NONRESONANT BIFURCATIONS OF HETEROCLINIC LOOPS WITH ONE INCLINATION FLIP

2011 ◽  
Vol 21 (01) ◽  
pp. 255-273 ◽  
Author(s):  
SHULIANG SHUI ◽  
JINGJING LI ◽  
XUYANG ZHANG
Keyword(s):  
Periodic Orbit ◽  
Homoclinic Orbit ◽  
Vector Fields ◽  
Heteroclinic Orbits ◽  
Dimensional Vector ◽  
Heteroclinic Loop ◽  
Stable Foliation ◽  
Local Coordinates ◽  
Inclination Flip

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.

BIFURCATION OF ROUGH HETEROCLINIC LOOP WITH ORBIT FLIPS

2012 ◽  
Vol 22 (11) ◽  
pp. 1250278 ◽  
Author(s):  
XINGBO LIU ◽  
ZHENZHEN WANG ◽  
DEMING ZHU
Keyword(s):  
Periodic Orbit ◽  
Coordinate System ◽  
Vector Fields ◽  
Poincaré Map ◽  
Local Coordinate System ◽  
Heteroclinic Orbit ◽  
Dimensional Vector ◽  
Homoclinic Loop ◽  
Heteroclinic Loop ◽  
Approximate Expressions

In this paper, heteroclinic loop bifurcations with double orbit flips are investigated in four-dimensional vector fields. We obtain the bifurcation equations by setting up a local coordinate system near the rough heteroclinic orbit and establishing the Poincaré map. By means of the bifurcation equations, we investigate the existence, coexistence and noncoexistence of periodic orbit, homoclinic loop and heteroclinic loop under some nongeneric conditions. The approximate expressions of corresponding bifurcation curves (or surfaces) are also given. An example of application is also given to demonstrate the existence of the heteroclinic loop with double orbit flips.

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 NONORIENTABLE MANIFOLDS IN THREE-DIMENSIONAL VECTOR FIELDS

2003 ◽  
Vol 13 (03) ◽  
pp. 553-570 ◽  
Author(s):  
HINKE M. OSINGA
Keyword(s):  
Vector Field ◽  
Periodic Orbit ◽  
Vector Fields ◽  
Invariant Manifolds ◽  
Three Dimensional ◽  
Period Doubling ◽  
Building Blocks ◽  
Two Dimensional ◽  
Dimensional Vector ◽  
Three Dimensional Vector Field

It is well known that a nonorientable manifold in a three-dimensional vector field is topologically equivalent to a Möbius strip. The most frequently used example is the unstable manifold of a periodic orbit that just lost its stability in a period-doubling bifurcation. However, there are not many explicit studies in the literature in the context of dynamical systems, and so far only qualitative sketches could be given as illustrations. We give an overview of the possible bifurcations in three-dimensional vector fields that create nonorientable manifolds. We mainly focus on nonorientable manifolds of periodic orbits, because they are the key building blocks. This is illustrated with invariant manifolds of three-dimensional vector fields that arise from applications. These manifolds were computed with a new algorithm for computing two-dimensional manifolds.

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.

scholarly journals Bifurcations of Nontwisted Heteroclinic Loop with Resonant Eigenvalues

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Yinlai Jin ◽  
Xiaowei Zhu ◽  
Zheng Guo ◽  
Han Xu ◽  
Liqun Zhang ◽  
...  
Keyword(s):  
Periodic Orbit ◽  
Variational Equation ◽  
Heteroclinic Orbits ◽  
Unperturbed System ◽  
Coordinate Systems ◽  
Homoclinic Loop ◽  
Heteroclinic Loop ◽  
Tubular Neighborhoods ◽  
Bifurcation Problems ◽  
Linear Variational Equation

By using the foundational solutions of the linear variational equation of the unperturbed system along the heteroclinic orbits to establish the local coordinate systems in the small tubular neighborhoods of the heteroclinic orbits, we study the bifurcation problems of nontwisted heteroclinic loop with resonant eigenvalues. The existence, numbers, and existence regions of 1-heteroclinic loop, 1-homoclinic loop, 1-periodic orbit, 2-fold 1-periodic orbit, and two 1-periodic orbits are obtained. Meanwhile, we give the corresponding bifurcation surfaces.

scholarly journals Bifurcation of Nongeneric Homoclinic Orbit Accompanied by Pitchfork Bifurcation

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Fengjie Geng ◽  
Song Li
Keyword(s):  
Periodic Orbit ◽  
Unstable Manifold ◽  
Homoclinic Orbit ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Pitchfork Bifurcation ◽  
Heteroclinic Orbits ◽  
The Difference

The bifurcation of a nongeneric homoclinic orbit (i.e., the orbit comes from the equilibrium along the unstable manifold instead of the center manifold) connecting a nonhyperbolic equilibrium is investigated, and the nonhyperbolic equilibrium undergoes a pitchfork bifurcation. The existence (resp., nonexistence) of a homoclinic orbit and an 1-periodic orbit are established when the pitchfork bifurcation does not happen, while as the nonhyperbolic equilibrium undergoes a pitchfork bifurcation, we obtain the sufficient conditions for the existence of homoclinic orbit and two or three heteroclinic orbits, and so forth. Moreover, we explore the difference between the bifurcation of the nongeneric homoclinic orbit and the generic one.

scholarly journals Homoclinic orbits and Lie rotated vector fields

2000 ◽  
Vol 24 (3) ◽  
pp. 187-192
Author(s):  
Jie Wang ◽  
Chen Chen
Keyword(s):  
Limit Cycles ◽  
Homoclinic Orbit ◽  
Vector Fields ◽  
Homoclinic Orbits ◽  
Singular Points ◽  
Definition Of

Based on the definition of Lie rotated vector fields in the plane, this paper gives the property of homoclinic orbit as parameter is changed and the singular points are fixed on Lie rotated vector fields. It gives the conditions of yielding limit cycles as well.

scholarly journals New Formulas and Results for 3-Dimensional Vector Fields

2021 ◽  
Vol 12 (11) ◽  
pp. 1058-1096
Author(s):  
Sadanand D. Agashe
Keyword(s):  
Vector Fields ◽  
Dimensional Vector ◽  
3 Dimensional
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