and attractive public infrastructures. We first show that there exists one long run efficient equilibrium which is saddle-point stable. If capital mobility is high enough, the small economy is always able to converge to a long run efficient state by a one-shot adjustment of the share of taxes allocated to infrastructure. If capital mobility is too low, convergence to long run efficiency may be impossible. In this case, an eventual economic collapse can occur.

show abstract

A Survey of Methods for Computing (Un)stable Manifolds of Vector Fields

Cited by 232 publications

References 33 publications

Basins of attraction for chimera states

Basins of attraction for chimera states

Geometry of Basins of Attraction and Heteroclinic Connections in Coupled Bistable Systems

On the long run sustainability of small jurisdictions

Contact Info

Product

Resources

About