Degenerate Hopf Bifurcation Research Articles

Anti-control of Hopf bifurcation in the new chaotic system with piecewise system and exponential system

In order to understand the complex three-dimensional dynamical system with the unstable nodes, we propose a nonlinear controller. The corresponding controlling system makes the codimension one, two, and three Hopf bifurcations happen. The mathematical deduction demonstrates that the system can be controlled to produce the degenerate Hopf bifurcation at desired location and stability of controllable bifurcation.

Biological self-heating in compost piles: A Semenov formulation

A uniformly distributed mathematical model (based on Semenov's theory of thermal explosions) is formulated to model the thermal behaviour of cellulosic materials in compost piles. The model consists of a mass balance equation for oxygen, a heat balance equation and incorporates the heat release due to biological activity within the pile. Singularity theory and degenerate Hopf bifurcation theory are used to investigate the generic properties of the model as well as to determine the loci of different singularities: the isola, cusp, double-limit points, boundary limit set, double-Hopf bifurcation, generalised Hopf (Bautin) bifurcation and Bogdanov–Takens bifurcation. These loci divide the secondary parameter plane into different regions of solution behaviours. Conditions under which biological activity can result in the initiation of an elevated-temperature branch within the compost pile which does not pose the risk of spontaneous ignition are identified. These are the ideal conditions for composting. The regions of parameter space when spontaneous ignition is likely are also determined.

A numerical study of the effect of degenerate Hopf bifurcations on the voltage stability in power systems

Bifurcation theory has been employed to qualitatively explain the mechanisms of voltage stability problems occurring in electric power systems by relating the voltage collapse and voltage oscillatory problems to the appearance of saddle node and Hopf bifurcations, respectively. The results obtained by bifurcation analysis have revealed the existence of degenerate Hopf bifurcations when two parameters of the system are varied simultaneously; however, the mechanisms of voltage stability problems when the system is operating near these kinds of bifurcations have not yet been investigated. Bearing this in mind, the main scope of this paper is to assess how the voltage stability of a power system is affected by the presence of codimension-2 degenerate Hopf bifurcations. From the multiparameter bifurcation analysis, the existence of degenerate Hopf bifurcation points are identified and a trajectory stability diagram is then computed for classifying the type of degenerate Hopf bifurcations. This classification is used for establishing the voltage dynamics’ behavior when the power system is operating near these bifurcations. Finally, the application of a StatCom is proposed to eliminate sustained voltage oscillations and the appearance of degenerate Hopf bifurcations.

Bifurcation analysis in a predator-prey model with constant-yield predator harvesting

In this paper we study the effect of constant-yield predator harvesting on the dynamics of a Leslie-Gower type predator-prey model. It is shown that the model has a Bogdanov-Takens singularity (cusp case) of codimension 3 or a weak focus of multiplicity two for some parameter values, respectively. Saddle-node bifurcation, repelling and attracting Bogdanov-Takens bifurcations, supercritical and subcritical Hopf bifurcations, and degenerate Hopf bifurcation are shown as the values of parameters vary. Hence, there are different parameter values for which the model has a homoclinic loop or two limit cycles. It is also proven that there exists a critical harvesting value such that the predator specie goes extinct for all admissible initial densities of both species when the harvest rate is greater than the critical value. These results indicate that the dynamical behavior of the model is very sensitive to the constant-yield predator harvesting and the initial densities of both species and it requires careful management in the applied conservation and renewable resource contexts. Numerical simulations, including the repelling and attracting Bogdanov-Takens bifurcation diagrams and corresponding phase portraits, two limit cycles, the coexistence of a stable homoclinic loop and an unstable limit cycle, and a stable limit cycle enclosing an unstable multiple focus with multiplicity one, are presented which not only support the theoretical analysis but also indicate the existence of Bogdanov-Takens bifurcation (cusp case) of codimension 3. These results reveal far richer and much more complex dynamics compared to the model without harvesting or with only constant-yield prey harvesting.

Complex Dynamics in Predator-prey Models with Nonmonotonic Functional Response and Harvesting

In this paper we study the complex dynamics of predator-prey systems with nonmonotonic functional response and harvesting. When the harvesting is constant-yield for prey, it is shown that various kinds of bifurcations, such as saddle-node bifurcation, degenerate Hopf bifurcation, and Bogdanov-Takens bifurcation, occur in the model as parameters vary. The existence of two limit cycles and a homoclinic loop is established by numerical simulations. When the harvesting is seasonal for both species, sufficient conditions for the existence of an asymptotically stable periodic solution and bifurcation of a stable periodic orbit into a stable invariant torus of the model are given. Numerical simulations are carried out to demonstrate the existence of bifurcation of a stable periodic orbit into an invariant torus and transition from invariant tori to periodic solutions, respectively, as the amplitude of seasonal harvesting increases.

DELAYED BIFURCATION IN FIRST-ORDER SINGULARLY PERTURBED PROBLEMS WITH A NONGENERIC TURNING POINT

In this paper, based on the method of upper and lower solutions, delayed bifurcation in first-order singularly perturbed problems with a nongeneric turning point is studied. The asymptotic behavior of the solutions is understood by constructing the upper and lower solutions with the desired dynamical properties. As an application of the obtained results, delayed phenomenon of degenerate Hopf bifurcation in a planar polynomial differential system with a slowly varying parameter is discussed in detail and the maximal delay is calculated. Numerical simulations are carried out to verify the theoretical results.

PERIODIC ORBITS NEAR EQUILIBRIA VIA AVERAGING THEORY OF SECOND ORDER

Lyapunov, Weinstein and Moser obtained remarkable theorems giving sufficient conditions for the existence of periodic orbits emanating from an equilibrium point of a differential system with a first integral. Using averaging theory of first order we established in [1] a similar result for a differential system without assuming the existence of a first integral. Now, using averaging theory of the second order, we extend our result to the case when the first order average is identically zero. Our result can be interpreted as a kind of degenerated Hopf bifurcation.

GENERALIZED HOPF BIFURCATION IN A FREQUENCY DOMAIN FORMULATION

The multiplicity problem of limit cycles arising from a weak focus is addressed. The proposed methodology is a combination of the frequency domain method to handle some degenerate Hopf bifurcations with the powerful tools of the singularity theory. The frequency domain approach uses the harmonic balance method to study the existence of periodic solutions. On the other hand, the singularity theory provides the conditions and formulas for the classification problem of the unfolding of the singularity in terms of the distinguished and auxiliary parameters. A classical example introduced by Bautin is shown in which the multiplicity of limit cycles is recovered by using this type of hybrid methodology and standard software in the continuation of periodic solutions (LOCBIF and XPPAUT). For small amplitude limit cycles, the proposed methodology gives accurate results.

Multiparametric bifurcations of an epidemiological model with strong Allee effect

In this paper we completely study bifurcations of an epidemic model with five parameters introduced by Hilker etal. (Am Nat 173:72-88, 2009), which describes the joint interplay of a strong Allee effect and infectious diseases in a single population. Existence of multiple positive equilibria and all kinds of bifurcation are examined as well as related dynamical behavior. It is shown that the model undergoes a series of bifurcations such as saddle-node bifurcation, pitchfork bifurcation, Bogdanov-Takens bifurcation, degenerate Hopf bifurcation of codimension two and degenerate elliptic type Bogdanov-Takens bifurcation of codimension three. Respective bifurcation surfaces in five-dimensional parameter spaces and related dynamical behavior are obtained. These theoretical conclusions confirm their numerical simulations and conjectures by Hilker etal., and reveal some new bifurcation phenomena which are not observed in Hilker etal. (Am Nat 173:72-88, 2009). The rich and complicated dynamics exhibit that the model is very sensitive to parameter perturbations, which has important implications for disease control of endangered species.

Stable periodic oscillations in a two-stage cancer model of tumor and immune system interactions

This paper presents qualitative and bifurcation analysis near the degenerate equilibrium in a two-stage cancer model of interactions between lymphocyte cells and solid tumor and contributes to a better understanding of the dynamics of tumor and immune system interactions. We first establish the existence of Hopf bifurcation in the 3-dimensional cancer model and rule out the occurrence of the degenerate Hopf bifurcation. Then a general Hopf bifurcation formula is applied to determine the stability of the limit cycle bifurcated from the interior equilibrium. Sufficient conditions on the existence of stable periodic oscillations of tumor levels are obtained for the two-stage cancer model. Numerical simulations are presented to illustrate the existence of stable periodic oscillations with reasonable parameters and demonstrate the phenomenon of long-term tumor relapse in the model.

BIFURCATION OF LIMIT CYCLES IN SMALL PERTURBATIONS OF A HYPER-ELLIPTIC HAMILTONIAN SYSTEM WITH TWO NILPOTENT SADDLES

In this paper we study the first-order Melnikov function for a planar near-Hamiltonian system near a heteroclinic loop connecting two nilpotent saddles. The asymptotic expansion of this Melnikov function and formulas for the first seven coefficients are given. Next, we consider the bifurcation of limit cycles in a class of hyper-elliptic Hamiltonian systems which has a heteroclinic loop connecting two nilpotent saddles. It is shown that this system can undergo a degenerate Hopf bifurcation and Poincare bifurcation, which emerges at most four limit cycles in the plane for sufficiently small positive e. The number of limit cycles which appear near the heteroclinic loop is discussed by using the asymptotic expansion of the first-order Melnikov function. Further more we give all possible distribution of limit cycles bifurcated from the period annulus.

Nonlinear analysis of a novel three-scroll chaotic system

This paper reports the nonlinear dynamics of a novel three-scroll chaotic system. The local stability of hyperbolic equilibrium and non-hyperbolic equilibrium are investigated by using center manifold theorem. Pitchfork bifurcation, degenerate pitchfork bifurcation and Hopf bifurcation are analyzed when the parameters are varied in the space of parameter. For a suitable choice of the parameters, the existence of singularly degenerate heteroclinic cycles and Hopf bifurcation without parameters are also investigated. Some numerical simulations are given to support the analytic results.

On the number of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems

This paper deals with small perturbations of a class of hyper-elliptic Hamiltonian systems of degree 6, which is a Liénard system of the form x ′ = y , y ′ = Q 1 ( x ) + ε y Q 2 ( x ) with Q 1 and Q 2 polynomials of degree 5 and 4, respectively. It is shown that this system can undergo degenerate Hopf bifurcation and Poincaré bifurcation, from which at most three limit cycles can emerge in the plane for ε sufficiently small. And the limit cycles can only surround an equilibrium inside, i.e. the system can have the configuration ( 3 , 0 , 0 ) of limit cycles for some values of the parameters, where ( 3 , 0 , 0 ) stands for three limit cycles surrounding an equilibrium and no limit cycles surrounding any of the two or three equilibria.

Dynamics and bifurcation of a model for hormonal control of the menstrual cycle with inhibin delay

A system of 13 ordinary differential equations with 42 parameters is presented to model hormonal regulation of the menstrual cycle. For an excellent fit to clinical data, the model requires a 36 h time delay for the effect of inhibin on the synthesis of follicle stimulating hormone. Biological and mathematical reasons for this delay are discussed. Bifurcations with respect to changes in three important parameters are examined. One parameter represents the level of estradiol adequate for significant synthesis of luteinizing hormone. Bifurcation diagrams with respect to this parameter reveal an interval of parameter values for which a unique stable periodic solution exists and this solution represents a menstrual cycle during which ovulation occurs. The second parameter measures mass transfer between the first two stages of ovarian development and is indicative of healthy follicular growth. The third parameter is the time delay. Changes in the second parameter and the time delay affect the size of the uniqueness interval defined with respect to the first parameter. Saddle-node, transcritical and degenerate Hopf bifurcations are studied.

Local and global bifurcations in a small power system

This paper presents a number of local and one global (homoclinic loop) bifurcation observed in a small power system involving a load tap changer transformer and an induction motor. Besides the well-known generic local bifurcations (saddle-node and generic Hopf), the paper presents examples of higher codimension bifurcations, such as fold with double zero eigenvalues, degenerate Hopf, and swallowtail bifurcation.

Takens–Bogdanov bifurcation of travelling-wave solutions in pipe flow

The appearance of travelling-wave-type solutions in pipe Poiseuille flow that are disconnected from the basic parabolic profile is numerically studied in detail. We focus on solutions in the twofold azimuthally-periodic subspace because of their special stability properties, but relate our findings to other solutions as well. Using time-stepping, an adapted Krylov–Newton method and Arnoldi iteration for the computation and stability analysis of relative equilibria, and a robust pseudo-arclength continuation scheme, we unfold a double-zero (Takens–Bogdanov) bifurcating scenario as a function of Reynolds number (Re) and wavenumber (κ). This scenario is extended, by the inclusion of higher-order terms in the normal form, to account for the appearance of supercritical modulated waves emanating from the upper branch of solutions at a degenerate Hopf bifurcation. We provide evidence that these modulated waves undergo a fold-of-cycles and compute some solutions on the unstable branch. These waves are shown to disappear in saddle-loop bifurcations upon collision with lower-branch solutions, in accordance with the bifurcation scenario proposed. The travelling-wave upper-branch solutions are stable within the subspace of twofold periodic flows, and their subsequent secondary bifurcations could contribute to the formation of the phase space structures that are required for turbulent dynamics at higher Re.

BIFURCATION OF LIMIT CYCLES IN SMALL PERTURBATIONS OF A CLASS OF HYPER-ELLIPTIC HAMILTONIAN SYSTEMS OF DEGREE 5 WITH A CUSP

This paper deals with small perturbations of a class of hyper-elliptic Hamiltonian system, which is a Li e nard system of the form \(\dot{x}=y,\) \(\;\dot{y}=Q_1(x)+\varepsilon yQ_2(x)\) with \(Q_1\) and \(Q_2\) polynomials of degree 4 and 3, respectively. It is shown that this system can undergo degenerated Hopf bifurcation and Poincar e bifurcation, which emerge at most three limit cycles for \(\varepsilon\) sufficiently small.

Time-delayed feedback control of unstable periodic orbits near a subcritical Hopf bifurcation

We show that the Pyragas delayed feedback control can stabilize an unstable periodic orbit (UPO) that arises from a generic subcritical Hopf bifurcation of a stable equilibrium in an n -dimensional dynamical system. This extends results of Fiedler et al. [B. Fiedler, V. Flunkert, M. Georgi, P. Hövel, E. Schöll, Refuting the odd-number limitation of time-delayed feedback control, Phys. Rev. Lett. 98(11) (2007) 114101], who demonstrated that such a feedback control can stabilize the UPO associated with a two-dimensional subcritical Hopf normal form. The Pyragas feedback requires an appropriate choice of a feedback gain matrix for stabilization, as well as knowledge of the period of the targeted UPO. We apply feedback in the directions tangent to the two-dimensional center manifold. We parameterize the feedback gain by a modulus and a phase angle, and give explicit formulae for choosing these two parameters given the period of the UPO in a neighborhood of the bifurcation point. We show, first heuristically, and then rigorously by a center manifold reduction for delay differential equations, that the stabilization mechanism involves a highly degenerate Hopf bifurcation problem that is induced by the time-delayed feedback. When the feedback gain modulus reaches a threshold for stabilization, both of the genericity assumptions associated with a two-dimensional Hopf bifurcation are violated: the eigenvalues of the linearized problem do not cross the imaginary axis as the bifurcation parameter is varied, and the real part of the cubic coefficient of the normal form vanishes. Our analysis reveals two qualitatively distinct cases when the degenerate bifurcation is unfolded in a two-parameter plane. In each case, the Pyragas-type feedback successfully stabilizes the branch of small-amplitude UPOs in a neighborhood of the original bifurcation point, provided that the feedback phase angle satisfies a certain restriction.

On the number of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems with one nilpotent saddle

In this paper, we make a complete study on small perturbations of Hamiltonian vector field with a hyper-elliptic Hamiltonian of degree five, which is a Liénard system of the form x ′ = y , y ′ = Q 1 ( x ) + ε y Q 2 ( x ) with Q 1 and Q 2 polynomials of degree respectively 4 and 3. It is shown that this system can undergo degenerated Hopf bifurcation and Poincaré bifurcation, which emerges at most three limit cycles in the plane for sufficiently small positive ε. And the limit cycles can encompass only an equilibrium inside, i.e. the configuration ( 3 , 0 ) of limit cycles can appear for some values of parameters, where ( 3 , 0 ) stands for three limit cycles surrounding an equilibrium and no limit cycles surrounding two equilibria.

Anti-control of Hopf bifurcation in the new chaotic system with two stable node-foci

In order to further understand a complex 3D dynamical system showing strange chaotic attractors with two stable node-foci near Hopf bifurcation point, we propose nonlinear control scheme to the system and the controlled system, depending on five parameters, can exhibit codimension one, two, and three Hopf bifurcations in a much larger parameter regain. The control strategy used keeps the equilibrium structure of the chaotic system and can be applied to degenerate Hopf bifurcation at the desired location with preferred stability.