Bifurcations toN-homoclinic orbits andN-periodic orbits in vector fields

Kisaka, Masashi; Kokubu, Hiroshi; Oka, Hiroe

doi:10.1007/bf01053164

Cited by 79 publications

(70 citation statements)

References 12 publications

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“…In order to meet the boundary conditions in item (iii), we will need to insert x ± = q(ξ) + u ± (µ)(ξ) + v ± (ξ) into the nonlinear equation (6.1). We find that v ± must solve the equations 12) in which the nonlinearities M ± are given by…”

Section: Lin's Methods For Mfdesmentioning

confidence: 99%

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Lin's method and homoclinic bifurcations for functional differential equations of mixed type

Hupkes

Lunel²

2009

Indiana Univ. Math. J.

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We extend Lin's method for use in the setting of parameter-dependent nonlinear functional differential equations of mixed type (MFDEs). We show that the presence of M -homoclinic and M -periodic solutions that bifurcate from a prescribed homoclinic connection can be detected by studying a finite dimensional bifurcation equation. As an application, we describe the codimension two orbit-flip bifurcation in the setting of MFDEs.

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Section: Lin's Methods For Mfdesmentioning

confidence: 99%

“…The former of these has been analyzed by several authors [7,12] using Lyapunov-Schmidt techniques, that unfortunately break down when studying the orbit-flip bifurcation. However, employing the adaptation of Lin's method discussed above, Sandstede obtained a general description of this bifurcation for ODEs in [16].…”

Section: Introductionmentioning

confidence: 99%

Lin's method and homoclinic bifurcations for functional differential equations of mixed type

Hupkes

Lunel²

2009

Indiana Univ. Math. J.

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“…See [6] for homoclinic bifurcation problems in generic systems with resonant eigenvalues. Condition (H 4) excludes an orbit flip condition (see [34]), while (2.1) is a generalization of the noninclination flip condition (see [23,19]). Note that these conditions are automatically satisfied if Fix G γ is two-dimensional, independent of the dimension of the entire space.…”

Section: Setting and Main Resultsmentioning

confidence: 99%

Bifurcation from codimension one relative homoclinic cycles

Homburg

Jukes

Knobloch

et al. 2011

Trans. Amer. Math. Soc.

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Abstract. We study bifurcations of relative homoclinic cycles in flows that are equivariant under the action of a finite group. The relative homoclinic cycles we consider are not robust, but have codimension one. We assume real leading eigenvalues and connecting trajectories that approach the equilibria along leading directions. We show how suspensions of subshifts of finite type generically appear in the unfolding. Descriptions of the suspended subshifts in terms of the geometry and symmetry of the connecting trajectories are provided.

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“…homoclinic orbits have been investigated by Homburg, Kokubu and Naudot [6], Kisaka, Kokubu and Oka [10,11] and Naudot [12]. For certain configurations of eigenvalues at the saddle point these homoclinic flip bifurcations cannot only cause homoclinic-doubling, but may also generate chaos; see Section 2 for more details.…”

Section: Introductionmentioning

confidence: 99%

“…The structure of the unfolding of a bifurcation of type ✬ was found in [10,11,6] for the inclination flip, and in [9] for the orbit flip. For an overview of the fan depicted in Figure 5 and the other possibilities for type ❸ see [19] and further references therein.…”

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confidence: 99%

Death of period-doublings: locating the homoclinic-doubling cascade

Oldeman

Krauskopf

Champneys

2000

Physica D: Nonlinear Phenomena

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General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. AbstractThis paper studies a natural mechanism, called a homoclinic-doubling cascade, for the disappearance of period-doubling cascades in vector fields. Simply put, an entire perioddoubling cascade collides with a saddle-type equilibrium. Homoclinic doubling cascades are known to have self-similar structure. In contrast to the well-known Feigenbaum constant, the scaling constants for homoclinic-doubling depend on the eigenvalues of the saddle equilibrium.Specifically, we present here for the first time a detailed study of homoclinic-doubling cascades in a smooth vector field, namely a three-dimensional polynomial model proposed by Sandstede. A numerical algorithm is presented for computing homoclinic-doubling cascades in general vector fields, which makes use of the program AUTO/HOMCONT. This allows us to compute two types of homoclinic-doubling cascades, one where the primary homoclinic orbit undergoes an inclination flip bifurcation and one where it undergoes an orbit flip bifurcation. Our results bring out the scaling constants in good agreement with analytical estimates obtained from one-dimensional maps.

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Bifurcations toN-homoclinic orbits andN-periodic orbits in vector fields

Cited by 79 publications

References 12 publications

Lin's method and homoclinic bifurcations for functional differential equations of mixed type

Lin's method and homoclinic bifurcations for functional differential equations of mixed type

Bifurcation from codimension one relative homoclinic cycles

Death of period-doublings: locating the homoclinic-doubling cascade

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