Computing One-Dimensional Stable Manifolds and Stable Sets of Planar Maps without the Inverse

England, James P.; Krauskopf, Bernd; Osinga, Hinke M.

doi:10.1137/030600131

Cited by 84 publications

(91 citation statements)

References 26 publications

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“…The map (6) has fixed points and periodic points, which correspond to periodic orbits of the associated vector field. If they are saddles then these points have stable and unstable invariant sets, which are the generalisations of stable and unstable manifolds to the context of noninvertible maps; see, for example, [16,17,32] for more details. Points on the stable set W s ( p) of a saddle periodic point p converge to p under iteration of f k where k is the (minimal) period of p; note that k = 1 if p is a fixed point.…”

Section: Wild Chaos In a Lorenz-type System Of Dimension Fivementioning

confidence: 99%

Chaos and Wild Chaos in Lorenz-Type Systems

Osinga

Krauskopf

Hittmeyer

2014

Springer Proceedings in Mathematics &Amp; Statistics

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This contribution provides a geometric perspective on the type of chaotic dynamics that one finds in the original Lorenz system and in a higher-dimensional Lorenz-type system. The latter provides an example of a system that features robustness of homoclinic tangencies; one also speaks of 'wild chaos' in contrast to the 'classical chaos' where homoclinic tangencies accumulate on one another, but do not occur robustly in open intervals in parameter space. Specifically, we discuss the manifestation of chaotic dynamics in the three-dimensional phase space of the Lorenz system, and illustrate the geometry behind the process that results in its description by a one-dimensional noninvertible map. For the higher-dimensional Lorenz-type system, the corresponding reduction process leads to a two-dimensional noninvertible map introduced in 2006 by Bamón, Kiwi, and Rivera-Letelier [arXiv 0508045] as a system displaying wild chaos. We present the geometric ingredientsin the form of different types of tangency bifurcations-that one encounters on the route to wild chaos.

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Section: Wild Chaos In a Lorenz-type System Of Dimension Fivementioning

confidence: 99%

Chaos and Wild Chaos in Lorenz-Type Systems

Osinga

Krauskopf

Hittmeyer

2014

Springer Proceedings in Mathematics &Amp; Statistics

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“…As an example, one can show that the two fixed points are saddle points for κ = 1, q = −0.6, g = 0.1, and ω = 0.1, To compute the growth of the stable and unstable manifolds we use the search circle algorithm [11,12]. This algorithm is implemented in the dynamical system software (DSTOOL) [13].…”

Section: Homoclinic Orbitsmentioning

confidence: 99%

Study of Localized Solutions of the Nonlinear Discrete Model for Dipolar BEC in an Optical Lattice by the Homoclinic Orbit Method

Messikh

Umarov

2013

J. Phys.: Conf. Ser.

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Abstract. We use homoclinic orbits to find solutions of a dynamical system of the dipolar Bose Einstein Condensate (BEC) in a deep optical lattice. The equation of motion is transformed to a two-dimensional map and its homoclinic orbits are computed numerically. Each homoclinic orbit leads to a different solution. These different solutions lead to different types of solitons. We also analyse the stability of the solutions.

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“…These have been "grown" numerically up to a finite length using the method of Krauskopf and Osinga 27 (see also England et al 28 ). As expected in a generic 3D mapping, the manifolds do not describe regular separatrices but have degenerated into heteroclinic tangles.…”

Section: A Structure Of the Basic Fieldmentioning

confidence: 99%

A generalized flux function for three-dimensional magnetic reconnection

Yeates

Hornig

2011

Physics of Plasmas

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The definition and measurement of magnetic reconnection in three-dimensional magnetic fields with multiple reconnection sites is a challenging problem, particularly in fields lacking null points. We propose a generalization of the familiar two-dimensional concept of a magnetic flux function to the case of a three-dimensional field connecting two planar boundaries. Using hyperbolic fixed points of the field line mapping, and their global stable and unstable manifolds, we define a unique flux partition of the magnetic field. This partition is more complicated than the corresponding (well-known) construction in a two-dimensional field, owing to the possibility of heteroclinic points and chaotic magnetic regions. Nevertheless, we show how the partition reconnection rate is readily measured with the generalized flux function. We relate our partition reconnection rate to the common definition of three-dimensional reconnection in terms of integrated parallel electric field. An analytical example demonstrates the theory, and shows how the flux partition responds to an isolated reconnection event.

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Computing One-Dimensional Stable Manifolds and Stable Sets of Planar Maps without the Inverse

Cited by 84 publications

References 26 publications

Chaos and Wild Chaos in Lorenz-Type Systems

Chaos and Wild Chaos in Lorenz-Type Systems

Study of Localized Solutions of the Nonlinear Discrete Model for Dipolar BEC in an Optical Lattice by the Homoclinic Orbit Method

A generalized flux function for three-dimensional magnetic reconnection

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