THE LIMIT CYCLE BIFURCATIONS OF A WHIRLING PENDULUM WITH PIECEWISE SMOOTH PERTURBATIONS

This paper deals with the problem of limit cycles for the whirling pendulum equation x=y,y=sin x(cosx-r)under piecewise smooth perturbations of polynomials of cos x,sin x and y of degree n with the switching line x=0.The upper bounds of the number of limit cycles in both the oscillatory and the rotary regions are obtained using the Picard-Fuchs equations,which the generating functions of the associated first order Melnikov functions satisfy.Furthermore,the exact bound of a special case is given using the Chebyshev system.At the end,some numerical simulations are given to illustrate the existence of limit cycles.

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.

Crossing Limit Cycles of Planar Piecewise Hamiltonian Systems with Linear Centers Separated by Two Parallel Straight Lines

In this paper, we have studied several classes of planar piecewise Hamiltonian systems with three zones separated by two parallel straight lines. Firstly, we give the maximal number of limit cycles in these classes of systems with a center in two zones and without equilibrium points in the other zone (or with a center in one zone and without equilibrium points in the other zones). In addition, we also give examples to illustrate that it can reach the maximal number.

Limit cycle problem for quadratic differential system x = -y + lx^2 + mxy, y =x(1 +ax +by)

The maximal number of limit cycles for a particular type Ⅲ system x = -y + lx2 + mxy, y =x(1 + ax + by) is studied and some errors that appeared in the paper by Suo Mingxia and Yue Xiting (Annals of Differential Equations, 2003,19(3):397-401) are corrected. By translating the system to be considered into the Lienard type and by using some related properties, we obtain several theorems with suitable conditions coefficients of the system, under which we prove that the system has at most two limit cycles. The conclusions improve the results given in Suo and Yue's paper mentioned above.

Limit Cycles in the Stability Analysis of a Chemical System

Aris and Amundson studied a chemical reactor and obtained the two equationsDaoud showed that at most one limit cycle may exist in the region of interest. Itis showed in this paper that other singular points exist and that a stable limitt cycle existsaround the singularity (1/2, 2) when K∈(9-δ, 9).

Limit cycle oscillation suppression of 2-DOF airfoil using nonlinear energy sink

This paper presents a novel mechanical attachment, i.e., nonlinear energy sink （NES）, for suppressing the limit cycle oscillation （LCO） of an airfoil. The dynamic responses of a two-degree-of-freedom （2-DOF） airfoil coupled with an NES are studied with the harmonic balance method. Different structure parameters of the NES, i.e., mass ratio between the NES and airfoil, NES offset, NES damping, and nonlinear stiffness in the NES, are chosen for studying the effect of the LCO suppression on an aeroelastic system with a supercritical Hopf bifurcation or subcritical Hopf bifurcation, respectively. The results show that the structural parameters of the NES have different influence on the supercritical Hopf bifurcation system and the subcritical Hopf bifurcation system.

CHAOTIC MOTIONS AND LIMIT CYCLE FLUTTER OF TWO-DIMENSIONAL WING IN SUPERSONIC FLOW

Based on the piston theory of supersonic flow and the energy method, the flutter motion equations of a two-dimensional wing with cubic stiffness in the pitching direction are established. The aeroelastic system contains both structural and aerodynamic nonlinearities. Hopf bifurcation theory is used to analyze the flutter speed of the system. The effects of system parameters on the flutter speed are studied. The 4th order Runge-Kutta method is used to calculate the stable limit cycle responses and chaotic motions of the aeroelastic system. Results show that the number and the stability of equilibrium points of the system vary with the increase of flow speed. Besides the simple limit cycle response of period 1, there are also period-doubling responses and chaotic motions in the flutter system. The route leading to chaos in the aeroelastic model used here is the period-doubling bifurcation. The chaotic motions in the system occur only when the flow speed is higher than the linear divergent speed and the initial condition is very small. Moreover, the flow speed regions in which the system behaves chaos axe very narrow.

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.

LIMIT CYCLES AND INVARIANT PARABOLA IN A KUKLES SYSTEM OF DEGREE THREE

In this article,the authors consider a class of Kukles planar polynomial differential system of degree three having an invariant parabola.For this class of second-order differential systems,it is shown that for certain values of the parameters the invariant parabola coexists with a center.For other values it can coexist with one,two or three small amplitude limit cycles which are constructed by Hopf bifurcation.This result gives an answer for the question given in[4],about the existence of limit cycles for such class of system.

LIMIT CYCLE PROBLEM OF QUADRATIC SYSTEM OF TYPE （ Ⅲ ）m=0, （ Ⅲ ）

To continue the discussion in （Ⅰ ） and （ Ⅱ ）,and finish the study of the limit cycle problem for quadratic system （ Ⅲ ）m=0 in this paper. Since there is at most one limit cycle that may be created from critical point O by Hopf bifurcation,the number of limit cycles depends on the different situations of separatrix cycle to be formed around O. If it is a homoclinic cycle passing through saddle S1 on 1 ＋ax-y = 0,which has the same stability with the limit cycle created by Hopf bifurcation,then the uniqueness of limit cycles in such cases can be proved. If it is a homoclinic cycle passing through saddle N on x= 0,which has the different stability from the limit cycle created by Hopf bifurcation,then it will be a case of two limit cycles. For the case when the separatrix cycle is a heteroclinic cycle passing through two saddles at infinity,the discussion of the paper shows that the number of limit cycles will change from one to two depending on the different values of parameters of system.

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.

Existence of Limit Cycles for a Cubic Kolmogorov System with a Hyperbolic Solution

This paper is concerned with a cubic Kolmogorov system with a solution of central quadratic curve which neither contacts with the coordinate axes, nor passes through the origin. The conclusion is that such a system may possess limit cycles.

ON NUMBER OF LIMIT CYCLES FOR THE QUADRATIC SYSTEMS WITH A WEAK FOCUS AND A STRONG FOCUS

It is proved that the quadratic system with a weak focus and a strong focus has at most one limit cycle around the strong focus, and as the weak focus is a 2nd order(or 3rd order) weak focus the quadratic system has at most two(one) limit cycles which have (1,1) distribution ((0,1) distribution).

NECESSARY AND SUFFICIENT CONDITIONS OF EXISTENCE AND UNIQUENESS OF LIMIT CYCLES FOR A CLASS OF POLYNOMIAL SYSTEM

In this paper, we discuss the limit cycles of the systemdx/dt=y·[1+(A(x)]oy/dt=(-x+δy+α_1x^2+α_2xy+α_5x^2y)[1+B(x)] (1)where A(x)=sum form i=1 to n(a_ix~), B(x)=sum form j=1 to m(β_jx^j) and 1+B(x)>0. We prove that (1) possesses at most one limit cycle and give out the necessary and sufficient conditions of existence and uniqueness of limit cycles.

Theoretical and Experimental Study of Hopf Bifurcation and Limit Cycles of Railway Vehicle Hunting

The nonlinear hunting stability of railway vehicles is studied theoretically and experimentally in this paper.The Hopf bifurcation point is determined through calculating the eigenvalues of the system linearization equations incorporating with the golden cut method.The bifurcated limit cycles are computed by use of the shooting method to solve the boundary value problem of the system differential equations.Experimental validation to the numerical results is carricd out by utilizing the full scale roller test rig.

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.

Reversed rotation of limit cycle oscillation and dynamics of low-intermediate-high confinement transition

The dynamics of the confinement transition from L mode to H mode（LH） is investigated in detail theoretically via the extended three-wave coupling model describing the interaction of turbulence and zonal flow（ZF） for the first time.Thereinto, turbulence is divided into a positive-frequency（PF） wave and a negative-frequency（NF） one, and the gradient of pressure is added as the auxiliary energy for the system. The LH confinement transition is observed for a sufficiently high input energy. Moreover, it is found that the rotation direction of the limit cycle oscillation（LCO） of PF wave and pressure gradient is reversed during the transition. The mechanism is illustrated by exploring the wave phases. The results presented here provide a new insight into the analysis of the LH transition, which is helpful for the experiments on the fusion devices.

LIMIT CYCLE BIFURCATIONS OF A PLANARNEAR-INTEGRABLE SYSTEM WITH TWO SMALL PARAMETERS

In this paper we consider a class of polynomial planar system with two small parameters,ε and λ,satisfying 0<ε《λ《1.The corresponding first order Melnikov function M_(1) with respect to ε depends on λ so that it has an expansion of the form M_(1)(h,λ)=∑k=0∞M_(1k)(h)λ^(k).Assume that M_(1k')(h) is the first non-zero coefficient in the expansion.Then by estimating the number of zeros of M_(1k')(h),we give a lower bound of the maximal number of limit cycles emerging from the period annulus of the unperturbed system for 0<ε《λ《1,when k'=0 or 1.In addition,for each k∈N,an upper bound of the maximal number of zeros of M_(1k)(h),taking into account their multiplicities,is presented.

NONLINEAR ANALYSIS OF EQUATORIAL EASTERN PACIFIC AIR-SEA COUPLING OSCILLATION AND A LIMIT-CYCLE THEORY FOR ENSO CYCLE

The troposphere and ocean mixed layer were considered as two components of a dynamic system operated by solar radiation as the constant source of energy, where upon an air-sea coupling self-exited coupling oscillation model was based with the aid of a locally averaged thermodynamic climate model, resulting mathematically in a closed self-governed dynamic system, a so-called El Nino-Southern Oscillation(ENSO) system. With the limit cycle solution of the system. It is shown that the essential physics of the coupled system can be described by the ENSO system. Compared with the observations, the theoretical limit cycle orbit matches the observed phase loop qualitatively. The ENSO system provides a useful theoretical framework for study of interannual variation of the tropical climate system.

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.