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 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 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 Non-accessible critical points of Cremer polynomials

2000 ◽  
Vol 20 (5) ◽  
pp. 1391-1403 ◽  
Author(s):  
JAN KIWI
Keyword(s):  
Periodic Orbit ◽  
Critical Point ◽  
Critical Points ◽  
Julia Set ◽  
Complex Polynomial

It is shown that a polynomial with a Cremer periodic orbit has a non-accessible critical point in its Julia set provided that the Cremer periodic orbit is approximated by small cycles. Also, this paper contains a new proof of the Douady–Shishikura inequality for the number of non-repelling cycles of a complex polynomial.

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.

Phase space correspondence between Jordan and Einstein frames

2021 ◽  
pp. 2150087
Author(s):  
Amin Salehi
Keyword(s):  
Phase Space ◽  
Critical Point ◽  
Critical Points ◽  
Dynamical Behavior ◽  
Cosmological Parameters ◽  
Jordan Frame ◽  
Field Equations ◽  
Theories Of Gravity ◽  
The Stability ◽  
Jordan And Einstein Frames

Scalar–tensor theories of gravity can be formulated in the Einstein frame or in the Jordan frame (JF) which are related with each other by conformal transformations. Although the two frames describe the same physics and are equivalent, the stability of the field equations in the two frames is not the same. Here, we implement dynamical system and phase space approach as a robustness tool to investigate this issue. We concentrate on the Brans–Dicke theory in a Friedmann–Lemaitre–Robertson–Walker universe, but the results can easily be generalized. Our analysis shows that while there is a one-to-one correspondence between critical points in two frames and each critical point in one frame is mapped to its corresponds in another frame, however, stability of a critical point in one frame does not guarantee the stability in another frame. Hence, an unstable point in one frame may be mapped to a stable point in another frame. All trajectories between two critical points in phase space in one frame are different from their corresponding in other ones. This indicates that the dynamical behavior of variables and cosmological parameters is different in two frames. Hence, for those features of the study, which focus on observational measurements, we must use the JF where experimental data have their usual interpretation.

UNIMODAL INTERVAL MAPS OBTAINED FROM THE MODIFIED CHUA EQUATIONS

1993 ◽  
Vol 03 (02) ◽  
pp. 323-332 ◽  
Author(s):  
MICHAŁ MISIUREWICZ
Keyword(s):  
Periodic Orbit ◽  
Critical Point ◽  
Real Line ◽  
Positive Measure ◽  
Schwarzian Derivative ◽  
Large Range ◽  
Unimodal Maps ◽  
Interval Maps ◽  
Central Piece ◽  
Piecewise Continuous

Following Brown [1992, 1993] we study maps of the real line into itself obtained from the modified Chua equations. We fix our attention on a one-parameter family of such maps, which seems to be typical. For a large range of parameters, invariant intervals exist. In such an invariant interval, the map is piecewise continuous, with most of pieces of continuity mapped in a monotone way onto the whole interval. However, on the central piece there is a critical point. This allows us to find sometimes a smaller invariant interval on which the map is unimodal. In such a way, we get one-parameter families of smooth unimodal maps, very similar to the well-known family of logistic maps x ↦ ax(1−x). We study more closely one of those and show that these maps have negative Schwarzian derivative. This implies the existence of at most one attracting periodic orbit. Moreover, there is a set of parameters of positive measure for which chaos occurs.

scholarly journals Global Phase Portrait and Bifurcation Diagram for a Multi-Parametric Linear System

2020 ◽  
Vol 55 (4) ◽  
Author(s):  
Jorge Rodríguez Contreras ◽  
Alberto Reyes Linero ◽  
Juliana Vargas Sánchez
Keyword(s):  
Linear System ◽  
Critical Point ◽  
Phase Portrait ◽  
Bifurcation Diagram ◽  
Critical Points ◽  
Global Dynamics ◽  
Parametric Surfaces ◽  
Critical Points At Infinity ◽  
Global Phase Portrait ◽  
The Stability

The goal of this article is to conduct a global dynamics study of a linear multiparameter system (real parameters (a,b,c) in R^3); for this, we take the different changes that these parameters present. First, we find the different parametric surfaces in which the space is divided, where the stability of the critical point is defined; we then create a bifurcation diagram to classify the different bifurcations that appear in the system. Finally, we determine and classify the critical points at infinity, considering the canonical shape of the Poincaré sphere, and thus, obtain a global phase portrait of the multiparametric linear system.

Bifurcation Analysis in a Class of Piecewise Nonlinear Systems with a Nonsmooth Heteroclinic Loop

2018 ◽  
Vol 28 (02) ◽  
pp. 1850026
Author(s):  
Yuanyuan Liu ◽  
Feng Li ◽  
Pei Dang
Keyword(s):  
Limit Cycles ◽  
Polynomial Systems ◽  
Melnikov Function ◽  
Heteroclinic Orbits ◽  
Phase Portraits ◽  
Unperturbed System ◽  
Piecewise Polynomial ◽  
Heteroclinic Loop ◽  
First Order ◽  
Nilpotent Critical Point

We consider the bifurcation in a class of piecewise polynomial systems with piecewise polynomial perturbations. The corresponding unperturbed system is supposed to possess an elementary or nilpotent critical point. First, we present 17 cases of possible phase portraits and conditions with at least one nonsmooth periodic orbit for the unperturbed system. Then we focus on the two specific cases with two heteroclinic orbits and investigate the number of limit cycles near the loop by means of the first-order Melnikov function, respectively. Finally, we take a quartic piecewise system with quintic piecewise polynomial perturbation as an example and obtain that there can exist ten limit cycles near the heteroclinic loop.

scholarly journals A method to estimate statistical errors of properties derived from charge-density modelling

2018 ◽  
Vol 74 (3) ◽  
pp. 170-183 ◽  
Author(s):  
Bertrand Fournier ◽  
Benoît Guillot ◽  
Claude Lecomte ◽  
Eduardo C. Escudero-Adán ◽  
Christian Jelsch
Keyword(s):  
Charge Density ◽  
Least Squares ◽  
Critical Point ◽  
Critical Points ◽  
Property Values ◽  
Density Model ◽  
Interaction Energies ◽  
Topological Properties ◽  
A Charge ◽  
Density Modelling

Estimating uncertainties of property values derived from a charge-density model is not straightforward. A methodology, based on calculation of sample standard deviations (SSD) of properties using randomly deviating charge-density models, is proposed with theMoProsoftware. The parameter shifts applied in the deviating models are generated in order to respect the variance–covariance matrix issued from the least-squares refinement. This `SSD methodology' procedure can be applied to estimate uncertainties ofanyproperty related to a charge-density model obtained by least-squares fitting. This includes topological properties such as critical point coordinates, electron density, Laplacian and ellipticity at critical points and charges integrated over atomic basins. Errors on electrostatic potentials and interaction energies are also available now through this procedure. The method is exemplified with the charge density of compound (E)-5-phenylpent-1-enylboronic acid, refined at 0.45 Å resolution. The procedure is implemented in the freely availableMoProprogram dedicated to charge-density refinement and modelling.

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