Codimension 3 Bifurcations of Homoclinic Orbits With Orbit Flips and Inclination Flips

Shui, Shuliang; Zhu, Deming

doi:10.1142/s0252959904000494

Cited by 3 publications

(3 citation statements)

References 15 publications

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“…Recently, some papers were devoted to the study on the bifurcations of the homoclinic or heteroclinic loops with an orbit flip or an inclination flip in higher dimensional vector fields, e.g. see Geng et al, 2009;Lu, 2009;Qiao et al, 2009;Shui & Zhu, 2004, 2005Wu & Sun, 2006;Xu et al, 2008;Xu et al, 2009;Zhang & Zhu, 2004, 2007, 2009. In this paper, we study the bifurcations of rough heteroclinic loops which are composed of two principal heteroclinic orbits but there is one stable foliation that involves the inclination flip in four-dimensional systems.…”

Section: Introductionmentioning

confidence: 99%

Nonresonant Bifurcations of Heteroclinic Loops With One Inclination Flip

Shui

Xuyang

2011

Int. J. Bifurcation Chaos

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Heteroclinic bifurcations in four-dimensional vector fields are investigated by setting up local coordinates near a heteroclinic loop. This heteroclinic loop consists of two principal heteroclinic orbits, but there is one stable foliation that involves an inclination flip. The existence, nonexistence, coexistence and uniqueness of the 1-heteroclinic loop, 1-homoclinic orbit, and 1-periodic orbit are studied. Also, the nonexistence, existence of the 2-homoclinic and 2-periodic orbit are demonstrated.

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Section: Introductionmentioning

confidence: 99%

Nonresonant Bifurcations of Heteroclinic Loops With One Inclination Flip

Shui

Xuyang

2011

Int. J. Bifurcation Chaos

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“…Because of the complexity, these non-principal homoclinic orbits and their associated bifurcations were mainly studied for 3-dimensional systems in the above mentioned references. Recently, we have considered codimension 3 homoclinic bifurcations in case that an orbit flip and an inclination flip hold simultaneously in [15]. We have also considered codimension 3 non-resonant bifurcations of homoclinic orbits with two inclination flips in [16].…”

Section: Introductionmentioning

confidence: 99%

“…Recently, we have considered codimension 3 homoclinic bifurcations in case that an orbit flip and an inclination flip hold simultaneously in [15]. We have also considered codimension 3 non-resonant bifurcations of homoclinic orbits with two inclination flips in [16]. In this paper, we study the codimension 3 bifurcations of rough heteroclinic loops that they are composed of a principal heteroclinic orbit and a non-principal heteroclinic orbit which takes orbit flip in 4-dimensional systems.…”

Section: Introductionmentioning

confidence: 99%

Codimension 3 Non-resonant Bifurcations of Rough Heteroclinic Loops with One Orbit Flip*

Shui

Zhu

2006

Chin. Ann. Math. Ser. B

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Heteroclinic bifurcations in four dimensional vector fields are investigated by setting up a local coordinates near a rough heteroclinic loop. This heteroclinic loop has a principal heteroclinic orbit and a non-principal heteroclinic orbit that takes orbit flip. The existence, nonexistence, coexistence and uniqueness of the 1-heteroclinic loop, 1-homoclinic orbit and 1-periodic orbit are studied. The existence of the two-fold or three-fold 1-periodic orbit is also obtained.

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Bifurcations of double homoclinic flip orbits with resonant eigenvalues

Zhang

Zhu

2007

Appl. Math. Mech.-Engl. Ed.

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Concerns double homoclinic loops with orbit flips and two resonant eigenvalues in a four-dimensional system. We use the solution of a normal form system to construct a singular map in some neighborhood of the equilibrium, and the solution of a linear variational system to construct a regular map in some neighborhood of the double homoclinic loops, then compose them to get the important Poincaré map. A simple calculation gives explicitly an expression of the associated successor function. By a delicate analysis of the bifurcation equation, we obtain the condition that the original double homoclinic loops are kept, and prove the existence and the existence regions of the large 1-homoclinic orbit bifurcation surface, 2-fold large 1-periodic orbit bifurcation surface, large 2-homoclinic orbit bifurcation surface and their approximate expressions. We also locate the large periodic orbits and large homoclinic orbits and their number.

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Codimension 3 Bifurcations of Homoclinic Orbits With Orbit Flips and Inclination Flips

Cited by 3 publications

References 15 publications

Nonresonant Bifurcations of Heteroclinic Loops With One Inclination Flip

Nonresonant Bifurcations of Heteroclinic Loops With One Inclination Flip

Codimension 3 Non-resonant Bifurcations of Rough Heteroclinic Loops with One Orbit Flip*

Bifurcations of double homoclinic flip orbits with resonant eigenvalues

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