Problem Of Limit Cycle Bifurcation Research Articles

On the number of limit cycles near a homoclinic loop with a nilpotent cusp of order m

This paper investigates the expansions of Melnikov functions near a homoclinic loop with a nilpotent cusp of order m. It presents a methodology for calculating all coefficients in these expansions, which can be employed to study the problem of limit cycle bifurcation. As an application, by utilizing the obtained results, the paper rigorously establishes that a polynomial Liénard system of degree n+1 has at least n+[n4] limit cycles near the homoclinic loop with a nilpotent cusp of order one. This work not only updates and generalizes existing results, but also provides a rigorous application of the obtained findings in the context of limit cycle bifurcation.

Hopf bifurcation problems near double positive equilibrium points for a class of quartic Kolmogorov model

<abstract><p>The Kolmogorov model is a class of significant ecological models and is initially introduced to describe the interaction between two species occupying the same ecological habitat. Limit cycle bifurcation problem is close to Hilbertis 16th problem. In this paper, we focus on investigating bifurcation of limit cycle for a class of quartic Kolmogorov model with two positive equilibrium points. Using the singular values method, we obtain the Lyapunov constants for each positive equilibrium point and investigate their limit cycle bifurcations behavior. Furthermore, based on the analysis of their Lyapunov constants' structure and Hopf bifurcation, we give the condition that each one positive equilibrium point of studied model can bifurcate 5 limit cycles, which include 3 stable limit cycles.</p></abstract>

Maximum number of limit cycles bifurcating from the period annulus of cubic polynomial systems

This paper is devoted to the limit cycle bifurcation problem for some cubic polynomial systems, whose unperturbed systems have a period annulus and two invariant lines. Using the first order Melnikov function and Chebyshev criterion, we obtain the maximum number of limit cycles bifurcating from the period annulus. It improves a known result given by Sui and Zhao [Internat. J. Bifur. Chaos Appl. Sci. Engrg. 28 (2018), 1850063].

Limit cycles appearing from a generalized heteroclinic loop with a cusp and a nilpotent saddle

This paper studies the limit cycle bifurcation problem of a class of piecewise smooth differential polynomial systems of degree n by perturbing a piecewise cubic polynomial system having a generalized heteroclinic loop with a cusp and a nilpotent saddle. First, we provide all possible phase portraits of the unperturbed system on the plane with crossing periodic orbits and obtain a condition for the appearance of a generalized heteroclinic loop with a cusp and a nilpotent saddle by qualitative theoretical knowledge. Then, we investigate the algebraic structure of the first order Melnikov function and give its asymptotic expansion near the generalized heteroclinic loop with the help of analytical skills. Finally, we employ the expansion together with its coefficients to obtain the existence of at least 3n−1 limit cycles.

Limit cycle bifurcations of planar piecewise differential systems with three zones

This paper considers the limit cycle bifurcation problem of planar piecewise differential systems with three zones. Some computation formulas studied the problem of limit cycle bifurcations are provided by introducing multiple parameters. As an application to the obtained method, the number of limit cycles of a piecewise linear system with three zones studied in Lima et al. (2017) is discussed and some more limit cycles are found.

Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters

<p style='text-indent:20px;'>In this paper, we concern with the problem of limit cycle bifurcation for a class of piecewise smooth cubic systems. Using the first order Melnikov function we prove that at least thirteen limit cycles can be bifurcated from periodic solutions surrounding the center.

LIMIT CYCLE BIFURCATIONS IN DISCONTINUOUS PLANAR SYSTEMS WITH MULTIPLE LINES

In this paper, the limit cycle bifurcation problem is investigated for a class of planar discontinuous perturbed systems with $ n $ parallel switch lines. Under the assumption that the unperturbed system has a family of periodic orbits crossing all of the lines, an explicit expression of the first order Melnikov function along the periodic orbits is presented, which plays an important role in studying the problem of limit cycle bifurcations. As an application of the established method, the maximal number of limit cycles of a discontinuous system is considered.

A class of reversible quadratic systems with piecewise polynomial perturbations

This paper investigates a class of reversible quadratic systems perturbed inside piecewise polynomial differential systems of arbitrary degree n. All possible phase portraits of the reversible quadratic systems on the plane are obtained by analytic techniques. Then, we present the algebraic structure of its corresponding first-order Melnikov function, which can be used to study the limit cycle bifurcation problem. In this direction, we use it to investigate the Hopf bifurcation of the perturbed systems, and obtain an upper bound for the Hopf cyclicity. Meanwhile, we employ it to discuss the maximal number of limit cycles emerging from the periodic orbits around the origin for a special case and also provide its upper bound.

Limit cycle bifurcations in perturbations of planar piecewise smooth systems with multiply lines of critical points

This paper investigates the limit cycle bifurcation problem of planar piecewise polynomial systems, whose unperturbed system has a center at the origin and may have invariant multiply lines. By Melnikov function method, this paper provides the upper and lower bounds for the Poincaré cyclicity. In particular, the exact cyclicities of two specific cases are given using analyzing method, one of which has been discussed in the paper [7] .

A class of quadratic reversible systems with a center of genus one

In this paper, we first investigate global phase portraits of a class of quadratic reversible systems with a center of genus one, and obtain six possible global phase portraits. We then perturb the phase portrait with a non-Morsean point, by using nth order polynomials, n=1,2,…,7. For the perturbed systems, we study the limit cycle bifurcation problem, obtaining some new results.

Limit cycle bifurcations by perturbing non-smooth Hamiltonian systems with 4 switching lines via multiple parameters

In this paper, we investigate the limit cycle bifurcation in perturbations of non-smooth Hamiltonian systems with 4 switching lines via multiple parameters. Using the first order Melnikov function, we derive some computational formulas, which can be used to study the bifurcation of limit cycles. As an application to the obtained results, we consider the limit cycle bifurcation problem of a non-smooth system studied in Wang and Han (2016). Comparing with the result in the above reference, we find n−1,n∈N+ more limit cycles by our method.

Limit Cycles of a Class of Piecewise Smooth Liénard Systems

In this paper, we study the problem of limit cycle bifurcation in two piecewise polynomial systems of Liénard type with multiple parameters. Based on the developed Melnikov function theory, we obtain the maximum number of limit cycles of these two systems.

Limit cycle bifurcations by perturbing a quadratic integrable system with a triangle

Finding cyclicity of homoclinic or heteroclinic loops in quadratic systems is an open problem, which is related to the second part of Hilbert's 16th problem. This paper mainly considers the problem of limit cycle bifurcation for quadratic polynomial systems near a triangle. It is proved that the cyclicity of the triangle in quadratic systems is at least three.

Behavior of limit cycle bifurcations for a class of quartic Kolmogorov models in a symmetrical vector field

In this study, we consider the limit cycle bifurcation problem for a class of quartic Kolmogorov models with five positive singular points, i.e., (1,1), (1,2), (2,1), (1,3), and (3,1), which lie in a symmetrical vector field relative to the line y=x. We classify these singular points. We show that points (1,2) and (2,1) can bifurcate into three small limit cycles by simultaneous Hopf bifurcation, and that points (1,3) and (3,1) can bifurcate into three small limit cycles by simultaneous Hopf bifurcation. In addition, we construct limit cycles for this model and we show that four positive singular points, i.e., (1,1), (1,2), (2,1), and (1,3), can bifurcate into eight limit cycles in total, among which six cycles may be stable. Few previous studies have considered a symmetrical Kolmogorov model with several positive singular points. Our results are good in terms of the Hilbert number for the Kolmogorov model.

Limit cycle bifurcations near homoclinic and heteroclinic loops via stability-changing of a homoclinic loop

This paper is concerned with the problem of limit cycle bifurcation near a homoclinic or heteroclinic loop for non-smooth systems. By establishing the Poincaré map, some stability criteria are derived for a homoclinic loop in the non-smooth system under study. Furthermore, based on the theory of stability-changing of a homoclinic loop, a new approach is proposed to find limit cycles for the non-smooth system. Finally, several examples are provided to illustrate the obtained results.

Limit Cycles in a Class of Quartic Kolmogorov Model with Three Positive Equilibrium Points

Limit cycle bifurcation problem of Kolmogorov model is interesting and significant both in theory and applications. In this paper, we will focus on investigating limit cycles for a class of quartic Kolmogorov model with three positive equilibrium points. Perturbed model can bifurcate three small limit cycles near (1, 2) or (2, 1) under a certain condition and can bifurcate one limit cycle near (1, 1). In addition, we have given some examples of simultaneous Hopf bifurcation and the structure of limit cycles bifurcated from three positive equilibrium points. The limit cycle bifurcation problem for Kolmogorov model with several positive equilibrium points are less seen in published references. Our result is good and interesting.

Limit cycles near a homoclinic loop by perturbing a class of integrable systems

In this paper, the problem of limit cycle bifurcation is investigated by perturbing a class of integrable systems with a homoclinic loop. Under the assumption that the homoclinic loop passes through a degenerate singular point at the origin, the asymptotic expansion of the Melnikov function along the level curves of the first integral inside the homoclinic loop is studied near the loop. Meanwhile, the formulas for the first coefficients in the expansion are given, which can be used to study the number of limit cycles near the homoclinic loop. Finally, an example is provided to demonstrate the obtained results.

Limit cycle bifurcations by perturbing piecewise smooth Hamiltonian systems with multiple parameters

This paper is concerned with the problem of limit cycle bifurcation for piecewise smooth near-Hamiltonian systems with multiple parameters. By the first Melnikov function, some novel criteria have been established for the existence of multiple limit cycles. Furthermore, an example is included to validate the obtained results by considering the maximum number of limit cycles for a piecewise quadratic system studied in Llibre and Mereu (2014) [12]. Compared with the result in the above reference, one more limit cycle is found by our method.

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.

THE NUMBER OF LIMIT CYCLES IN A Z2-EQUIVARIANT LIÉNARD SYSTEM

In this paper, we consider the problem of limit cycle bifurcation near center points and a Z2-equivariant compound cycle in a polynomial Liénard system. Using the methods of Hopf, homoclinic and heteroclinic bifurcation theory, we found some new and better lower bounds of the maximal number of limit cycles for this system.