Bifurcation of limit cycles from a compound loop with five saddles

We concern the number of limit cycles of a polynomial system with degree nine. We prove that under different conditions, the system can have 12 and 20 limit cycles bifurcating from a compound loop with five saddles. Our method relies upon the Melnikov function method and the method of stability-changing of a double homoclinic loop proposed by the authors[J. Yang, Y. Xiong and M. Han, {\em Nonlinear Anal-Theor.}, 2014, 95, 756--773].     

Bifurcation of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems of degree 5 with a cusp

This paper deals with small perturbations of a class of hyper-elliptic Hamiltonian system, which is a Li é nard system of the form \(\dot{x}=y,\)  \(\;\dot{y}=Q_1(x)+\varepsilon yQ_2(x)\) with \(Q_1\) and \(Q_2\) polynomials of degree 4 and 3, respectively. It is shown that this system can undergo degenerated Hopf bifurcation and Poincar é bifurcation, which emerge at most three limit cycles for \(\varepsilon\) sufficiently small.     

On the number of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems with one nilpotent saddle

In this paper, we make a complete study on small perturbations of Hamiltonian vector field with a hyper-elliptic Hamiltonian of degree five, which is a Liénard system of the form x^′=y, y^′=Q₁(x)+εyQ₂(x) with Q₁ and Q₂ polynomials of degree respectively 4 and 3. It is shown that this system can undergo degenerated Hopf bifurcation and Poincaré bifurcation, which emerges at most three limit cycles in the plane for sufficiently small positive ε. And the limit cycles can encompass only an equilibrium inside, i.e. the configuration (3,0) of limit cycles can appear for some values of parameters, where (3,0) stands for three limit cycles surrounding an equilibrium and no limit cycles surrounding two equilibria.     

Bifurcation of limit cycles from a heteroclinic loop with a cusp

In this article, we study the expansion of the first Melnikov function of a near-Hamiltonian system near a heteroclinic loop with a cusp and a saddle or two cusps, obtaining formulas to compute the first coefficients of the expansion. Then we use the results to study the problem of limit cycle bifurcation for two polynomial systems.     

Bifurcation of limit cycles by perturbing a class of hyper-elliptic Hamiltonian systems of degree five

In this paper, we investigate a class of hyper-elliptic Hamiltonian systems of degree five under the polynomial perturbation of degree m+1$m+1$. First, we study the number of different phase portraits of the unperturbed system when it has a class of family of periodic orbits and prove that the number is 40. Then, we consider the limit cycle bifurcations and obtain some new results on the lower bound of the maximal number of limit cycles for these systems.     

Bifurcation of limit cycles in piecewise smooth systems via Melnikov function

In this short paper, we present some remarks on the role of the rstorder Melnikov functions in studying the number of limit cycles of piecewisesmooth near-Hamiltonian systems on the plane.     

Bifurcation of limit cycles by perturbing piecewise non-Hamiltonian systems with nonlinear switching manifold

This paper is devoted to the study of limit cycles that can bifurcate of a perturbation of piecewise non-Hamiltonian systems with nonlinear switching manifold. We derive the first order Melnikov function to these systems. As application, the sharp upper bound of the number of bifurcated limit cycles of two concrete systems, whose switching manifolds are algebraic curves, is presented.     

Bifurcation of limit cycles at a nilpotent critical point in a septic Lyapunov system

In this paper, we characterize local behavior of an isolated nilpotent critical point for a class of septic polynomial differential systems, including center conditions and bifurcation of limit cycles. With the help of computer algebra system-MATHEMATICA 12.0, the first 15 quasi-Lyapunov constants are deduced. As a result, necessary and sufficient conditions of such system having a center are obtained. We prove that there exist 16 small amplitude limit cycles created from the third-order nilpotent critical point. And then we give a lower bound of cyclicity of third-order nilpotent critical point for septic Lyapunov systems.     

Limit cycles for perturbing a piecewise linear Hamiltonian system with one or two saddles

We study bifurcations of limit cycles arising after perturbations of linear piecewise Hamiltonian systems. In this paper we find bounds for the numbers of limit cycles for several families of which phase portraits were classified in Xiong and Han (2013) [14].     

Bifurcation of limit cycles in a quintic Hamiltonian system under a sixth-order perturbation

This paper intends to explore the bifurcation of limit cycles for planar polynomial systems with even number of degrees. To obtain the maximum number of limit cycles, a sixth-order polynomial perturbation is added to a quintic Hamiltonian system, and both local and global bifurcations are considered. By employing the detection function method for global bifurcations of limit cycles and the normal form theory for local degenerate Hopf bifurcations, 31 and 35 limit cycles and their configurations are obtained for different sets of controlled parameters. It is shown that: H(6) ⩾ 35 = 6² − 1, where H(6) is the Hilbert number for sixth-degree polynomial systems.     

Bifurcation of limit cycles in a quintic Hamiltonian system under a sixth-order perturbation

This paper intends to explore the bifurcation of limit cycles for planar polynomial systems with even number of degrees. To obtain the maximum number of limit cycles, a sixth-order polynomial perturbation is added to a quintic Hamiltonian system, and both local and global bifurcations are considered. By employing the detection function method for global bifurcations of limit cycles and the normal form theory for local degenerate Hopf bifurcations, 31 and 35 limit cycles and their configurations are obtained for different sets of controlled parameters. It is shown that: H(6)  35 = 6² − 1, where H(6) is the Hilbert number for sixth-degree polynomial systems.     

Bifurcation of limit cycles for a quartic near-Hamiltonian system by perturbing a nilpotent center

Hopf bifurcation which produces oscillations is a very important phenomena in the theory and application of dynamical systems. Almost all works available about Hopf bifurcations are related to a non-degenerate focus or center. For the case of a degenerate focus or center, the study of the bifurcations becomes challenge. In this paper, we consider the bifurcation of limit cycles for a quartic near-Hamiltonian system by perturbing a nilpotent center. We take coefficients as parameters, then we can get six limit cycles.     

Bifurcations of limit cycles for a cubic Hamiltonian system under quartic perturbations

This paper is concerned with the number of limit cycles of a cubic system with quartic perturbations. Fifteen limit cycles are found and their distributions are studied by using the methods of bifurcation theory and qualitative analysis. It gives rise to the conclusion: H(4)15, where H(n) is the Hilbert number for the second part of Hilbert's 16th problem.     

On the number of limit cycles for a quintic Li\""{e}nard system under polynomial perturbations

In this paper, we mainly study the number of limit cycles for a quintic Li\"{e}nard system under polynomial perturbations. In some cases, we give new estimations for the lower bound of the maximal number of limit cycles.     

Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loop

In this paper, we study limit cycle bifurcations for a kind of non-smooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with the center at the origin and a homoclinic loop around the origin. By using the first Melnikov function of piecewise near-Hamiltonian systems, we give lower bounds of the maximal number of limit cycles in Hopf and homoclinic bifurcations, and derive an upper bound of the number of limit cycles that bifurcate from the periodic annulus between the center and the homoclinic loop up to the first order in ε$\epsilon$. In the case when the degree of perturbing terms is low, we obtain a precise result on the number of zeros of the first Melnikov function.     

Bifurcation of limit cycles of the nongeneric quadratic reversible system with discontinuous perturbations

By using the Picard-Fuchs equation and the property of the Chebyshev space to the discontinuous differential system, we obtain an upper bound of the number of limit cycles for the nongeneric quadratic reversible system when it is perturbed inside all discontinuous polynomials with degree n.