BIFURCATIONS OF GENERIC HETEROCLINIC LOOP ACCOMPANIED BY TRANSCRITICAL BIFURCATION

2008 ◽  
Vol 18 (04) ◽  
pp. 1069-1083 ◽  
Author(s):  
FENGJIE GENG ◽  
DAN LIU ◽  
DEMING ZHU
Keyword(s):  
Periodic Orbit ◽  
Heteroclinic Orbit ◽  
Transcritical Bifurcation ◽  
Homoclinic Loop ◽  
Heteroclinic Loop ◽  
Higher Dimensional ◽  
Hyperbolic Saddle

The bifurcations of generic heteroclinic loop with one nonhyperbolic equilibrium p1and one hyperbolic saddle p2are investigated, where p1is assumed to undergo transcritical bifurcation. Firstly, we discuss bifurcations of heteroclinic loop when transcritical bifurcation does not happen, the persistence of heteroclinic loop, the existence of homoclinic loop connecting p1(resp. p2) and the coexistence of one homoclinic loop and one periodic orbit are established. Secondly, we analyze bifurcations of heteroclinic loop accompanied by transcritical bifurcation, namely, nonhyperbolic equilibrium p1splits into two hyperbolic saddles [Formula: see text] and [Formula: see text], a heteroclinic loop connecting [Formula: see text] and p2, homoclinic loop with [Formula: see text] (resp. p2) and heteroclinic orbit joining [Formula: see text] and [Formula: see text] (resp. [Formula: see text] and p2; p2and [Formula: see text]) are found. The results achieved here can be extended to higher dimensional systems.

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.

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.

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.

On the Number of Limit Cycles Bifurcated from Some Hamiltonian Systems with a Double Homoclinic Loop and a Heteroclinic Loop

2017 ◽  
Vol 27 (04) ◽  
pp. 1750055 ◽  
Author(s):  
Pegah Moghimi ◽  
Rasoul Asheghi ◽  
Rasool Kazemi
Keyword(s):  
Hamiltonian System ◽  
Lower Bound ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Homoclinic Loop ◽  
Bifurcation Of Limit Cycles ◽  
Heteroclinic Loop ◽  
Nilpotent Saddle ◽  
Hyperbolic Saddle

In this paper, we study the number of bifurcated limit cycles from near-Hamiltonian systems where the corresponding Hamiltonian system has a double homoclinic loop passing through a hyperbolic saddle surrounded by a heteroclinic loop with a hyperbolic saddle and a nilpotent saddle, and obtain some new results on the lower bound of the maximal number of limit cycles for these systems. In particular, we study the bifurcation of limit cycles of the following system [Formula: see text] as an application of our results, where [Formula: see text] is a polynomial of degree five.

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.

Bifurcations of Heteroclinic Loop with Twisted Conditions

2017 ◽  
Vol 27 (08) ◽  
pp. 1750120
Author(s):  
Yinlai Jin ◽  
Suoling Yang ◽  
Yuanyuan Liu ◽  
Dandan Xie ◽  
Nana Zhang
Keyword(s):  
Periodic Orbit ◽  
Critical Point ◽  
Critical Points ◽  
Unperturbed System ◽  
Homoclinic Loop ◽  
Heteroclinic Loop ◽  
Principal Eigenvalues ◽  
Bifurcation Problems

The bifurcation problems of twisted heteroclinic loop with two hyperbolic critical points are studied for the case [Formula: see text], [Formula: see text], [Formula: see text], where [Formula: see text] and [Formula: see text] are the pair of principal eigenvalues of unperturbed system at the critical point [Formula: see text], [Formula: see text]. Under the transversal conditions, the authors obtained some results of the existence and the number of 1-homoclinic loop, 1-periodic orbit, double 1-periodic orbit, 2-homoclinic loop and 2-periodic orbit. Moreover, the relative bifurcation surfaces and the existence regions are given, and the corresponding bifurcation graphs are drawn.

scholarly journals Bifurcations and Stability of Nondegenerated Homoclinic Loops for Higher Dimensional Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Yinlai Jin ◽  
Feng Li ◽  
Han Xu ◽  
Jing Li ◽  
Liqun Zhang ◽  
...  
Keyword(s):  
Periodic Orbit ◽  
Homoclinic Orbit ◽  
Variational Equation ◽  
Small Neighborhood ◽  
Unperturbed System ◽  
Homoclinic Loop ◽  
Resonant Condition ◽  
Local Current ◽  
The Stability ◽  
Linear Variational Equation

By using the foundational solutions of the linear variational equation of the unperturbed system along the homoclinic orbit as the local current coordinates system of the system in the small neighborhood of the homoclinic orbit, we discuss the bifurcation problems of nondegenerated homoclinic loops. Under the nonresonant condition, existence, uniqueness, and incoexistence of 1-homoclinic loop and 1-periodic orbit, the inexistence ofk-homoclinic loop andk-periodic orbit is obtained. Under the resonant condition, we study the existence of 1-homoclinic loop, 1-periodic orbit, 2-fold 1-periodic orbit, and two 1-periodic orbits; the coexistence of 1-homoclinic loop and 1-periodic orbit. Moreover, we give the corresponding existence fields and bifurcation surfaces. At last, we study the stability of the homoclinic loop for the two cases of non-resonant and resonant, and we obtain the corresponding criterions.

CODIMENSION 3 HETEROCLINIC BIFURCATIONS WITH ORBIT AND INCLINATION FLIPS IN REVERSIBLE SYSTEMS

2008 ◽  
Vol 18 (12) ◽  
pp. 3689-3701 ◽  
Author(s):  
YANCONG XU ◽  
DEMING ZHU ◽  
FENGJIE GENG
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Homoclinic Orbit ◽  
Heteroclinic Orbit ◽  
Homoclinic Bifurcation ◽  
Reversible Systems ◽  
Reversible System ◽  
Bifurcation Diagrams ◽  
Symmetric Periodic Orbits ◽  
The Continuum

Heteroclinic bifurcations with orbit-flips and inclination-flips are investigated in a four-dimensional reversible system by using the method originally established in [Zhu, 1998; Zhu & Xia, 1998]. The existence and coexistence of R-symmetric homoclinic orbit and R-symmetric heteroclinic orbit, R-symmetric homoclinic orbit and R-symmetric periodic orbit are obtained. The double R-symmetric homoclinic bifurcation is found, and the continuum of R-symmetric periodic orbits accumulating into a homoclinic orbit is also demonstrated. Moreover, the bifurcation surfaces and the existence regions are given, and the corresponding bifurcation diagrams are drawn.

scholarly journals Dynamical behaviour of a compound elastic pendulum

2018 ◽  
Vol 145 ◽  
pp. 01003
Author(s):  
Svetoslav Nikolov ◽  
Daniela Zaharieva
Keyword(s):  
Hamiltonian System ◽  
Periodic Orbit ◽  
Chaotic Dynamics ◽  
Degrees Of Freedom ◽  
Invariant Manifolds ◽  
Heteroclinic Orbit ◽  
Dynamical Behaviour ◽  
Complex Behavior ◽  
Elastic Pendulum ◽  
Comprehensive Study

The aim of the paper is a comprehensive study of the compound elastic pendulum (CEP) with two degrees of freedom to point out the main complex (chaotic) dynamics that it can exhibit. The simplest way to find complex behavior in a nonintegrable Hamiltonian system is firstly to look for homoclinic (heteroclinic) orbit(s). Here, under suitable assumptions, we detect the existence of a homoclinic orbit of CEP and present the equation for it. Moreover, we show that for any value of the small parameter the system has a hyperbolic periodic orbit, whose invariant manifolds intersect themselves transversally.

ON THE STUDY OF LIMIT CYCLES OF THE GENERALIZED RAYLEIGH–LIENARD OSCILLATOR

2004 ◽  
Vol 14 (08) ◽  
pp. 2905-2914 ◽  
Author(s):  
YUHAI WU ◽  
MAOAN HAN ◽  
XIANFENG CHEN
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Saddle Connection ◽  
Homoclinic Loop ◽  
Heteroclinic Loop

The Hopf bifurcation, saddle connection loop bifurcation and Poincaré bifurcation of the generalized Rayleigh–Liénard oscillator Ẍ+aX+2bX3+ε(c3+c2X2+c1X4+c4Ẋ2)Ẋ=0 are studied. It is proved that for the case a<0, b>0 the system has at most six limit cycles bifurcated from Hopf bifurcation or has at least seven limit cycles bifurcated from the double homoclinic loop. For the case a>0, b<0 the system has at most three limit cycles bifurcated from Hopf bifurcation or has three limit cycles bifurcated from the heteroclinic loop.

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