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.

HOMOCLINIC BIFURCATIONS IN A PLANAR DYNAMICAL SYSTEM

2001 ◽  
Vol 11 (04) ◽  
pp. 1183-1191 ◽  
Author(s):  
FOTIOS GIANNAKOPOULOS ◽  
TASSILO KÜPPER ◽  
YONGKUI ZOU
Keyword(s):  
Dynamical System ◽  
Numerical Methods ◽  
Bifurcation Diagram ◽  
Homoclinic Orbit ◽  
Homoclinic Orbits ◽  
Intersection Point ◽  
Homoclinic Bifurcation ◽  
Numerical Techniques ◽  
Planar Dynamical System ◽  
Homoclinic Bifurcations

The homoclinic bifurcation properties of a planar dynamical system are analyzed and the corresponding bifurcation diagram is presented. The occurrence of two Bogdanov–Takens bifurcation points provides two local existing curves of homoclinic orbits to a saddle excluding the separatrices not belonging to the homoclinic orbits. Using numerical techniques, these curves are continued in the parameter space. Two further curves of homoclinic orbits to a saddle including the separatrices not belonging to the homoclinic orbits are calculated by numerical methods. All these curves of homoclinic orbits have a unique intersection point, at which there exists a double homoclinic orbit. The local homoclinic bifurcation diagram of both the double homoclinic orbit point and the points of homoclinic orbits to a saddle-node are also gained by numerical computation and simulation.

scholarly journals NUMERICAL DETECTION AND CONTINUATION OF CODIMENSION-TWO HOMOCLINIC BIFURCATIONS

1994 ◽  
Vol 04 (04) ◽  
pp. 785-822 ◽  
Author(s):  
A.R. CHAMPNEYS ◽  
Yu. A. KUZNETSOV
Keyword(s):  
Ad Hoc ◽  
Autonomous Systems ◽  
Numerical Procedure ◽  
Homoclinic Orbits ◽  
Homoclinic Loop ◽  
Codimension Two ◽  
Codimension One ◽  
Homoclinic Bifurcations ◽  
Nagumo Equations ◽  
Boundary Value Method

A numerical procedure is presented for the automatic accurate location of certain codimension-two homoclinic singularities along curves of codimension-one homoclinic bifurcations to hyperbolic equilibria in autonomous systems of ordinary differential equations. The procedure also allows for the continuation of multiple-codimension homoclinic orbits in the relevant number of free parameters. A systematic treatment is given of codimension-two bifurcations that involve a unique homoclinic orbit. In each case the known theoretical results are reviewed and a regular test-function is derived for a truncated problem. In particular, the test-functions for global degeneracies involving the orientation of a homoclinic loop are presented. It is shown how such a procedure can be incorporated into an existing boundary-value method for homoclinic continuation and implemented using the continuation code AUTO. Several examples are studied, including Chua’s electronic circuit and the FitzHugh-Nagumo equations. In each case, the method is shown to reproduce codimension-two bifurcation points that have previously been found using ad hoc methods, and, in some cases, to obtain new results.

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.

Bifurcation Analysis of a Dynamical Model for the Innate Immune Response to Initial Pulmonary Infections

2020 ◽  
Vol 30 (16) ◽  
pp. 2050252
Author(s):  
Shujing Shi ◽  
Jicai Huang ◽  
Jing Wen ◽  
Shigui Ruan
Keyword(s):  
Hopf Bifurcation ◽  
Bifurcation Analysis ◽  
Polymorphonuclear Leukocytes ◽  
Homoclinic Orbits ◽  
Organ Damage ◽  
Innate Immune ◽  
Dynamical Model ◽  
System Response ◽  
Pulmonary Infections ◽  
Theoretical Results

It has been reported that COVID-19 patients had an increased neutrophil count and a decreased lymphocyte count in the severe phase and neutrophils may contribute to organ damage and mortality. In this paper, we present the bifurcation analysis of a dynamical model for the initial innate system response to pulmonary infection. The model describes the interaction between a pathogen and neutrophilis (also known as polymorphonuclear leukocytes). It is shown that the system undergoes a sequence of bifurcations including subcritical and supercritical Bogdanov–Takens bifurcations, Hopf bifurcation, and degenerate Hopf bifurcation as the parameters vary, and the model exhibits rich dynamics such as the existence of multiple coexistent periodic oscillations, homoclinic orbits, bistability and tristability, etc. Numerical simulations are presented to explain the theoretical results.

Differential Equations with Bifocal Homoclinic Orbits

1997 ◽  
Vol 07 (01) ◽  
pp. 27-37 ◽  
Author(s):  
Paul Glendinning
Keyword(s):  
Differential Equations ◽  
Homoclinic Orbit ◽  
Stationary Point ◽  
Piecewise Linear ◽  
Global Bifurcation ◽  
Homoclinic Orbits ◽  
Bifurcation Phenomena ◽  
Global Bifurcation Theory ◽  
Linear Differential ◽  
Theoretical Results

Global bifurcation theory can be used to understand complicated bifurcation phenomena in families of differential equations. There are many theoretical results relating to systems having a homoclinic orbit biasymptotic to a stationary point at some value of the parameters, and these results depend upon the eigenvalues of the Jacobian matrix of the flow evaluated at the stationary point. Three important cases arise in the theoretical analysis, and there are many examples of systems which illustrate two of these three cases. We describe a construction which can be used to produce examples of the third case (bifocal homoclinic orbits), and use this construction to prove the existence of a bifocal homoclinic orbit in a simple piecewise linear differential equation.

scholarly journals A non-transverse homoclinic orbit to a saddle-node equilibrium

1996 ◽  
Vol 16 (3) ◽  
pp. 431-450 ◽  
Author(s):  
Alan R. Champneys ◽  
Jörg Härterich ◽  
Björn Sandstede
Keyword(s):  
Homoclinic Orbit ◽  
Homoclinic Orbits ◽  
Bifurcation Curve ◽  
Numerical Computations ◽  
Parameter Region ◽  
Codimension Two ◽  
Saddle Node ◽  
Unstable Manifolds ◽  
The Creation ◽  
Shift Dynamics

AbstractA homoclinic orbit is considered for which the center-stable and center-unstable manifolds of a saddle-node equilibrium have a quadratic tangency. This bifurcation is of codimension two and leads generically to the creation of a bifurcation curve defining two independent transverse homoclinic orbits to a saddle-node. This latter case was shown by Shilnikov to imply shift dynamics. It is proved here that in a large open parameter region of the codimension-two singularity, the dynamics are completely described by a perturbation of the Hénon-map giving strange attractors, Newhouse sinks and the creation of the shift dynamics. In addition, an example system admitting this bifurcation is constructed and numerical computations are performed on it.

HOMOCLINIC BIFURCATION IN SEMI-CONTINUOUS DYNAMIC SYSTEMS

2012 ◽  
Vol 05 (06) ◽  
pp. 1250059 ◽  
Author(s):  
CHUANJUN DAI ◽  
MIN ZHAO ◽  
LANSUN CHEN
Keyword(s):  
Periodic Solution ◽  
Dynamic System ◽  
Dynamic Systems ◽  
Vector Fields ◽  
Homoclinic Bifurcation ◽  
Homoclinic Bifurcations ◽  
Continuous Dynamic System ◽  
Continuous Dynamic ◽  
Theoretical Results

In this paper, a class of homoclinic bifurcations in semi-continuous dynamic systems are investigated. On the basis of rotated vector fields theory, existence of order-1 periodic solution and the rotated vector fields of the semi-continuous dynamic system are discussed. Furthermore, homoclinic cycles and homoclinic bifurcations are described. Finally, an example is provided to show the validity of our theoretical results.

Codimension-Two Bifurcation Analysis of a Vehicle Suspension System Moving Over Rough Surface

2018 ◽  
Vol 18 (12) ◽  
pp. 1871012
Author(s):  
Chun-Cheng Chen ◽  
Shun-Chang Chang
Keyword(s):  
Bifurcation Analysis ◽  
Dynamical Behavior ◽  
Homoclinic Orbits ◽  
Suspension System ◽  
Nonlinear Dynamical ◽  
Codimension Two ◽  
Quarter Car Model ◽  
Road Profile ◽  
Vehicle Suspension System ◽  
Codimension Two Bifurcation

This paper examines the dynamics of a nonlinear semi-active suspension system using a quarter-car model moving over rough road profiles. The bifurcation analysis of the nonlinear dynamical behavior of this system is performed. Codimension-two bifurcation and homoclinic orbits can be discovered in this system. When the external force of a road profile was added to this system as a parameter with a certain range of values, a strange attractor can be found using the numerical simulation. Finally, the Lyapunov exponent is adopted to identify the onset of chaotic motion and verify the bifurcation analysis.

Approximations of the Homoclinic Orbits Near a Double-Zero Bifurcation with Symmetry of Order Two

2017 ◽  
Vol 27 (07) ◽  
pp. 1750109 ◽  
Author(s):  
Carmen Rocşoreanu ◽  
Mihaela Sterpu
Keyword(s):  
Normal Form ◽  
Blow Up ◽  
Homoclinic Orbits ◽  
Homoclinic Bifurcation ◽  
Second Order ◽  
Dimensional System ◽  
Double Zero ◽  
Codimension Two ◽  
Two Dimensional System ◽  
Codimension Two Bifurcation

The two-dimensional system of differential equations corresponding to the normal form of the double-zero bifurcation with symmetry of order two is considered. This is a codimension two bifurcation. The associated dynamical system exhibits, among others, a homoclinic bifurcation. In this paper, we obtain second order approximations both for the curve of parametric values of homoclinic bifurcation and for the homoclinic orbits. To perform this task, we reduce first the normal form to a perturbed Hamiltonian system, using a blow-up technique. Then, by means of a perturbation method, we determine explicit first and second order approximations of the homoclinic orbits. The solutions obtained theoretically are compared with those obtained numerically for several cases. Finally, an application of the obtained results is presented.

Homoclinic Bifurcations and Chaos in the Fishing Principle for the Lorenz-like Systems

2020 ◽  
Vol 30 (08) ◽  
pp. 2050124 ◽  
Author(s):  
G. A. Leonov ◽  
R. N. Mokaev ◽  
N. V. Kuznetsov ◽  
T. N. Mokaev
Keyword(s):  
Numerical Analysis ◽  
Homoclinic Orbit ◽  
Analytical Method ◽  
Qualitative Description ◽  
Homoclinic Bifurcations ◽  
Different Types

In this article using an analytical method called Fishing principle we obtain the region of parameters, where the existence of a homoclinic orbit to a zero saddle equilibrium in the Lorenz-like system is proved. For a qualitative description of the different types of homoclinic bifurcations, a numerical analysis of the obtained region of parameters is organized, which leads to the discovery of new bifurcation scenarios.

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