High-Order Analysis of Global Bifurcations in a Codimension-Three Takens–Bogdanov Singularity in Reversible Systems

2020 ◽  
Vol 30 (01) ◽  
pp. 2050017 ◽  
Author(s):  
Bo-Wei Qin ◽  
Kwok-Wai Chung ◽  
Antonio Algaba ◽  
Alejandro J. Rodríguez-Luis
Keyword(s):  
Transformation Method ◽  
High Order ◽  
Numerical Continuation ◽  
Reversible Systems ◽  
Efficient Manner ◽  
Codimension Two ◽  
Codimension One ◽  
Melnikov Functions ◽  
Global Connections ◽  
Nonlinear Time Transformation

A codimension-three Takens–Bogdanov bifurcation in reversible systems has been very recently analyzed in the literature. In this paper, we study with the help of the nonlinear time transformation method, the codimension-one and -two homoclinic and heteroclinic connections present in the corresponding unfolding. The algorithm developed allows to obtain high-order approximations for the global connections, in such a way that it supplies in a very efficient manner the coefficients that would be obtained with high-order Melnikov functions. As we show, all our analytical predictions have excellent agreement with the numerical results. In particular we remark that, for the two different codimension-two points, the theoretical approximation coincides in six decimal digits with the numerical continuation, even being quite far from the codimension-three point. The better approximations we provide in this work will help in the study of reversible systems that exhibit this codimension-three Takens–Bogdanov bifurcation.

High-Order Analysis of Canard Explosion in the Brusselator Equations

2020 ◽  
Vol 30 (05) ◽  
pp. 2050078 ◽  
Author(s):  
Bo-Wei Qin ◽  
Kwok-Wai Chung ◽  
Antonio Algaba ◽  
Alejandro J. Rodríguez-Luis
Keyword(s):  
Order Approximation ◽  
Transformation Method ◽  
High Order ◽  
Numerical Continuation ◽  
Chemical System ◽  
High Order Approximation ◽  
Order Analysis ◽  
First Order Approximation ◽  
Strongly Agree ◽  
Nonlinear Time Transformation

The aim of this paper is to obtain a high-order approximation of the canard explosion in the Brusselator equations. This classical chemical system has been extensively studied but, until now, only first-order approximation to the canard explosion has been provided. Here, with the help of the nonlinear time transformation method, we are able to obtain an approximation to any desired order. Our results strongly agree with those obtained by numerical continuation.

Codimension-One and -Two Bifurcations of a Three-Dimensional Discrete Game Model

2021 ◽  
Vol 31 (02) ◽  
pp. 2150023
Author(s):  
Zohreh Eskandari ◽  
Javad Alidousti ◽  
Reza Khoshsiar Ghaziani
Keyword(s):  
Center Manifold ◽  
Continuation Method ◽  
Three Dimensional ◽  
Period Doubling ◽  
Numerical Continuation ◽  
Game Model ◽  
Codimension Two ◽  
Codimension One ◽  
Two Parameters ◽  
Discrete Game

In this paper, bifurcation analysis of a three-dimensional discrete game model is provided. Possible codimension-one (codim-1) and codimension-two (codim-2) bifurcations of this model and its iterations are investigated under variation of one and two parameters, respectively. For each bifurcation, normal form coefficients are calculated through reduction of the system to the associated center manifold. The bifurcations detected in this paper include transcritical, fold, flip (period-doubling), Neimark–Sacker, period-doubling Neimark–Sacker, resonance 1:2, resonance 1:3, resonance 1:4 and fold-flip bifurcations. Moreover, we depict bifurcation diagrams corresponding to each bifurcation with the aid of numerical continuation method. These bifurcation curves not only confirm our analytical results, but also reveal a richer dynamics of the model especially in the higher iterations.

Bifurcations of Symmetric Periodic Orbits Near Equilibrium in Reversible Systems

1997 ◽  
Vol 07 (03) ◽  
pp. 569-584 ◽  
Author(s):  
Chih-Wen Shih
Keyword(s):  
Normal Form ◽  
Real Number ◽  
Periodic Solutions ◽  
Simple Form ◽  
Reversible Systems ◽  
Codimension Two ◽  
Codimension One ◽  
Nonzero Real Number ◽  
Symmetric Periodic Orbits ◽  
Complex Basis

Consider a family of reversible systems [Formula: see text] with the origin being an equilibrium for each μ. Suppose Dxf(0, 0) has only purely imaginary eigenvalues ±iw1,…,±iwk. We investigate the typical bifurcations of symmetric periodic solutions near the origin. A suitable complex basis is chosen so that Dxf(0, 0) and the involution are in respective simple form. Incorporated with putting f into normal form, a modified version of Lyapunov–Schmidt reduction can be applied to obtain the reduced bifurcation equations. We then focus on the cases in resonance, that is, wj = njw0, where w0 is a nonzero real number and nj is an integer for each j. Some codimension-two bifurcations are illustrated for the system in non-semisimple resonance with nj = 1, 2. A few codimension-one cases are also given for comparison with earlier works by other researchers.

Global Bifurcation Analysis of a Population Model with Stage Structure and Beverton–Holt Saturation Function

2015 ◽  
Vol 25 (12) ◽  
pp. 1550170 ◽  
Author(s):  
Li Fan ◽  
Sanyi Tang
Keyword(s):  
Bifurcation Analysis ◽  
Population Model ◽  
Birth Rate ◽  
Global Bifurcation ◽  
Stage Structure ◽  
Phase Portraits ◽  
Approximation Techniques ◽  
Codimension Two ◽  
Codimension One ◽  
Stage Transition

In the present paper, we perform a complete bifurcation analysis of a two-stage population model, in which the per capita birth rate and stage transition rate from juveniles to adults are density dependent and take the general Beverton–Holt functions. Our study reveals a rich bifurcation structure including codimension-one bifurcations such as saddle-node, Hopf, homoclinic bifurcations, and codimension-two bifurcations such as Bogdanov–Takens (BT), Bautin bifurcations, etc. Moreover, by employing the polynomial analysis and approximation techniques, the existences of equilibria, Hopf and BT bifurcations as well as the formulas for calculating their bifurcation sets have been provided. Finally, the complete bifurcation diagrams and associate phase portraits are obtained not only analytically but also confirmed and extended numerically.

Unknotting tori in codimension one and spheres in codimension two

1965 ◽  
Vol 61 (3) ◽  
pp. 659-664 ◽  
Author(s):  
C. T. C. Wall
Keyword(s):  
Euclidean Space ◽  
Unit Sphere ◽  
Piecewise Linear ◽  
Differential Topology ◽  
Standard Unit ◽  
Codimension Two ◽  
Codimension One

We shall present this paper in the framework and terminology of differential topology though all our arguments are valid in the piecewise linear ease also, under local un-knottedness hypotheses. In particular we use Rp for Euclidean space of dimension p, Sp−1 for the standard unit sphere in it, and Dp for the disc which it bounds.

scholarly journals A NUMERICAL TOOLBOX FOR HOMOCLINIC BIFURCATION ANALYSIS

1996 ◽  
Vol 06 (05) ◽  
pp. 867-887 ◽  
Author(s):  
A.R. CHAMPNEYS ◽  
YU. A. KUZNETSOV ◽  
B. SANDSTEDE
Keyword(s):  
Numerical Analysis ◽  
Chemical Kinetics ◽  
Bifurcation Analysis ◽  
Homoclinic Orbit ◽  
Homoclinic Orbits ◽  
Homoclinic Bifurcation ◽  
Codimension Two ◽  
Codimension One ◽  
Homoclinic Bifurcations ◽  
Theoretical Results

This paper presents extensions and improvements of recently developed algorithms for the numerical analysis of orbits homoclinic to equilibria in ODEs and describes the implementation of these algorithms within the standard continuation package AUTO86. This leads to a kind of toolbox, called HOMCONT, for analysing homoclinic bifurcations either as an aid to producing new theoretical results, or to understand dynamics arising from applications. This toolbox allows the continuation of codimension-one homoclinic orbits to hyperbolic or non-hyperbolic equilibria as well as detection and continuation of higher-order homoclinic singularities in more parameters. All known codimension-two cases that involve a unique homoclinic orbit are supported. Two specific example systems from ecology and chemical kinetics are analysed in some detail, allowing the reader to understand how to use the the toolbox for themselves. In the process, new results are also derived on these two particular models.

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.

scholarly journals The n -component nonlinear Schrödinger equations: dark–bright mixed N - and high-order solitons and breathers, and dynamics

2018 ◽  
Vol 474 (2215) ◽  
pp. 20170688 ◽  
Author(s):  
Guoqiang Zhang ◽  
Zhenya Yan
Keyword(s):  
Numerical Simulations ◽  
Explicit Formula ◽  
Soliton Solutions ◽  
Transformation Method ◽  
High Order ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations

The general n -component nonlinear Schrödinger equations are systematically investigated with the aid of the Darboux transformation method and its extension. Firstly, we explore the condition of the existence for dark–bright mixed soliton solutions and derive an explicit formula of dark–bright mixed multi-soliton solutions in terms of the determinant. Secondly, we present the formula of dark–bright mixed high-order semi-rational solitons, and predict their general N th-order wave structures. Thirdly, we investigate the wing-shaped structures of breather. Finally, we perform the numerical simulations for some representative solitons to study their dynamical behaviours.

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