scholarly journals Bifurcations of a Homoclinic Orbit to Saddle-Center in Reversible Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-12
Author(s):  
Zhiqin Qiao ◽  
Yancong Xu
Keyword(s):  
Periodic Orbit ◽  
Homoclinic Orbit ◽  
Moving Frame ◽  
Reversible Systems ◽  
Reversible System ◽  
Melnikov Functions ◽  
Existence And Nonexistence ◽  
Local Moving Frame

The bifurcations near a primary homoclinic orbit to a saddle-center are investigated in a 4-dimensional reversible system. By establishing a new kind of local moving frame along the primary homoclinic orbit and using the Melnikov functions, the existence and nonexistence of 1-homoclinic orbit and 1-periodic orbit, including symmetric 1-homoclinic orbit and 1-periodic orbit, and their corresponding codimension 1 or codimension 3 surfaces, are obtained.

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.

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.

BIFURCATIONS OF 2-2-1 HETERODIMENSIONAL CYCLES UNDER TRANSVERSALITY CONDITION

2012 ◽  
Vol 22 (08) ◽  
pp. 1250191
Author(s):  
DAN LIU ◽  
MAOAN HAN ◽  
WEIPENG ZHANG
Keyword(s):  
Periodic Orbit ◽  
Sufficient Conditions ◽  
Transversality Condition ◽  
Moving Frame ◽  
Two Dimensional ◽  
Poincare Mapping ◽  
Unstable Manifolds ◽  
Local Moving Frame ◽  
Vector Formula ◽  
Homoclinic Cycle

Bifurcations of generic 2-2-1 heterodimensional cycles connecting to three saddles, in which two of them have two-dimensional unstable manifolds, are studied by setting up a local moving frame. Under a certain transversal condition, we firstly present the existence, uniqueness and noncoexistence of a 3-point heterodimensional cycle, 2-point heterodimensional or equidimensional cycle, 1-homoclinic cycle and 1-periodic orbit bifurcated from the 3-point heterodimensional cycle, and the bifurcation surfaces and bifurcation regions are located when the u-component [Formula: see text] of the vector [Formula: see text] under the Poincaré mapping [Formula: see text] is nonzero. Conversely, we obtain some sufficient conditions such that the bifurcation of a 2-fold 1-periodic orbit occurs and a 1-periodic orbit coexists with the surviving heterodimensional cycle, showing some new bifurcation behaviors different from the well-known equidimensional cycles.

SUBSIDIARY HOMOCLINIC ORBITS TO A SADDLE-FOCUS FOR REVERSIBLE SYSTEMS

1994 ◽  
Vol 04 (06) ◽  
pp. 1447-1482 ◽  
Author(s):  
A.R. CHAMPNEYS
Keyword(s):  
Homoclinic Orbit ◽  
Water Waves ◽  
Numerical Experiments ◽  
Homoclinic Orbits ◽  
Reversible Systems ◽  
Reversible System ◽  
Type Analysis ◽  
Reversible Transformation ◽  
Positive Integers ◽  
Good Agreement

A dynamical system is said to be reversible if there is an involution of phase space that reverses the direction of the flow. Examples are classical Hamiltonian systems with quadratic kinetic energy. For reversible systems, homoclinic orbits that are invariant under the reversible transformation typically persist as parameters are varied. This paper concerns reversible systems for which a primary homoclinic orbit to a saddle-focus is assumed to exist. The problem under investigation is a characterisation of the subsidiary homoclinic orbits which then exist in a neighbourhood of the primary one. Such orbits have applications as solitary water waves and as buckling solutions of nonlinear struts. A Shil’nikov-type analysis is performed for four-dimensional linearly reversible systems. It is shown that each subsidiary homoclinic orbit can be labelled by a symmetric string of positive integers. All possible strings of length one, two or three correspond to the existence of a homoclinic orbit, whereas only certain of those of length four or greater do. This situation contrasts with known results if the reversible system is also Hamiltonian. The analysis is supported by performing careful numerical experiments on the equation [Formula: see text] where P and α are parameters; a good agreement with the theory is found.

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.

Bifurcations of Double Homoclinic Loops in Reversible Systems

2020 ◽  
Vol 30 (16) ◽  
pp. 2050246
Author(s):  
Yuzhen Bai ◽  
Xingbo Liu
Keyword(s):  
Coordinate System ◽  
Periodic Orbits ◽  
Homoclinic Orbit ◽  
Poincaré Map ◽  
Local Coordinate System ◽  
Reversible Systems ◽  
Bifurcation Equation ◽  
Bifurcation Phenomena ◽  
New Type ◽  
Symmetric Periodic Orbits

This paper is devoted to the study of bifurcation phenomena of double homoclinic loops in reversible systems. With the aid of a suitable local coordinate system, the Poincaré map is constructed. By means of the bifurcation equation, we perform a detailed study to obtain fruitful results, and demonstrate the existence of the R-symmetric large homoclinic orbit of new type near the primary double homoclinic loops, the existence of infinitely many R-symmetric periodic orbits accumulating onto the R-symmetric large homoclinic orbit, and the coexistence of R-symmetric large homoclinic orbit and the double homoclinic loops. The homoclinic bellow can also be found under suitable perturbation. The relevant bifurcation surfaces and the existence regions are located.

Dynamical Behavior of Supernonlinear Positron-Acoustic Periodic Waves and Chaos in Nonextensive Electron-Positron-Ion Plasmas

2019 ◽  
Vol 74 (6) ◽  
pp. 499-511 ◽  
Author(s):  
Jharna Tamang ◽  
Asit Saha
Keyword(s):  
Periodic Orbit ◽  
Homoclinic Orbit ◽  
Dynamical Behavior ◽  
Periodic Wave ◽  
Phase Portraits ◽  
Periodic Waves ◽  
Periodic Force ◽  
Positive Ions ◽  
Qualitative Approaches ◽  
Electron Positron

AbstractPropagation of nonlinear and supernonlinear positron-acoustic periodic waves is examined in an electron-positron-ion plasma composed of static positive ions, mobile cold positrons, and q-nonextensive electrons and hot positrons. Employing the phase plane theory of planar dynamical systems, all qualitatively different phase portraits that include nonlinear positron-acoustic homoclinic orbit, nonlinear positron-acoustic periodic orbit, supernonlinear positron-acoustic homoclinic orbit, and supernonlinear positron-acoustic periodic orbit are demonstrated subjected to the parameters $q,{\mu_{1}},{\mu_{2}},{\sigma_{1}},{\sigma_{2}}$, and V. The nonlinear and supernonlinear positron-acoustic periodic wave solutions are reported for different situations through numerical computations. It is observed that the nonextensive parameter (q) acts as a controlling parameter in the dynamic motion of nonlinear and supernonlinear positron-acoustic periodic waves. The dynamic motions for the positron-acoustic traveling waves with the influence of an extrinsic periodic force are investigated through distinct qualitative approaches, such as phase portrait analysis, sensitivity analysis, time series analysis, and Poincaré section. The results of this paper may be applicable in understanding nonlinear, supernonlinear positron-acoustic periodic waves, and their chaotic motion in space plasma environments.

BIFURCATIONS OF HOMOCLINIC ORBIT CONNECTING TWO NONLEADING EIGENDIRECTIONS

2007 ◽  
Vol 17 (03) ◽  
pp. 823-836 ◽  
Author(s):  
TIANSI ZHANG ◽  
DEMING ZHU
Keyword(s):  
Periodic Orbit ◽  
Homoclinic Orbit ◽  
Period Doubling ◽  
Homoclinic Bifurcation ◽  
Period Doubling Bifurcation ◽  
Dimensional System ◽  
Bifurcation Surface

Bifurcations of homoclinic orbit connecting the strong stable and strong unstable directions are investigated for four-dimensional system. The existence, numbers, co-existence and incoexistence of 1-homoclinic orbit, 2n-homoclinic orbit, 1-periodic orbit and 2n-periodic orbit are obtained, and the bifurcation surfaces (including codimension-1 homoclinic bifurcation surfaces, double periodic orbit bifurcation surfaces, homoclinic-doubling bifurcation surfaces, period-doubling bifurcation surfaces and codimension-2 triple periodic orbit bifurcation surface, and homoclinic and double periodic orbit bifurcation surface) and the existence regions are also located.

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.

Sign in / Sign up

Export Citation Format

Share Document