Using Lin's method to solve Bykov's problems

Knobloch, Jürgen; Lamb, Jeroen S. W.; Webster, Kevin

doi:10.1016/j.jde.2014.06.006

Cited by 22 publications

(52 citation statements)

References 30 publications

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“…In the same way as in [18] we make some further hypotheses ensuring that the T-point has codimension two:…”

Section: Ses (H 4)-(h 8)mentioning

confidence: 99%

“…Meanwhile T-points have been found to appear in many further applications such as Kuramoto-Sivashinsky systems, electronic oscillators, semiconductor lasers, magneto convection, and travelling waves in reaction-diffusion dynamics. For precise references concerning these applications we refer to [18].…”

Section: Introductionmentioning

confidence: 99%

“…[3] and references therein. More references can also be found in [18] and [14]. It turns out that the complexity of the nonwandering nearby dynamics depends to a large extent on the leading eigenvalues of the equilibria.…”

Section: Introductionmentioning

confidence: 99%

“…It turns out that the complexity of the nonwandering nearby dynamics depends to a large extent on the leading eigenvalues of the equilibria. If the leading eigenvalues of both equilibria are real then the corresponding dynamics is rather tame, whereas shift dynamics occurs in the presence of complex leading eigenvalues [3,18]. However, in [3] it is also claimed that double T-points give rise to shift dynamics also in the case of real leading eigenvalues.…”

Section: Introductionmentioning

confidence: 99%

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Shift dynamics near non-elementary T-points with real eigenvalues

Knobloch

Lamb

Webster

2017

Journal of Difference Equations and Applications

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“…In the same way as in [18] we make some further hypotheses ensuring that the T-point has codimension two:…”

Section: Ses (H 4)-(h 8)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Shift dynamics near non-elementary T-points with real eigenvalues

Knobloch

Lamb

Webster

2017

Journal of Difference Equations and Applications

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“…There are many other such T-points in the Lorenz system, corresponding to increasingly complicated windings of W u (0) around p ± before connecting to p ± ; see [18] for details. The bifurcation structure near a T-point has been studied, for example, in [40,43,59,66] but not in the context of the loss of the foliation condition. The bifurcation curves of the main homoclinic and heteroclinic bifurcations of p ± both terminate at a T-point.…”

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confidence: 99%

Finding First Foliation Tangencies in the Lorenz System

Creaser¹,

Krauskopf²,

Osinga³

2017

SIAM J. Appl. Dyn. Syst.

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Abstract. Classical studies of chaos in the well-known Lorenz system are based on reduction to the onedimensional Lorenz map, which captures the full behavior of the dynamics of the chaotic Lorenz attractor. This reduction requires that the stable and unstable foliations in a particular Poincaré section are transverse locally near the chaotic Lorenz attractor. We study when this so-called foliation condition fails for the first time and the classic Lorenz attractor becomes a quasi-attractor. This transition is characterized by the creation of tangencies between the stable and unstable foliations and the appearance of hooked horseshoes in the Poincaré return map. We consider how the threedimensional phase space is organized by the global invariant manifolds of saddle equilibria and saddle periodic orbits-before and after the loss of the foliation condition. We compute these global objects as families of orbit segments, which are found by setting up a suitable two-point boundary value problem (BVP). We then formulate a multi-segment BVP to find the first tangency between the stable foliation and the intersection curves in the Poincaré section of the two-dimensional unstable manifold of a periodic orbit. It is a distinct advantage of our BVP setup that we are able to detect and readily continue the locus of first foliation tangency in any plane of two parameters as part of the overall bifurcation diagram. Our computations show that the region of existence of the classic Lorenz attractor is bounded in each parameter plane. It forms a slanted (unbounded) cone in the three-parameter space with a curve of terminal-point, or T-point, bifurcations on the locus of first foliation tangency; we identify the tip of this cone as a codimension-three T-point-Hopf bifurcation point, where the curve of T-point bifurcations meets a surface of Hopf bifurcation. Moreover, we are able to find other first foliation tangencies for larger values of the parameters that are associated with additional T-point bifurcations: each tangency adds an extra twist to the central region of the quasi-attractor.

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Unfolding a Bykov Attractor: From an Attracting Torus to Strange Attractors

Rodrigues

2020

J Dyn Diff Equat

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Using Lin's method to solve Bykov's problems

Cited by 22 publications

References 30 publications

Shift dynamics near non-elementary T-points with real eigenvalues

Shift dynamics near non-elementary T-points with real eigenvalues

Finding First Foliation Tangencies in the Lorenz System

Unfolding a Bykov Attractor: From an Attracting Torus to Strange Attractors

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