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.

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.

Stability and Perturbations of Homoclinic Loops in a Class of Piecewise Smooth Systems

2015 ◽  
Vol 25 (09) ◽  
pp. 1550114 ◽  
Author(s):  
Shuang Chen ◽  
Zhengdong Du
Keyword(s):  
Limit Cycles ◽  
Homoclinic Orbit ◽  
Piecewise Linear ◽  
Small Neighborhood ◽  
Homoclinic Orbits ◽  
Piecewise Smooth ◽  
Piecewise Smooth Systems ◽  
Planar System ◽  
The Stability ◽  
First Time

Like for smooth systems, a typical method to produce multiple limit cycles for a given piecewise smooth planar system is via homoclinic bifurcation. Previous works only focused on limit cycles that bifurcate from homoclinic orbits of piecewise-linear systems. In this paper, we consider for the first time the same problem for a class of general nonlinear piecewise smooth systems. By introducing the Dulac map in a small neighborhood of the hyperbolic saddle, we obtain the approximation of the Poincaré map for the nonsmooth homoclinic orbit. Then, we give conditions for the stability of the homoclinic orbit and conditions under which one or two limit cycles bifurcate from it. As an example, we construct a nonlinear piecewise smooth system with two limit cycles that bifurcate from a homoclinic orbit.

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 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.

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.

On the Stability of Rotors in Cylindrical Journal Bearings

1962 ◽  
Vol 84 (4) ◽  
pp. 521-531 ◽  
Author(s):  
G. M. Rentzepis ◽  
B. Sternlicht
Keyword(s):  
Experimental Evidence ◽  
Carrying Capacity ◽  
Variational Equation ◽  
Equation Of Motion ◽  
Journal Bearings ◽  
Load Carrying Capacity ◽  
Load Carrying ◽  
Regions Of Stability ◽  
The Stability ◽  
Linear Variational Equation

The regions of stability for plain cylindrical journal bearings have been determined analytically here. The linear “variational” equation of motion has been employed to obtain the stability regions bounded by families of load-carrying capacity and operating eccentricity curves. The results were applied to the “quasi-static” equilibrium case for gas lubricated cylindrical journal bearings of L/D = 2. They show that there exists a “minimum” in the stability curves, a prediction supported by experimental evidence. The results of this work seem to bridge together observation on stability at very small clearances and large ones.

Stability of Forced Oscillations With Nonlinear Second-Order Terms

1959 ◽  
Vol 26 (4) ◽  
pp. 499-502
Author(s):  
Chi-Neng Shen
Keyword(s):  
Continued Fraction ◽  
Variational Equation ◽  
Second Order ◽  
Iteration Procedure ◽  
Forced Oscillations ◽  
The Stability ◽  
Boundary Of Stability

Abstract A solution is obtained for forced oscillations with nonlinear second-order terms. The stability of this solution is given by its variational equation. The boundary of stability is analyzed by both the perturbation and continued fraction methods. The amplitude of osclllation with damping terms is also determined by the iteration procedure.

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 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.

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