Saddle Invariant Objects and Their Global Manifolds in a Neighborhood of a Homoclinic Flip Bifurcation of  Case B

Giraldo, Andrus; Krauskopf, Bernd; Osinga, Hinke M.

doi:10.1137/16m1097419

Cited by 18 publications

(29 citation statements)

References 40 publications

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“…Inclination flip bifurcations have been linked to emerging stability windows of stable equilibria and periodic orbits in the Shimizu-Morioka system [67]. These numerous inclination flip bifurcations, of which there exist several types [40], and their role for the organization of nearby chaotic dynamics and stability windows can be studied in the spirit of recent investigations [3,30]. This is an interesting direction for future work.…”

Section: Further First Foliation Tangencies In Thementioning

confidence: 96%

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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Section: Further First Foliation Tangencies In Thementioning

confidence: 96%

Finding First Foliation Tangencies in the Lorenz System

Creaser¹,

Krauskopf²,

Osinga³

2017

SIAM J. Appl. Dyn. Syst.

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“…This type of bifurcation of a homoclinic orbit to a real hyperbolic saddle-a special trajectory that converges both in forward and backward time to the saddle equilibrium-occurs when a stable or unstable manifold, when followed along the homoclinic orbit, transitions from being orientable to being non-orientable, or vice versa. While such a change of orientability may occur in higher-dimensional phase spaces, the characterization of homoclinic flip bifurcations and their unfoldings have been studied in detail mostly for the lowest-dimensional case of a three-dimensional systems, both from a theoretical [10,18,19,20,22,31] and a numerical point of view [1,8,15,16,21].…”

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confidence: 99%

Computing connecting orbits to infinity associated with a homoclinic flip bifurcation

Giraldo¹,

Krauskopf²,

Osinga³

2020

Journal of Computational Dynamics

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We consider the bifurcation diagram in a suitable parameter plane of a quadratic vector field in R 3 that features a homoclinic flip bifurcation of the most complicated type. This codimension-two bifurcation is characterized by a change of orientability of associated two-dimensional manifolds and generates infinite families of secondary bifurcations. We show that curves of secondary n-homoclinic bifurcations accumulate on a curve of a heteroclinic bifurcation involving infinity.We present an adaptation of the technique known as Lin's method that enables us to compute such connecting orbits to infinity. We first perform a weighted directional compactification of R 3 with a subsequent blow-up of a non-hyperbolic saddle at infinity. We then set up boundary-value problems for two orbit segments from and to a common two-dimensional section: the first is to a finite saddle in the regular coordinates, and the second is from the vicinity of the saddle at infinity in the blown-up chart. The so-called Lin gap along a fixed one-dimensional direction in the section is then brought to zero by continuation. Once a connecting orbit has been found in this way, its locus can be traced out as a curve in a parameter plane.

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“…Simultaneous zeros of the gaps correspond to the global objects sought. Lin's method has been implemented also as a numerical technique for finding heteroclinic and homoclinic connections; see for instance [18,33,38,46]. More recently, the approach has been successfully applied in the slow-fast context; specifically, for detecting so-called connecting canard orbits arising as codimension-zero intersections between the twodimensional unstable manifold of a saddle-focus equilibrium and a two-dimensional repelling slow manifold in a model with a singular Hopf bifurcation [45].…”

mentioning

confidence: 99%

“…of the end points u a (1) and u r (0) in Σ onto n Z , respectively. Specifically, the boundary conditions (17) and (18) introduce the parameters β a and β r that represent these projections.…”

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A Lin's method approach for detecting all canard orbits arising from a folded node

Mujica¹,

Krauskopf²,

Osinga³

2017

Journal of Computational Dynamics

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Canard orbits are relevant objects in slow-fast dynamical systems that organize the spiraling of orbits nearby. In three-dimensional vector fields with two slow and one fast variables, canard orbits arise from the intersection between an attracting and a repelling two-dimensional slow manifold. Special points called folded nodes generate such intersections: in a suitable transverse two-dimensional section Σ, the attracting and repelling slow manifolds are counter-rotating spirals that intersect in a finite number of points. We present an implementation of Lin's method that is able to detect all of these intersection points and, hence, all of the canard orbits arising from a folded node. With a boundary-value-problem setup we compute orbit segments on each slow manifold up to Σ, where we require that the corresponding end points in Σ lie in a one-dimensional subspace known as the Lin space Z. The Lin space Z must be transverse to the slow manifolds and it remains fixed during the detection of canard orbits as zeros of the signed distance along Z. During the computation, a tangency of Z with one of the intersection curves in Σ may arise. To overcome this, we update the Lin space at an intermediate continuation step to detect a double tangency of Z to both curves in Σ, after which the canard detection is able to continue. Our method is demonstrated with the examples of the normal form for a folded node and of the Koper model.

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Saddle Invariant Objects and Their Global Manifolds in a Neighborhood of a Homoclinic Flip Bifurcation of Case B

Cited by 18 publications

References 40 publications

Finding First Foliation Tangencies in the Lorenz System

Finding First Foliation Tangencies in the Lorenz System

Computing connecting orbits to infinity associated with a homoclinic flip bifurcation

A Lin's method approach for detecting all canard orbits arising from a folded node

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