Two-dimensional global manifolds of vector fields

Krauskopf, Bernd; Osinga, Hinke M.

doi:10.1063/1.166450

Cited by 85 publications

(103 citation statements)

References 12 publications

(12 reference statements)

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“…Recent works have presented the computation of two-dimensional invariant manifolds. [39][40][41][42] The computation of stable manifolds of much higher dimensions, as in our case, is still a challenging task. However, we can approximate one-dimensional projections of segments of the stable manifold ͑SM͒ of the mediating orbit close to the tangency points using the stable manifold of SCS, to determine the local boundary between regions B and S, as shown by the dashed lines in Fig.…”

Section: Fig 4 Three-dimensional Projectionmentioning

confidence: 98%

Analysis of chaotic saddles in high-dimensional dynamical systems: The Kuramoto–Sivashinsky equation

Rempel

Chian

Macau

et al. 2004

Chaos: An Interdisciplinary Journal of Nonlinear Science

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This paper presents a methodology to study the role played by nonattracting chaotic sets called chaotic saddles in chaotic transitions of high-dimensional dynamical systems. Our methodology is applied to the Kuramoto-Sivashinsky equation, a reaction-diffusion partial differential equation. The paper describes a novel technique that uses the stable manifold of a chaotic saddle to characterize the homoclinic tangency responsible for an interior crisis, a chaotic transition that results in the enlargement of a chaotic attractor. The numerical techniques explained here are important to improve the understanding of the connection between low-dimensional chaotic systems and spatiotemporal systems which exhibit temporal chaos and spatial coherence. © 2004 American Institute of Physics. ͓DOI: 10.1063/1.1759297͔In the past decades chaos theory has been revealed as a powerful way to explain the physics of low-dimensional dynamical systems, that is, dynamical systems with a small number of state variables. However, most problems of practical interest in physics and engineering are described by partial differential equations "PDEs…, and it is still unclear in which situations the physical mechanisms responsible for the onset of chaos in low-dimensional systems are applicable to systems described by PDEs. Several authors have tried to develop a dynamical systems theory for high-dimensional dynamical systems, frequently described by sets of coupled ordinary differential equations obtained as approximations to the original PDEs. 1-7 The simplest approach consists in first ''borrowing'' the tools developed for low-dimensional dynamical systems and applying them to spatially extended systems in regimes that display characteristics of lowdimensional chaos. Then, it is possible to develop the tools to be used in more complex regimes. In this paper we try to improve the understanding of the connection between low-dimensional chaotic systems and a certain class of spatiotemporal systems: the one that exhibits temporal chaos and spatial coherence. We focus on the development of a methodology to study the role played by nonattracting chaotic sets called chaotic saddles in chaotic transitions of high-dimensional dynamical systems.

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Section: Fig 4 Three-dimensional Projectionmentioning

confidence: 98%

Analysis of chaotic saddles in high-dimensional dynamical systems: The Kuramoto–Sivashinsky equation

Rempel

Chian

Macau

et al. 2004

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…Its formulation in terms of geodesic level sets further develops previous work in [22] for the specific case when k = 2 and n = 3. The method is now implemented for the case k = 2 and any n. This includes the case of two-dimensional stable and unstable manifolds of periodic orbits.…”

mentioning

confidence: 75%

“…A trivial case is that the manifold converges to a regular attractor, such as an equilibrium or periodic orbit. Our implementation deals with this case by growing the manifold with different speeds in different directions, as is explained in [22].…”

Section: Background and Conceptsmentioning

confidence: 99%

“…A number of algorithms have been devised for this task [4,5,7,9,10,12,16,17,20,22,25,26]. The method in [4,5] is special as it computes a box covering of W u (x 0 ); it has been implemented for arbitrary n and is independent of k. All other algorithms are implemented specifically for the case of two-dimensional manifolds and produce a triangulation of the manifold that is built up by starting near the saddle point.…”

mentioning

confidence: 99%

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Computing Geodesic Level Sets on Global (Un)stable Manifolds of Vector Fields

Krauskopf¹,

Osinga²

2003

SIAM J. Appl. Dyn. Syst.

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104

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Abstract. Many applications give rise to dynamical systems in the form of a vector field with a phase space of moderate dimension. Examples are the Lorenz equations, mechanical and other oscillators, and models of spiking neurons. The global dynamics of such a system is organized by the stable and unstable manifolds of the saddle points, of the saddle periodic orbits, and, more generally, of all compact invariant manifolds of saddle type. Except in very special circumstances the (un)stable manifolds are global objects that cannot be found analytically but need to be computed numerically. This is a nontrivial task when the dimension of the manifold is larger than one. In this paper we present an algorithm to compute the k-dimensional unstable manifold of an equilibrium or periodic orbit (or a more general normally hyperbolic invariant manifold) of a vector field with an n-dimensional phase space, where 1 < k < n. Stable manifolds are computed by considering the flow for negative time. The key idea is to view the unstable manifold as a purely geometric object, hence disregarding the dynamics on the manifold, and compute it as a list of approximate geodesic level sets, which are (topological) (k − 1)-spheres. Starting from a (k − 1)-sphere in the linear eigenspace of the equilibrium or periodic orbit, the next geodesic level set is found in a local (and changing) coordinate system given by hyperplanes perpendicular to the last geodesic level set. In this setup the mesh points defining the approximation of the next geodesic level set can be found by solving boundary value problems. By appropriately adding or removing mesh points it is ensured that the mesh that represents the computed manifold is of a prescribed quality.The algorithm is presently implemented to compute two-dimensional manifolds in a phase space of arbitrary dimension. In this case the geodesic level sets are topological circles and the manifold is represented as a list of bands between consecutive level sets. We use color to distinguish between consecutive bands or to indicate geodesic distance from the equilibrium or periodic orbit, and we also show how geodesic level sets change with increasing geodesic distance. This is very helpful when one wants to understand the often very complicated embeddings of two-dimensional (un)stable manifolds in phase space.The properties and performance of our method are illustrated with several examples, including the stable manifold of the origin of the Lorenz system, a two-dimensional stable manifold in a fourdimensional phase space arising in a problem in optimal control, and a stable manifold of a periodic orbit that is a Möbius strip. Each illustration is accompanied by an animation (supplied with this paper).

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“…For example, bifurcation to chaos can be identified by tracing the evolution of stable and unstable manifolds in the phase space with parameters. Recently, some algorithms for computing two-dimensional stable and unstable manifolds in a three-dimensional phase space are proposed [8,9,10] and applied to visualize the structure of chaos [11]. Since UPOs are closely related to invariant manifolds, a basic issue in their relations is describing how local invariant manifolds rotate around UPOs in the phase space.…”

Section: Introductionmentioning

confidence: 99%

Rotation Numbers of Invariant Manifolds Around Unstable Periodic Orbits for the Diamagnetic Kepler Problem

2008

Fractals

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In this paper, a method to construct topological template in terms of symbolic dynamics for the diamagnetic Kepler problem is proposed. To confirm the topological template, rotation numbers of invariant manifolds around unstable periodic orbits in a phase space are taken as an object of comparison. The rotation numbers are determined from the definition and connected with symbolic sequences encoding the periodic orbits in a reduced Poincaré section. Only symbolic codes with inverse ordering in the forward mapping can contribute to the rotation of invariant manifolds around the periodic orbits. By using symbolic ordering, the reduced Poincaré section is constricted along stable manifolds and a topological template, which preserves the ordering of forward sequences and can be used to extract the rotation numbers, is established. The rotation numbers computed from the topological template are the same as those computed from their original definition.

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Two-dimensional global manifolds of vector fields

Cited by 85 publications

References 12 publications

Analysis of chaotic saddles in high-dimensional dynamical systems: The Kuramoto–Sivashinsky equation

Analysis of chaotic saddles in high-dimensional dynamical systems: The Kuramoto–Sivashinsky equation

Computing Geodesic Level Sets on Global (Un)stable Manifolds of Vector Fields

Rotation Numbers of Invariant Manifolds Around Unstable Periodic Orbits for the Diamagnetic Kepler Problem

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