Degenerate Hopf Bifurcation Research Articles

Degenerate Hopf bifurcations in the Lü system

In this Letter we study the stability and local codimension one, two and three Hopf bifurcations which occur in the Lü system. A description of the regions in the parameter space for which multiple small periodic solutions arise through the Hopf bifurcations at the equilibria is given.

Self-localized structures in vertical-cavity surface-emitting lasers with external feedback

In this paper, we analyze a model of broad area vertical-cavity surface-emitting lasers subjected to frequency-selective optical feedback. In particular, we analyze the spatio-temporal regimes arising above threshold and the existence and dynamical properties of cavity solitons. We build the bifurcation diagram of stationary self-localized states, finding that branches of cavity solitons emerge from the degenerate Hopf bifurcations marking the homogeneous solutions with maximal and minimal gain. These branches collide in a saddle-node bifurcation, defining a maximum pump current for soliton existence that lies below the threshold of the laser without feedback. The properties of these cavity solitons are in good agreement with those observed in recent experiments.

Strange Attractors in Periodically-Kicked Degenerate Hopf Bifurcations

We prove that spiral sinks (stable foci of vector fields) can be transformed into strange attractors exhibiting sustained, observable chaos if subjected to periodic pulsatile forcing. We show that this phenomenon occurs in the context of periodically-kicked degenerate supercritical Hopf bifurcations. The results and their proofs make use of a k-parameter version of the theory of rank one maps.

The Effect of Infinitesimal Damping on the Dynamic Instability Mechanism of Conservative Systems

The local instability of 2 degrees of freedom (DOF) weakly damped systems is thoroughly discussed using the Liénard‐Chipart stability criterion. The individual and coupling effect of mass and stiffness distribution on the dynamic asymptotic stability due to mainly infinitesimal damping is examined. These systems may be as follows: (a) unloaded (free motion) and (b) subjected to a suddenly applied load of constant magnitude and direction with infinite duration (forced motion). The aforementioned parameters combined with the algebraic structure of the damping matrix (being either positive semidefinite or indefinite) may have under certain conditions a tremendous effect on the Jacobian eigenvalues and then on the local stability of these autonomous systems. It was found that such systems when unloaded may exhibit periodic motions or a divergent motion, while when subjected to the above step load may experience either a degenerate Hopf bifurcation or periodic attractors due to a generic Hopf bifurcation. Conditions for the existence of purely imaginary eigenvalues leading to global asymptotic stability are fully assessed. The validity of the theoretical findings presented herein is verified via a nonlinear dynamic analysis.

Coexistence of Limit Cycles and Homoclinic Loops in a SIRS Model with a Nonlinear Incidence Rate

Recently, Ruan and Wang [J. Differential Equations, 188 (2003), pp. 135–163] studied the global dynamics of a SIRS epidemic model with vital dynamics and a nonlinear saturated incidence rate. Under certain conditions they showed that the model undergoes a Bogdanov–Takens bifurcation; i.e., it exhibits saddle-node, Hopf, and homoclinic bifurcations. They also considered the existence of none, one, or two limit cycles. In this paper, we investigate the coexistence of a limit cycle and a homoclinic loop in this model. One of the difficulties is to determine the multiplicity of the weak focus. We first prove that the maximal multiplicity of the weak focus is 2. Then feasible conditions are given for the uniqueness of limit cycles. The coexistence of a limit cycle and a homoclinic loop is obtained by reducing the model to a universal unfolding for a cusp of codimension 3 and studying degenerate Hopf bifurcations and degenerate Bogdanov–Takens bifurcations of limit cycles and homoclinic loops of order 2.

A BIPARAMETRIC BIFURCATION IN 3D CONTINUOUS PIECEWISE LINEAR SYSTEMS WITH TWO ZONES: APPLICATION TO CHUA'S CIRCUIT

In this paper, a possible degeneration of the focus-center-limit cycle bifurcation for piecewise smooth continuous systems is analyzed. The case of continuous piecewise linear systems with two zones is considered, and the coexistence of two limit cycles for certain values of parameters is justified. Finally, the Chua's circuit is shown to exhibit the analyzed bifurcation. The obtained bifurcation set in the parameter plane is similar to the degenerate Hopf bifurcation for differentiable systems.

The complex cubic–quintic Ginzburg–Landau equation: Hopf bifurcations yielding traveling waves

In this paper we use a traveling wave reduction or a so-called spatial approximation to comprehensively investigate the periodic solutions of the complex cubic–quintic Ginzburg–Landau equation. The primary tools used here are Hopf bifurcation theory and perturbation theory. Explicit results are obtained for the post-bifurcation periodic orbits and their stability. Generalized and degenerate Hopf bifurcations are also briefly considered to track the emergence of global structure such as homoclinic orbits.

Bifurcations of an epidemic model with non-linear incidence and infection-dependent removal rate

An epidemic model with a generalized non-linear incidence is extended to incorporate the effect of an infection-dependent removal strategy, which is defined as a function of the number of infected individuals. It is assumed that the removal rate decreases from a maximum capacity for removing infected individuals as their number increases. The existence and stability of the associated equilibria are analysed, and the basic reproductive number (R0) is formulated. It is shown that R0 is independent of the functional form of the incidence, but depends on the removal rate. Normal forms are derived to show the different types of bifurcation the model undergoes, including transcritical, generalized Hopf (Bautin), saddle-node and Bogdanov-Takens. A degenerate Hopf bifurcation at the Bautin point, where the first Lyapunov coefficient vanishes, is discussed. Sotomayor's theorem is applied to establish a saddle-node bifurcation at the turning point of backward bifurcation. The Bogdanov-Takens normal form is derived, from which the local bifurcation curve for a family of homoclinic orbits is formulated. Bifurcation diagrams and numerical simulations, using parameter values estimated for some infectious diseases, are also presented to provide more intuition to the theoretical findings. The results show that sufficiently increasing the removal rate can reduce R0 below a subthreshold domain, which leads to disease eradication.

First Liapunov coefficient for coupled identical oscillators. Application to coupled demand–supply model

AbstractA general formula for the computation of the first Liapunov coefficient corresponding to the Hopf bifurcation in a four‐dimensional system of two coupled identical oscillators is performed for two cases. Only bi‐dimensional vectors are involved. Then a model of two coupled demand–supply systems, depending on four parameters is considered. A study of the Hopf bifurcation is done around one of the symmetrical equilibrium, as the parameters vary. The loci in the parameter space of the parameters values corresponding to subcritical, supercritical or degenerated Hopf bifurcation are found. The computation of the Liapunov coefficients is done using the derived formula. Numerical plots emphasizing the existence of different types of limit cycles are developed. Copyright © 2006 John Wiley & Sons, Ltd.

Neimark–Sacker bifurcations in a non-standard numerical scheme for a class of positivity-preserving ODEs

We discuss the nature of Neimark–Sacker bifurcations occurring in a non-standard numerical scheme, for a class of positivity-preserving systems of ordinary differential equations (ODEs) which undergoes a corresponding Hopf bifurcation. Extending previous work (Alexander & Moghadas 2005 a Electron. J. Diff. Eqn. Conf . 12 , 9–19), it is shown that the type of Neimark–Sacker bifurcation (supercritical or subcritical) may be affected by the integration time-step . The general form of the scheme in the vicinity of a fixed point is given, from which the expression for the first Lyapunov coefficient for two-dimensional systems, valid for arbitrary time-step, is explicitly derived. The analysis shows that this coefficient undergoes an shift with respect to the corresponding coefficient of the original ODE. This could lead to a type of bifurcation which differs from the corresponding Hopf bifurcation in the ODE, due to changes in the sign of the first Lyapunov coefficient as varies. This is especially problematic in the vicinity of certain types of degenerate Hopf bifurcation, at which this coefficient may vanish. We also present a general method to eliminate the possible shift in the bifurcation parameter of the scheme; however, the first Lyapunov coefficient may still be subjected to an shift, leading to a possibly erroneous type of bifurcation. Examples are given to illustrate the theoretical results of the paper with applications to mathematical biology.

Hamiltonian weakly damped autonomous systems exhibiting periodic attractors

The dynamic local stability of autonomous Hamiltonian, weakly damped, lumped-mass (discrete) systems is reconsidered. For such potential(conservative) systems conditions for the existence of limit cycles are discussed by studying the effect of the damping matrix on the Jacobian eigenvalues. New findings that contradict existing results are presented. Thus, undamped stable symmetric systems with the inclusion of slight damping may experience: (a) a double zero eigenvalue bifurcation, a degenerate Hopf bifurcation and a generic (usual) Hopf bifurcation, and (b) a limit cycle (dynamic) mode of instability prior to the static (divergence) mode of instability (failure of Zieglers kinetic criterion). A variety of numerical examples verified by a nonlinear analysis confirm the validity of the theoretical findings presented herein.

Multiple Focus and Hopf Bifurcations in a Predator-Prey System with Nonmonotonic Functional Response

In this paper, we develop a criterion to calculate the multiplicity of a multiple focus for general predator-prey systems. As applications of this criterion, we calculate the largest multiplicity of a multiple focus in a predator-prey system with nonmonotonic functional response $p(x)=\frac{x}{ax^2+bx+1}$ studied by Zhu, Campbell, and Wolkowicz [SIAM J. Appl. Math., 63 (2002), pp. 636-682] and prove that the degenerate Hopf bifurcation is of codimension two. Furthermore, we show that there exist parameter values for which this system has a unique positive hyperbolic stable equilibrium and exactly two limit cycles, the inner one unstable and outer one stable. Numerical simulations for the existence of the two limit cycles bifurcated from the multiple focus are also given in support of the criterion.

Traveling wavetrains in the complex cubic–quintic Ginzburg–Landau equation

In this paper we use a traveling wave reduction or a so-called spatial approximation to comprehensively investigate the periodic solutions of the complex cubic-quintic Ginzburg-Landau equation. The primary tools used here are Hopf bifurcation theory and perturbation theory. Explicit results are obtained for the post-bifurcation periodic orbits and their stability. Generalized and degenerate Hopf bifurcations are also briefly considered to track the emergence of global structure such as homoclinic orbits.

Bifurcation of limit cycles in a quintic Hamiltonian system under a sixth-order perturbation

Abstract This paper intends to explore the bifurcation of limit cycles for planar polynomial systems with even number of degrees. To obtain the maximum number of limit cycles, a sixth-order polynomial perturbation is added to a quintic Hamiltonian system, and both local and global bifurcations are considered. By employing the detection function method for global bifurcations of limit cycles and the normal form theory for local degenerate Hopf bifurcations, 31 and 35 limit cycles and their configurations are obtained for different sets of controlled parameters. It is shown that: H (6) ⩾ 35 = 6 2 − 1, where H (6) is the Hilbert number for sixth-degree polynomial systems.

MULTIPARAMETRIC BIFURCATIONS IN AN ENZYME-CATALYZED REACTION MODEL

An exhaustive analysis of local and global bifurcations in an enzyme-catalyzed reaction model is carried out. The model, given by a planar five-parameter system of autonomous ordinary differential equations, presents a great richness of bifurcations. This enzyme-catalyzed model has been considered previously by several authors, but they only detected a minimal part of the dynamical and bifurcation behavior exhibited by the system.First, we study local bifurcations of equilibria up to codimension-three (saddle-node, cusps, nondegenerate and degenerate Hopf bifurcations, and nondegenerate and degenerate Bogdanov–Takens bifurcations) by using analytical and numerical techniques. The numerical continuation of curves of global bifurcations allows to improve the results provided by the study of local bifurcations of equilibria and to detect new homoclinic connections of codimension-three. Our analysis shows that such a system exhibits up to sixteen different kinds of homoclinic orbits and thirty different configurations of equilibria and periodic orbits. The coexistence of up to five periodic orbits is also pointed out. Several bifurcation sets are sketched in order to show the dynamical behavior the system exhibits. The different codimension-one and -two bifurcations are organized around five codimension-three degeneracies.

Hopf bifurcations of ODE systems along the singular direction in the parameter plane

This paper considers Hopf bifurcation of ordinary differential systems along the singular direction in the parameter plane. Hopf Bifurcation of the same system along non-singular directions have been studied recently in [9]. As an application of our main results, we also obtain results of existence and non-existence for a type of degenerate Hopf bifurcation (i.e., the type without transversality) for one parameter ODE systems.

Hopf bifurcation in a system of two coupled advertising oscillators

A system of two symmetrical coupled identical oscillators, each of them being an advertising model, is considered. It possesses 4 variables and it depends on 3 parameters. The existence of five equilibrium points is found and the locus in the parameter space of values corresponding to non-hyperbolic singularities is emphasized. The topological type of the symmetric equilibrium point, which is the origin, is determined. Then the parameter values of non-degenerated or degenerated Hopf bifurcation corresponding to the origin are found by computing the first Liapunov coefficient. The center manifolds associated with these values are determined. Some 2D and 3D projections of specific phase portraits are plotted. Numerical computations in accordance with the theoretical results show the presence of stable limit cycles with 8 regimes of behavior.

Bifurcation Theory: An Introduction With Applications to PBEs (Applied Mathematical Sciences 156 Series)

9R3. Bifurcation Theory: An Introduction With Applications to PBEs (Applied Mathematical Sciences 156 Series). - Edited by H Kielhofer (Inst for Math, Univ of Augsburg, Universitatsstr 14, Raum 2011, Augsburg D-86135 Germany). Springer-Verlag, New York. 2004. 346 pp. ISBN 0-387-40401-5. $69.95.Reviewed by HW Haslach Jr (Dept of Mech Eng, Univ. of Maryland, College Park MD 20742-3035).This unified exposition of the single-parameter bifurcation response of an operator on infinite dimensional space is organized into three sections: an abstract development of local and then global bifurcation and applications to elliptic and parabolic partial differential equations. The local bifurcation theory, taking up about half the book, depends on the breakdown of the implicit function theorem and is based on the Lyapunov-Schmidt reduction for infinite dimensional spaces. The case that the Fre´chet derivative has a one-dimensional kernel includes the saddle-node and the various types of pitchfork bifurcations. A principle of exchange of stability is proved. A Hopf bifurcation theorem for periodic solutions is first proved for ordinary differential equations and then for retarded functional differential equations. A center manifold theorem is used for Hopf bifurcation of Hamiltonian, reversible, and conservative systems. A principle of exchange of stability for Hopf bifurcations is proved, and the stability of global continuation of solutions is examined. A principle of exchange of stability for period-doubling bifurcations is developed. The Newton polygon method is described for problems with a one-dimensional kernel in both the degenerate and nondegenerate cases. Floquet exponents are used to create a principle of exchange of stability for degenerate Hopf bifurcations. Some results are given for higher dimensional kernels of the Fre´chet derivative and multiparameter bifurcation. The global bifurcation theory, subject of the second section, is based on the local index of the operator F, obtained from the Leray-Schauder degree which takes the place of the Brouwer degree of finite-dimensional problems. The crossing number, which is related to a change in Morse index, is defined from the index of the operator. Then if DxF0,λ has an odd crossing number for a particular value of the bifurcation parameter λ, that value of the parameter gives a bifurcation point. A degree is defined for a class of Fredholm operators that includes some classes of elliptic operators and for potential operators, and global bifurcation theorems proved, often using Rabinowitz’s theorem. The applications forming the third section begin with results on elliptic operators of second order acting on scalar functions. An example using the degenerate Newton polygon is developed. The particular case of the Cahn-Hilliard energy that describes a binary alloy is solved by taking the mass as the bifurcation parameter and applying a symmetry constraint to obtain a one-dimensional kernel. Then local bifurcations of the free nonlinear vibrations described by the nonlinear wave equation with the period as bifurcation parameter are examined in the one-dimensional case and some extensions are given to higher dimensions. Results are also obtained for the wave equation on the unit sphere. A Hopf bifurcation is analyzed for a nonlinear parabolic problem in which the principle of exchange of stability holds. Finally global bifurcation and continuation results are given for quasi-linear elliptic problems and are applied to the Euler-Lagrange equation for the Cahn-Hilliard energy problem. This mathematics book requires substantial mathematical background. The book is directed to advanced readers rather than beginners so that some basic ideas are not defined. The author has separated the abstract description in the first two sections from the applications with the goal of clarity and allowing the results to be available for any application, but at the risk of making it harder to understand the significance of some of the ideas. In many applications to models of physical behavior, the hypotheses of the various theorems are quite difficult to verify. The book is very useful as a reference because it collects and organizes the bifurcation analysis of infinite-dimensional operators. It could also be used as a text in an advanced course on bifurcation theory with an emphasis on partial differential equations.

Discrete May–Leonard Competition Models III

A discretization method attributed to Kahan is used to approximate the May–Leonard (M–L) competition model for three species. The local dynamics of this discrete-time M–L model are analyzed. This model differs from the discrete M–L models being studied previously. This discrete model shows dynamical consistency with the continuous M–L model. Numerically, we showed that the discrete M–L model has a degenerate Hopf-bifurcation, which is consistent with the continuous M–L model.

Onset of Degenerate Hopf Bifurcation of a Vibro-Impact Oscillator

An analytical method of the degenerate Hopf bifurcation is proposed for vibro-impact systems. The phenomenon of the bifurcation and its complicated dynamics are observed. This type of bifurcation originates multi-coexisting solutions dependent of the initial state of the system.