Degenerate bifurcations of heterodimensional cycle with orbit flip and inclination flip

2013 ◽  
Vol 43 (11) ◽  
pp. 1113-1129
Author(s):  
XingBo LIU
Keyword(s):  
Orbit Flip ◽  
Inclination Flip

scholarly journals Resonant Homoclinic Flips Bifurcation in Principal Eigendirections

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Tiansi Zhang ◽  
Xiaoxin Huang ◽  
Deming Zhu
Keyword(s):  
Periodic Orbit ◽  
Coordinate System ◽  
Homoclinic Orbit ◽  
Principal Eigenvalue ◽  
Small Neighborhood ◽  
Homoclinic Bifurcation ◽  
Bifurcation Equation ◽  
Orbit Flip ◽  
Inclination Flip ◽  
Poincaré Return Map

A codimension-4 homoclinic bifurcation with one orbit flip and one inclination flip at principal eigenvalue direction resonance is considered. By introducing a local active coordinate system in some small neighborhood of homoclinic orbit, we get the Poincaré return map and the bifurcation equation. A detailed investigation produces the number and the existence of 1-homoclinic orbit, 1-periodic orbit, and double 1-periodic orbits. We also locate their bifurcation surfaces in certain regions.

scholarly journals Bifurcations of Orbit and Inclination Flips Heteroclinic Loop with Nonhyperbolic Equilibria

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Fengjie Geng ◽  
Junfang Zhao
Keyword(s):  
Periodic Orbits ◽  
Homoclinic Orbits ◽  
Transcritical Bifurcation ◽  
Heteroclinic Orbits ◽  
Heteroclinic Loop ◽  
Orbit Flip ◽  
Inclination Flip ◽  
Hyperbolic Saddle

The bifurcations of heteroclinic loop with one nonhyperbolic equilibrium and one hyperbolic saddle are considered, where the nonhyperbolic equilibrium is supposed to undergo a transcritical bifurcation; moreover, the heteroclinic loop has an orbit flip and an inclination flip. When the nonhyperbolic equilibrium does not undergo a transcritical bifurcation, we establish the coexistence and noncoexistence of the periodic orbits and homoclinic orbits. While the nonhyperbolic equilibrium undergoes the transcritical bifurcation, we obtain the noncoexistence of the periodic orbits and homoclinic orbits and the existence of two or three heteroclinic orbits.

Homoclinic and Heteroclinic Bifurcations Close to a Twisted Heteroclinic Cycle

1998 ◽  
Vol 08 (02) ◽  
pp. 359-375 ◽  
Author(s):  
Martín G. Zimmermann ◽  
Mario A. Natiello
Keyword(s):  
Reaction Diffusion ◽  
Homoclinic Orbits ◽  
Heteroclinic Orbits ◽  
Reaction Diffusion Equation ◽  
Heteroclinic Cycle ◽  
Flip Bifurcation ◽  
Double Loop ◽  
Orbit Flip ◽  
Inclination Flip ◽  
Two Parameters

We study the interaction of a transcritical (or saddle-node) bifurcation with a codimension-0/codimension-2 heteroclinic cycle close to (but away from) the local bifurcation point. The study is motivated by numerical observations on the traveling wave ODE of a reaction–diffusion equation. The manifold organization is such that two branches of homoclinic orbits to each fixed point are created when varying the two parameters controlling the codimension-2 loop. It is shown that the homoclinic orbits may become degenerate in an orbit-flip bifurcation. We establish the occurrence of multi-loop homoclinic and heteroclinic orbits in this system. The double-loop homoclinic orbits are shown to bifurcate in an inclination-flip bifurcation, where a Smale's horseshoe is found.

Inclination-flip homoclinic orbits arising from orbit-flip

Nonlinearity ◽  
2001 ◽  
Vol 14 (2) ◽  
pp. 379-393 ◽  
Author(s):  
C A Morales ◽  
M J Pacifico
Keyword(s):  
Homoclinic Orbits ◽  
Orbit Flip ◽  
Inclination Flip

Bifurcation Complexity from Orbit-Flip Homoclinic Orbit of Weak Type

2016 ◽  
Vol 26 (04) ◽  
pp. 1650059 ◽  
Author(s):  
Qiuying Lu ◽  
Vincent Naudot
Keyword(s):  
Homoclinic Orbit ◽  
Weak Type ◽  
Blow Up ◽  
Three Dimensional ◽  
Homoclinic Orbits ◽  
Unperturbed System ◽  
Orbit Flip ◽  
Inclination Flip ◽  
Three Dimensional Vector Field ◽  
Degenerate Version

In this paper, we study the unfolding of a three-dimensional vector field having an orbit-flip homoclinic orbit of weak type. Such a homoclinic orbit is a degenerate version of the so-called orbit-flip homoclinic orbit. We show the existence of inclination-flip homoclinic orbits of arbitrary higher order bifurcating from the unperturbed system. Our strategy consists of using the local moving coordinates method and blow up in the parameter space. In addition, the numerical existence of the orbit-flip homoclinic orbit of weak type is presented based on the truncated Taylor expansion and the numerical computation for both the strong stable manifold and unstable manifold.

Bifurcations Near the Weak Type Heterodimensional Cycle

2014 ◽  
Vol 24 (09) ◽  
pp. 1450112 ◽  
Author(s):  
Xingbo Liu
Keyword(s):  
Coordinate System ◽  
Periodic Orbits ◽  
Weak Type ◽  
Local Coordinate System ◽  
Heteroclinic Orbit ◽  
Homoclinic Orbits ◽  
Small Perturbations ◽  
Orbit Flip ◽  
Bifurcation Phenomena ◽  
Inclination Flip

The aim of this paper is to show the bifurcation phenomena near the weak type heterodimensional cycle when the orbit flip and inclination flip occur simultaneously in its nontransversal heteroclinic orbit. With the aid of a suitable local coordinate system, the Poincaré map is constructed. By means of the bifurcation equations, the persistence of heterodimensional cycles, the coexistence of the heterodimensional cycle and periodic orbits or homoclinic orbits, and the existence of bifurcation surfaces of homoclinic orbits or the periodic orbits are discussed under small perturbations. Moreover, an example is given to show the existence of the system which has a heterodimensional cycle with orbit flip and inclination flip.

scholarly journals ECCENTRICITY GROWTH AND ORBIT FLIP IN NEAR-COPLANAR HIERARCHICAL THREE-BODY SYSTEMS

2014 ◽  
Vol 785 (2) ◽  
pp. 116 ◽  
Author(s):  
Gongjie Li ◽  
Smadar Naoz ◽  
Bence Kocsis ◽  
Abraham Loeb
Keyword(s):  
Orbit Flip ◽  
Three Body

HETERODIMENSIONAL CYCLE BIFURCATION WITH ORBIT-FLIP

2010 ◽  
Vol 20 (02) ◽  
pp. 491-508 ◽  
Author(s):  
QIUYING LU ◽  
ZHIQIN QIAO ◽  
TIANSI ZHANG ◽  
DEMING ZHU
Keyword(s):  
Periodic Orbit ◽  
Bifurcation Analysis ◽  
Homoclinic Orbit ◽  
Heteroclinic Orbit ◽  
Moving Frame ◽  
Homoclinic Loop ◽  
Orbit Flip ◽  
Local Moving Frame

The local moving frame approach is employed to study the bifurcation of a degenerate heterodimensional cycle with orbit-flip in its nontransversal orbit. Under some generic hypotheses, we provide the conditions for the existence, uniqueness and noncoexistence of the homoclinic orbit, heteroclinic orbit and periodic orbit. And we also present the coexistence conditions for the homoclinic orbit and the periodic orbit. But it is impossible for the coexistence of the periodic orbit and the persistent heterodimensional cycle or the coexistence of the homoclinic loop and the persistent heterodimensional cycle. Moreover, the double and triple periodic orbit bifurcation surfaces are established as well. Based on the bifurcation analysis, the bifurcation surfaces and the existence regions are located. An example of application is also given to demonstrate our main results.

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