Example of a quadratic system with two cycles appearing in a homoclinic loop bifurcation

We give here a planar quadratic differential system depending on two parameters, λ, δ. There is a curve in the λ-δ space corresponding to a homoclinic loop bifurcation (HLB). The bifurcation is degenerate at one point of the curve and we get a narrow tongue in which we have two limit cycles. This is the first example of such a bifurcation in planar quadratic differential systems. We propose also a model for the bifurcation diagram of a system with two limit cycles appearing at a singular point from a degenerate Hopf bifurcation, and dying in a degenerate HLB. This model shows a deep duality between degenerate Hopf bifurcations and degenerate HLBs. We give a bound for the maximal number of cycles that can appear in certain simultaneous Hopf and homoclinic loop bifurcations. We also give an example of quadratic system depending on three parameters which has at one place a degenerate Hopf bifurcation of order 3, and at another place a Hopf bifurcation of order 2 together with a HLB. We characterize the planar quadratic systems which are integrable in the neighbourhood of a homoclinic loop.     

POINCAR BIFURCATION FOR QUADRATIC SYSTEMS WITH A CENTER REGION AND AN UNBOUNDED TRIANGULAR REGION

In this paper, we discuss the Poincare bifurcation for a class of quadratic systems with an unbounded triangular region and a center region. It is proved, by Poincare bifurcation, that inside the center region quadratic system perturbed by quadratic polynomial perturbation may generate three limit cycles.     

Ten limit cycles around a center-type singular point in a 3-d quadratic system with quadratic perturbation

In this paper, we show that perturbing a simple 3-d quadratic system with a center-type singular point can yield at least 10 small-amplitude limit cycles around a singular point. This result improves the 7 limit cycles obtained recently in a simple 3-d quadratic system around a Hopf singular point. Compared with Bautin’s result for quadratic planar vector fields, which can only have 3 small-amplitude limit cycles around an elementary center or focus, this result of 10 limit cycles is surprisingly high. The theory and methodology developed in this paper can be used to consider bifurcation of limit cycles in higher-dimensional systems.     

Bifurcation analysis of a Lyapunov-based controlled boost converter

Lyapunov-based controlled boost converters have a unique equilibrium point, which is globally asymptotically stable, for known resistive loads. This article investigates the dynamic behaviors that appear in the system when the nominal load differs from the actual one and no action is taken by the controller to compensate for the mismatch. Exploiting the fact that the closed-loop system is, in fact, planar and quadratic, one may provide not only local but also global stability results: specifically, it is proved that the number of equilibria of the converter may grow up to three and that, in any case, the system trajectories are always bounded, i.e. it is a bounded quadratic system. The possible phase portraits of the closed-loop system are also characterized in terms of the selected bifurcation parameters, namely, the actual load value and the gain of the control law. Accordingly, the analysis allows the numerical illustration of many bifurcation phenomena that appear in bounded quadratic systems through a physical example borrowed from power electronics.     

Limit cycle bifurcations by perturbing a class of integrable systems with a polycycle

In this paper, we deal with the problem of limit cycle bifurcation near a 2-polycycle or 3-polycycle for a class of integrable systems by using the first order Melnikov function. We first get the formal expansion of the Melnikov function corresponding to the heteroclinic loop and then give some computational formulas for the first coefficients of the expansion. Based on the coefficients, we obtain a lower bound for the maximal number of limit cycles near the polycycle. As an application of our main results, we consider quadratic integrable polynomial systems, obtaining at least two limit cycles.     

Codiagonalization of Matrices and Existence of Multiple Homoclinic Solutions

The purpose of this paper is twofold. First, we use Lagrange''s method and the generalized eigenvalue problem to study systems of two quadratic equations. We find exact conditions so the system can be codiagonalized and can have up to $4$ solutions. Second, we use this result to study homoclinic bifurcations for a periodically perturbed system. The homoclinic bifurcation is determined by $3$ bifurcation equations. To the lowest order, they are $3$ quadratic equations, which can be simplified by the codiagonalization of quadratic forms. We find that up to $4$ transverse homoclinic orbits can be created near the degenerate homoclinic orbit.     

Hopf Cyclicity of a Family of Generic Reversible Quadratic Systems with One Center

This paper is concerned with small quadratic perturbations to one parameter family of generic reversible quadratic vector fields with a simple center. The first objective is to show that this system exhibits two small amplitude limit cycles emerging from a Hopf bifurcation. The second one we prove that the system has no limit cycle around the weak focus of order two. The results may be viewed as a contribution to proving the conjecture on cyclicity proposed by Iliev (1998).     

BIFURCATION OF CUBIC INTEGRABLE SYSTEM UNDER CUBIC PERTURBATION

In this paper, we consider the bifurcation for a class of cubic integrable system under cubic perturbation. Using bifurcation theory and qualitative analysis, we obtain a complete bifurcation diagram of the system in a neighbourhood of the origin for parameter plane.     

Numerical Approximation of Transcritical Simple Bifurcation Point of the Navier-Stokes Equations

The extended system of nondegenerate simple bifurcation point of the Navier-Stokes equations is constructed in this paper, due to its derivative has a block lower triangular form, we design a Newton-like method, using the extended system and splitting iterative technique to compute transcritical nondegenerate simple bifurcation point, we not only reduces computational complexity, but also obtain quadratic convergence of algorithm.     

Stability and bifurcation analysis of a buckled beam via active control

In this paper, we focus on applying active control to nonlinear dynamical beam system to eliminate its vibration. We analyzed stability using frequency-response equations and bifurcation. The analytical solution of the nonlinear differential equations describing the above system is investigated using multiple time scale method (MTSM). All resonance cases were extracted from second order approximations. Numerical solutions of the system are included. The effects of most system parameters were investigated. The results demonstrated that proposed controller is efficient to suppress the vibrations. Increasing the quadratic stiffness coefficient term vanished the multi-valued solution. Bifurcation diagrams refiled the effects of various system parameters on its stability showing different bifurcation cases. Finally, we conclude that for low values of natural frequencies dynamical system, the controller is more effective. The results show that the analytical solutions of the system are in good agreement with the numerical solutions.     

Analysis of a delayed predator-prey model with ratio-dependent functional response and quadratic harvesting

In this paper, we investigate the dynamics of a ratio dependent predator-prey model with quadratic harvesting. We examine the existence of the positive equilibria, the related dynamical behaviors of the model, as well as the boundedness and permanence property of the system. We also study the global stability of the interior equilibrium without time delay. Finally some bifurcation analysis is carried out for the system with delay and the results are illustrated numerically.     

BIFURCATION THEORY OF QUADRATIC DIFFERENTIAL SYSTEMS

This paper deals with the number of limit cycles and bifurcation problem of quadratic differential systems. Under conditions $a<0,b+2l>0,l+1<0$, the author draws three bifurcation diagrams of the system (1.18) below in the (\delta,m) plane, which show that the maximum number of limit cycles around a focus is two in this case.     

Bifurcation analysis in a simplified tri-neuron BAM network model with multiple delays

In this paper, we consider a three-dimensional delayed differential equation representing a bidirectional associate memory (BAM) neural network with three neurons and two discrete delays. By analyzing the number and stability of equilibria, the pitchfork bifurcation curve of the system is obtained. Furthermore, on the pitchfork bifurcation curve, by using the sum of two delays as the bifurcation parameter, we find that the system can undergo a Hopf bifurcation at the origin and the three-dimensional ordinary differential equation describing the flow on the center manifold is given.     

Control of Hopf bifurcation in a simple plankton population model with a non-integer exponent of closure

In this paper, control of Hopf bifurcation for a plankton population model with a non-integer exponent of closure is studied. For the plankton model with the closure term varies continuously between the commonly used linear and quadratic forms, washout filter aided feedback controllers are employed to control the stability of the Hopf bifurcation, and thus the chaos could be eliminated. By employing the stability theorems derived, the stability of a control system is investigated. Finally numerical simulations are provided to show the effectiveness and feasibility of the developed methods.     

A quadratic system with two parallel straight-line-isoclines

In this paper, a quadratic system with two parallel straight-line-isoclines is considered. This system corresponds to the system of class II in the classification of Ye Yanqian [Ye Yanqian et al., Theory of Limit Cycles, Transl. Math. Monogr., vol. 66, American Mathematical Society, Providence, RI, 1986]. Using the field rotation parameters of the constructed canonical system and geometric properties of the spirals filling the interior and exterior domains of its limit cycles, we prove that the maximum number of limit cycles in a quadratic system with two parallel straight-line-isoclines and two finite singular points is equal to two. Besides, we obtain the same result in a different way: applying the Wintner–Perko termination principle for multiple limit cycles and using the methods of global bifurcation theory developed in [V.A. Gaiko, Global Bifurcation Theory and Hilbert’s Sixteenth Problem, Kluwer, Boston, 2003].     

THE POINCAR BIFURCATION OF QUADRATIC SYSTEMS HAVING A REGION CONSISTING OF PERIODIC CYCLES BOUNDED BY A HYPERBOLA AND AN ARC OF EQUATOR

The usual bifureation problems oeeurred in the study of the Planar differential systems having a region eonsisting of Periodie eycles are as follows:15 there any closed orbit whieh generates limit eyeles after a small perturbation?How many limit eyel     

Epidemiological models with quadratic equation for endemic equilibria—A bifurcation atlas

The existence and occurrence, especially by a backward bifurcation, of endemic equilibria is of utmost importance in determining the spread and persistence of a disease. In many epidemiological models, the equation for the endemic equilibria is quadratic, with the coefficients determined by the parameters of the model. Despite its apparent simplicity, such an equation can describe an amazing number of dynamical behaviors. In this paper, we shall provide a comprehensive survey of possible bifurcation patterns, deriving explicit conditions on the equation's parameters for the occurrence of each of them, and discuss illustrative examples.     

Hopf-Takens-Bogdanov interaction for a fluid-conveying tube

We investigate the dynamics after loss of stability of the downhanging configuration of a fluid conveying tube with a small end mass and an elastic support. By varying the fluid flow rate and the stiffness and location of the elastic support, different degenerate bifurcation scenarios can be observed. In this article we investigate the bifurcating solution branches of the codimension 3 interaction between a Hopf bifurcation and a Bogdanov-Takens bifurcation. A complete discussion of the primary and secondary solution branches was already given by W. F. Langford and K. Zhan. After reducing the system to the three-dimensional Normal Form equations we apply a numerical continuation procedure to locate the expected higher order bifurcation branches and detect more complicated dynamics, like Shilnikov orbits. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bifurcations of Rough Heteroclinic Loops with Three Saddle Points

In this paper, we study the bifurcation problems of rough heteroclinic loops connecting three saddle points for a higher-dimensional system. Under some transversal conditions and the nontwisted condition, the existence, uniqueness, and incoexistence of the 1-heteroclinic loop with three or two saddle points, 1-homoclinic orbit and 1-periodic orbit near Γ are obtained. Meanwhile, the bifurcation surfaces and existence regions are also given. Moreover, the above bifurcation results are extended to the case for heteroclinic loop with l saddle points. Received January 4, 2001, Accepted July 3, 2001.     

On the Limit Cycles Bifurcating from a Quadratic Reversible Center of Genus One

The paper is concerned with the bifurcation of limit cycles in perturbations of a quadratic reversible system with a center of genus one. By studying the properties of the auxiliary curve and centroid curve defined by the Abelian integrals, we have proved that under small quadratic perturbations, at most two limit cycles arise from the period annulus surrounding the quadratic reversible center, and the bound is sharp. This partially verifies Conjecture 1 given in Gautier et al. (Discrete Contin Dyn Syst 25:511–535, 2009).