Problem Of Limit Cycles Research Articles

On the number of limit cycles for a perturbed cubic reversible Hamiltonian system.

This paper is concerned with the limit cycle problem of a cubic reversible Hamiltonian system under perturbation of polynomials of degree n with a switching line x=0. The upper and lower bounds of the number of limit cycles are obtained using the first order Melnikov function and its expansion. The method for calculating the Melnikov function relies upon some iterative formulas, which differs from other approaches.

Limit cycle bifurcations near a double 2-polycycle on the cylinder

In this paper, we consider the bifurcation problem of limit cycles near a double 2-polycycle for a cylinder near-Hamiltonian system. We first study the stability of a general homoclinic loop of type II on the cylinder, and then obtain certain results on the number and distribution of limit cycles for the cylinder near-Hamiltonian system near the double 2-polycycle by using the Melnikov function method and the method of stability-changing of a homoclinic loop of type I or II. Furthermore, we provide a way to find alien limit cycles for cylinder near-Hamiltonian system. As applications, we obtain the number of limit cycles of a class of cylinder near-Hamiltonian systems.

General study on limit cycle bifurcation near a double homoclinic loop

In this paper, we study the bifurcation problem of limit cycles near a general double homoclinic loop. We establish a general theory to obtain a lower bound of the maximal number of limit cycles near the double homoclinic loop. As an application, we prove that a near-Hamiltonian system of the form x˙=y(y2−1)+ε∑i=0naix2i+1,y˙=−x has at least [52n] limit cycles for 1≤n≤56. This number is maximal that we can find so far for the system.

Limit Cycle Bifurcations in a Class of Piecewise Smooth Polynomial Systems

In this paper, we consider the bifurcation problem of limit cycles for a class of piecewise smooth cubic systems separated by the straight line [Formula: see text]. Using the first order Melnikov function, we prove that at least [Formula: see text] limit cycles can bifurcate from an isochronous cubic center at the origin under perturbations of piecewise polynomials of degree [Formula: see text]. Further, the maximum number of limit cycles bifurcating from the center of the unperturbed system is at least [Formula: see text] if the origin is the unique singular point under perturbations.

Limit cycle oscillator in nonlinear systems with multiple time delays

This paper concerns with the problem of generation stable limit cycles in a class of nonlinear time-delay systems with multiple time delays. With the help of the Lyapunov–Krasovskii functional approach, a state-feedback controller is constructed based on the backstepping technique. According to the recursive procedure of the backstepping technique, at first, the desired limit cycle is generated in the second-order subsystem by utilizing the Lyapunov theorem analysis of the positive limit sets. Then, this method is expanded for higher-order systems. By introducing an appropriate Lyapunov-Krasovskii functional, the complexities raised by unknown time-delays are solved in the control design process. Based on this scheme, a delay-independent controller is explicitly designed which does not need the precise knowledge of time delays. The proposed method rigorously guarantees the practical stability of the closed-loop system and also ensures the convergence of the phase trajectories of the closed-loop system to the target limit cycle. Finally, to prove the theoretical achievement and also to illustrate the effective performance of the proposed approach, the simulation results are provided for several examples.

Homoclinic bifurcation of limit cycles in near-Hamiltonian systems on the cylinder

In this paper, we study a near-Hamiltonian system on the cylinder. First, we establish some general methods on the existence of limit cycles bifurcating from closed orbits of type II by the Melnikov function method, then we derive the expansions of the first order Melnikov function and consider the bifurcation problem of limit cycles near a double homoclinic loop. As an application, we discuss the number of limit cycles of a class of cylinder pendulum-like systems.

The Limit Cycles of the Higgins–Selkov Systems

In this paper, we investigate the problem of limit cycles for general Higgins–Selkov systems with degree $$n+1$$ . In particular, we first prove the uniqueness of limit cycles for a general Liénard system, which allows for discontinuity. Then, by changing the Higgins–Selkov systems into Liénard systems, theorems and some techniques for Liénard systems can be applied. After, we prove the nonexistence of limit cycles if the bifurcation parameter is outside an open interval. Finally, we complete the analysis of limit cycles for the Higgins–Selkov systems showing its uniqueness.

Limit Cycle Problem for Quadratic System with some Applications of a Class I

This paper is part of a wider study limit cycle problems of a class and planar system; we study the existence of limit cycle for polynomial planar system. In this paper, we present a proof of a result on the existence of limit cycle of the Quadratic System:

Existence of Limit Cycles for Liénard System with Application

This paper is part of a wider study limit cycle problems and planar system; The aims of this is to study the existence of limit cycle for Liénard system. We followed the historical analytical mathematical method to present a proof of a result on the existence of limit cycle for Liénard system form x ̇=y-F(x) ,y ̇=-g(x)

LIMIT CYCLE BIFURCATIONS IN A CLASS OF PIECEWISE SMOOTH DIFFERENTIAL SYSTEMS UNDER NON-SMOOTH PERTURBATIONS

<abstract> This paper deals with the problem of limit cycles of a class of piecewise smooth integrable differential systems with switching line $ x = 0 $. The generating functions of the associated first order Melnikov function satisfy two different Picard-Fuchs equations. By using the property of Chebyshev space, we obtain an upper bound for the number of limit cycles bifurcating from the period annulus under non-smooth perturbations of polynomials of degree $ n $. Finally, we present a concrete example to illustrate the theoretical result. </abstract>

Limit cycles and global dynamics of planar piecewise linear refracting systems of focus–focus type

The limit cycle problem of planar piecewise linear refracting systems has been solved except the focus–focus (FF) type. Here we concern this remaining FF type, and prove that such systems have at most one limit cycle and have 14 topologically different global phase portraits.

Discontinuity-induced limit cycles in a general planar piecewise linear system of saddle–focus type

The aim of this paper is to deal with the problem of limit cycles for a general planar piecewise linear differential system of saddle–focus type. By using the Liénard-like canonical form with five parameters and dividing the total parameter space into several regions, the number of limit cycles is discussed in detail. In particular, we give parameter regions where there are at least two limit cycles. Moreover, we investigate the existence and stability of exactly two nested limit cycles in some parameter regions.

Integrability and limit cycles in cubic Kukles systems with a nilpotent singular point

In this paper, integrability problem and bifurcation of limit cycles for cubic Kukles systems which are assumed to have a nilpotent origin are investigated. A complete classification is given on the integrability conditions and proven to have a total of 7 cases. Bifurcation of limit cycles is also discussed; six or seven limit cycles can be obtained by two different perturbation methods. Integrability problem and bifurcation of limit cycles for the cubic Kukles systems with a nilpotent origin have been completely solved.

Bifurcation of limit cycles and center problem for [formula omitted] homogeneous weight systems

The quasi-homogeneous polynomial differential systems have been studied from many different points of view. In this paper, center problem and bifurcation of limit cycles for p:q homogeneous weight systems are studied. Some properties of successive function and focus values are discussed. Furthermore, the method of computing focal values is given. As an example, center problem and bifurcation of limit cycles for 2:3 homogeneous weight system are studied, three or five limit cycles in the neighborhood of origin can be obtained by different perturbations.

On-Line Fault-Tolerant Fuzzy-Based Path Planning and Obstacles Avoidance Approach for Manipulator Robots

Manipulator robots are widely used in many fields to replace humans in complex and risky environments. However, in some particular environments the robot is prone to failure, resulting in decreased performance. In such environments, it is extremely difficult to repair the robot which interrupts the execution process. Therefore, fault tolerance plays an important role in industrial manipulators applications. In this paper, the key problems related to fault-tolerance and path planning of manipulator robots under joints failures are handled within an on-line fault-tolerant fuzzy-logic based path planning approach for high degree-of-freedom robots. This approach provides an alternative to using mathematical models to control such robots, and improves tolerance to certain faults and mechanical failures. The controller consists of two fuzzy units (i) the first unit, Fuzzy_Path_Planner, is responsible of path planning; (ii) the second unit, Fuzzy_Obstacle_Avoidance, is conceived for obstacles avoidance. Moreover, the proposed approach is capable of repelling the manipulator away from both local minima and limit cycle problems. Finally, to validate the proposed approach and show its performances and effectiveness, different tests are carried out on two six degree-of-freedom manipulator robots (ULM and PUMA560 robots), accomplishing point-to-point tasks, with and without considering some joints failures.

Bi-center problem and bifurcation of limit cycles from nilpotent singular points in Z2-equivariant cubic vector fields

In this paper, bi-center problem and bifurcation of limit cycles from nilpotent singular points in Z2-equivariant cubic vector fields are studied. First, the system is simplified by using some proper transformations and the first five Lyapunov constants at a nilpotent singular point are calculated by applying the inverse integrating factor method. Then, sufficient and necessary conditions are obtained for two nilpotent singular points of the system being centers. A new perturbation scheme is present to prove the existence of 12 small-amplitude limit cycles in cubic Z2-equivariant vector fields, which bifurcate from two nilpotent singular points. This is a new lower bound of the number of limit cycles bifurcating in such systems.

一类三次微分系统的异宿环分支

本文运用分支的方法，通过分析未扰系统的异宿环在小扰动下的稳定流形和不稳定流形之间的相对距离，研究了一类三次微分系统的异宿环分支极限环的问题，给出了系统至少产生一个极限环的条件。 In this paper, by using the bifurcation method to analyze the relative distance between the stable manifold and the unstable manifold after the heteroclinic loop of the unperturbed system was perturbed to break, the authors studied the existence problem of limit cycles of a class of cubic differential system and obtained the conditions to ensure the system has at least one limit cycle.

Limit Cycle Bifurcations by Perturbing a Piecewise Hamiltonian System with a Double Homoclinic Loop

This paper is concerned with the bifurcation problem of limit cycles by perturbing a piecewise Hamiltonian system with a double homoclinic loop. First, the derivative of the first Melnikov function is provided. Then, we use it, together with the analytic method, to derive the asymptotic expansion of the first Melnikov function near the loop. Meanwhile, we present the first coefficients in the expansion, which can be applied to study the limit cycle bifurcation near the loop. We give sufficient conditions for this system to have [Formula: see text] limit cycles in the neighborhood of the loop. As an application, a piecewise polynomial Liénard system is investigated, finding six limit cycles with the help of the obtained method.

On Periodic Perturbations of Asymmetric Duffing–Van-der-Pol Equation

Time-periodic perturbations of an asymmetric Duffing–Van-der-Pol equation close to an integrable equation with a homoclinic "figure-eight" of a saddle are considered. The behavior of solutions outside the neighborhood of "figure-eight" is studied analytically. The problem of limit cycles for an autonomous equation is solved and resonance zones for a nonautonomous equation are analyzed. The behavior of the separatrices of a fixed saddle point of the Poincaré map in the small neighborhood of the unperturbed "figure-eight" is ascertained. The results obtained are illustrated by numerical computations.

Limit Cycles Bifurcations for a Class of Kolmogorov Model in Symmetrical Vector Field

The problem of limit cycles for Kolmogorov model is interesting and significant both in theory and applications. Our work is concerned with limit cycles bifurcations problem for a class of quartic Kolmogorov model with two positive singular points (i.e. (1, 2) and (2, 1)). The investigated model is symmetrical with regard to y = x. We show that each one of points (1, 2) and (2, 1) can bifurcate five small limit cycles at the same step under a certain condition. Hence, the two positive singular points can bifurcate ten limit cycles in sum, in which six cycles can be stable. In terms of symmetrical Kolmogorov model, published references are less. In terms of the Hilbert Number of Kolmogorov model, our results are new.