HOPF BIFURCATION AND UNIQUENESS OF LIMIT CYCLE FOR A CLASS OF QUARTIC SYSTEM

This paper studies a class of quartic system which is more general and realistic than the quartic accompanying system.x＇=-y＋ex＋lx^2＋mxy＋ny^2,y＇=x（1-Ay）（1＋Cy^2）,（＊）where C 〉 0. Sufficient conditions are obtained for the uniqueness of limit cycle of system （＊） and some more in-depth conclusion such as Hopf bifurcation.

Limit cycles and homoclinic orbits and their bifurcation of Bogdanov-Takens system

A quantitative analysis of limit cycles and homoclinic orbits, and the bifurcation curve for the Bogdanov-Takens system are discussed. The parameter incremental method for approximate analytical-expressions of these problems is given. These analytical-expressions of the limit cycle and homoclinic orbit are shown as the generalized harmonic functions by employing a time transformation. Curves of the parameters and the stability characteristic exponent of the limit cycle versus amplitude are drawn. Some of the limit cycles and homoclinic orbits phase portraits are plotted. The relationship curves of parameters μ and A with amplitude a and the bifurcation diagrams about the parameter are also given. The numerical accuracy of the calculation results is good.

BIFURCATIONS OF SUBHARMONIC SOLUTIONS IN PERIODIC PERTURBATION OF A HYPERBOLIC LIMIT CYCLE

Bifurcations of subharmonic solutions of order m of a planar periodic perturbed system near a hyperbolic limit cycle are discussed. By using a Poincare map and the method of rescaling a discriminating condition for the existence of subharmonic solutions of order m is obtained. An example is given in the end of the paper.

Bifurcation and Limit Cycle of a Ratio-dependent Predator-prey, System with Refuge on Prey

Influences of prey refuge on the dynamics of a predator-prey model with ratio-dependent functional response are investigated. The local and global stability of positive equilibrium of the system are considered. Theoretical analysis indicates that constant refuge leads to the system undergo supercritical Hopf bifurcation twice with the birth rate of prey species changing continuously.

Limit Cycle Bifurcations in a Class of Cubic System near a Nilpotent Center

In this paper we deal with a cubic near-Hamiltonian system whose unperturbed system is a simple cubic Hamiltonian system having a nilpotent center. We prove that the system can have 5 limit cycles by using bifurcation theory.

Amplitude control of limit cycle in a van der Pol Duffing system

This paper applies washout filter technology to amplitude control of limit cycles emerging from Hopf bifurcation of the van der Pol-Duffing system. The controlling parameters for the appearance of Hopf bifurcation are given by the Routh-Hurwitz criteria. Noticeably, numerical simulation indicates that the controllers control the amplitude of limit cycles not only of the weakly nonlinear van der Pol-Duffing system but also of the strongly nonlinear van der Pol-Duffing system. In particular, the emergence of Hopf bifurcation can be controlled by a suitable choice of controlling parameters. Gain-amplitude curves of controlled systems are also drawn.

Hopf Bifurcation Control of a Hyperchaotic Circuit System

This paper is concerned with the Hopf bifurcation control of a new hyperchaotic circuit system. The stability of the hyperchaotie circuit system depends on a selected control parameter is studied, and the critical value of the system parameter at which Hopf bifurcation occurs is investigated. Theoretical analysis give the stability of the Hopf bifurcation. In particular, washout filter aided feedback controllers are designed for delaying the bifurcation point and ensuring the stability of the bifurcated limit cycles. An important feature of the control laws is that they do not result in any change in the set of equilibria. Computer simulation results are presented to confirm the analytical predictions.

The Dynamics and Bifurcation Control of a Singular Biological Economic Model

The objective of this paper is to study systematically the dynamics and control strategy of a singular biological economic model that is described by a differential-algebraic equation. It is shown that when the economic profit passes through zero, this model exhibits the transcritical bifurcation, the Hopf bifurcation, and the limit cycle. In particular, the system undergoes the singularity induced bifurcation at the positive equilibrium, which can result in impulse. Then, state feedback controllers closer to the actual control strategies are designed to eliminate the unexpected singularity induced bifurcation and stabilize the positive equilibrium under the positive profit. Finally, numerical simulations verify the results and illustrate the effectiveness of the controllers. Also, the model with positive economic profit is shown numerically to have different dynamics.

Non-linear 2-DOF model and centre manifold theory to study limit cycle oscillations caused by drum-brake judder

This paper presents the research on the laws of systematic-parameter dependent variation in the vibration amplitude of drum-brake limit cycle oscillations (LCO). We established a two-degree non-linear dynamic model to describe the low-frequency vibration of the drum brake, applied the centre manifold theory to simplify the system, and obtained the LCO amplitude by calculating the normal form of the simplified system at the Hopf bifurcation point. It is indicated that when the friction coefficient is smaller than the friction coefficient at the bifurcation point, the amplitude decreases; whereas with a friction coefficient larger than the friction coefficient of bifurcation point, LCO occurs. The results suggest that it is applicable to suppress the LCO amplitude by changing systematic parameters, and thus improve the safety and ride comfort when applying brake. These findings can be applied to guiding the design of drum brakes.

The Number and Distributions of Limit Cycles of a Cubic Hamiltonian System with Z_2-symmetry Perturbation

This paper is concerned with the number and distributions of limit cycles of a cubic Z2-symmetry Hamiltonian system under quintic perturbation.By using qualitative analysis of differential equation,bifurcation theory of dynamical systems and the method of detection function,we obtain that this system exists at least 14 limit cycles with the distribution C91 [C11 ＋ 2（C32 2C12）].

Existence of Two Limit Cycles in Zeeman’s Class 30 for 3D Lotka-Volterra Competitive System

Gyllenberg and Yan(Discrete Contin Dyn Syst Ser B 11(2):347–352,2009)presented a system in Zeeman’s class 30 of 3-dimensional Lotka-Volterra(3D LV)competitive systems to admit at least two limit cycles,one of which is generated by the Hopf bifurcation and the other is obtained by the Poincaré-Bendixson theorem.Yu et al.(J Math Anal Appl 436:521–555,2016,Sect.3.4)recalculated the first Liapunov coefficient of Gyllenberg and Yan’s system to be positive,rather than negative as in Gyllenberg and Yan(2009),and pointed out that the Poincaré-Bendixson theorem is not applicable for that system.Jiang et al.(J Differ Equ 284:183–218,2021,p.213)proposed an open question:“whether Zeeman’s class 30 can be rigorously proved to admit at least two limit cycles by the Hopf theorem and the Poincaré-Bendixson theorem?”This paper provides four systems in Zeeman’s class 30 to admit at least two limit cycles by the Hopf theorem and the Poincaré-Bendixson theorem and gives an answer to the above question.

Existence of a Limit Cycle in an Intraguild Food Web Model with Holling Type II and Logistic Growth for the Common Prey

In this paper, we prove the existence of a limit cycle for a given system of differential equations corresponding to an asymmetrical intraguild food web model with functional responses Holling type II for the middle and top predators and logistic grow for the (common) prey. The existence of such limit cycle is guaranteed, via the first Lyapunov coefficient and the Andronov-Hopf bifurcation theorem, under certain conditions for the parameters involved in the system.

Bifurcations of Limit Cycles in a Z_4-Equivariant Quintic Planar Vector Field

In this paper, a Z4-equivariant quintic planar vector field is studied. The Hopf bifurcation method and polycycle bifurcation method are combined to study the limit cycles bifurcated from the compounded cycle with 4 hyperbolic saddle points. It is found that this special quintic planar polynomial system has at least four large limit cycles which surround all singular points. By applying the double homoclinic loops bifurcation method and Hopf bifurcation method, we conclude that 28 limit cycles with two different configurations exist in this special planar polynomial system. The results acquired in this paper are useful for studying the weakened 16th Hilbert＇s Problem.

BIFURCATION OF LIMIT CYCLES FROM A DOUBLE HOMOCLINIC LOOP WITH A ROUGH SADDLE

This paper concerns with the bifurcation of limit cycles from a double homoclinic loop under multiple parameter perturbations for general planar systems. The existence conditions of 4 homoclinic bifurcation curves and small and large limit cycles are especially investigated.

Bifurcation of limit cycles near equivariant compound cycles

In this paper we study some equivariant systems on the plane. We first give some criteria for the outer or inner stability of compound cycles of these systems. Then we investigate the number of limit cycles which appear near a compound cycle of a Hamiltonian equivariant system under equivariant perturbations. In the last part of the paper we present an application of our general theory to show that a Z3 equivariant system can have 13 limit cycles.

LIMIT CYCLES AND BIFURCATION CURVES FOR THE QUADRATIC DIFFERENTIAL SYSTEM (III)_(m=0) HAVING THREE ANTI-SADDLES (I)

For the quadratic system: x=-y+δx + lx2 + ny2, y=x(1+ax-y) under conditions -1<l<0,n+l - 1>0 the author draws in the (a, ()) parameter plane the global bifurcationdiagram of trajectories around O(0,0). Notice that when na2+l < 0 the system has one saddleN(0,1/n) and three anti-saddles.

Bifurcation of Limit Cycles and the Cusp of Order n

The author first investigates the limit cycles bifurcating from a center for general two dimensional systems,and then proves the conjecture that any unfolding of the cusp of order n has at most n-1 limit cycles.

LIMIT CYCLES AND BIFURCATION CURVES FOR THE QUADRATIC DIFFERENTIAL SYSTEM(III)_(m=0)HAVING THREE ANTI-SADDLES(II)

As a continuation of,the author studies the limit cycle bifurcation around the focus S_(1)other than O(0,0)for the system(1)asδvaries.A conjecture on the mon-existence of limit cycles around S_(1),and another one on the non-coexistence of limit cycles ariund both O and S_(1)are given,together with some numerical examples.

Center Conditions and Bifurcation of Limit Cycles at Nilpotent Critical Point in a Quintic Lyapunov System

In this paper, center conditions and bifurcation of limit cycles at the nilpotent critical point in a class of quintic polynomial differential system are investigated. With the help of computer algebra system MATHEMATICA, the first 8 quasi Lyapunov constants are deduced. As a result, the necessary and sufficient conditions to have a center are obtained. The fact that there exist 8 small amplitude limit cycles created from the three-order nilpotent critical point is also proved. Henceforth we give a lower bound of cyclicity of three-order nilpotent critical point for quintic Lyapunov systems.

CENTER CONDITIONS AND BIFURCATION OF LIMIT CYCLES FOR A CLASS OF FIFTH DEGREE SYSTEMS

The center conditions and bifurcation of limit cycles for a class of fifth degree systems are investigated.Two recursive formulas to compute singular quantities at infinity and at the origin are given.The first nine singular point quantities at infinity and first seven singular point quantities at the origin for the system are given in order to get center conditions and study bifurcation of limit cycles.Two fifth degree systems are constructed.One allows the appearance of eight limit cycles in the neighborhood of infinity,which is the first example that a polynomial differential system bifurcates eight limit cycles at infinity.The other perturbs six limit cycles at the origin.