Bifurcation of limit cycles from generalized homoclinic loops in planar piecewise smooth systems

The generalized homoclinic loop appears in the study of dynamics on piecewise smooth differential systems during the past two decades. For planar piecewise smooth differential systems, there are concrete examples showing that under suitable perturbations of a generalized homoclinic loop one or two limit cycles can appear. But up to now there is no a general theory to study the cyclicity of a generalized homoclinic loop, that is, the maximal number of limit cycles which are bifurcated from it.     

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.     

Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth systems with a cusp

For a piecewise analytical Hamiltonian system with a cusp on a switch line, which has a family of periodic orbits near a generalized homoclinic loop, we study the maximum number of limit cycles bifurcating from the periodic orbits. For doing so, we first obtain the asymptotic expressions of the Melnikov functions near the loop. Finally we present two examples illustrating applications of the main results.     

The loop quantities and bifurcations of homoclinic loops

The stability and bifurcations of a homoclinic loop for planar vector fields are closely related to the limit cycles. For a homoclinic loop of a given planar vector field, a sequence of quantities, the homoclinic loop quantities were defined to study the stability and bifurcations of the loop. Among the sequence of the loop quantities, the first nonzero one determines the stability of the homoclinic loop. There are formulas for the first three and the fifth loop quantities. In this paper we will establish the formula for the fourth loop quantity for both the single and double homoclinic loops. As applications, we present examples of planar polynomial vector fields which can have five or twelve limit cycles respectively in the case of a single or double homoclinic loop by using the method of stability-switching.     

On the number of limit cycles by perturbing a piecewise smooth Hamilton system with two straight lines of separation

This paper deals with the problem of limit cycle bifurcations for a piecewise smooth Hamilton system with two straight lines of separation. By analyzing the obtained first order Melnikov function, we give upper and lower bounds of the number of limit cycles bifurcating from the period annulus between the origin and the generalized homoclinic loop. It is found that the first order Melnikov function is more complicated than in the case with one straight line of separation and more limit cycles can be bifurcated.     

Limit cycle bifurcations near a double homoclinic loop with a nilpotent saddle of order m

In this paper, we study the explicit expansion of the first order Melnikov function near a double homoclinic loop passing through a nilpotent saddle of order m in a near-Hamiltonian system. For any positive integer $m(m\ge 1)$, we derive the formulas of the coefficients in the expansion, which can be used to study the limit cycle bifurcations for near-Hamiltonian systems. In particular, for $m=2$, we use the coefficients to consider the limit cycle bifurcations of general near-Hamiltonian systems and give the existence conditions for 10, 11, 13, 15 and 16 (11, 13 and 16, respectively) limit cycles in the case that the homoclinic loop is of cuspidal type (smooth type, respectively) and their distributions. As an application, we consider a near-Hamiltonian system with a nilpotent saddle of order 2 and obtain the lower bounds of the maximal number of limit cycles.     

On Melnikov functions of a homoclinic loop through a nilpotent saddle for planar near-Hamiltonian systems

The first-order Melnikov function of a homoclinic loop through a nilpotent saddle for general planar near-Hamiltonian systems is considered. The asymptotic expansion of this Melnikov function and formulas for its first coefficients are given. The number of limit cycles which appear near the homoclinic loop is discussed by using the asymptotic expansion of the first-order Melnikov function. An example is presented as an application of the main results.     

On the limit cycles of perturbed discontinuous planar systems with 4 switching lines

Limit cycle bifurcations for a class of perturbed planar piecewise smooth systems with 4 switching lines are investigated. The expressions of the first order Melnikov function are established when the unperturbed system has a compound global center, a compound homoclinic loop, a compound 2-polycycle, a compound 3-polycycle or a compound 4-polycycle, respectively. Using Melnikov’s method, we obtain lower bounds of the maximal number of limit cycles for the above five different cases. Further, we derive upper bounds of the number of limit cycles for the later four different cases. Finally, we give a numerical example to verify the theory results.     

LIMIT CYCLES NEAR A DOUBLE HOMOCLINIC LOOP

In this article,we study the expansion of the first order Melnikov function in a near-Hamiltonian system on the plane near a double homoclinic loop.We obtain an explicit formula to compute the first four coeffcients,and then identify the method of finding at least 7 limit cycles near the double homoclinic loop using these coefficients.Finally,we present some interesting applications.     

Limit cycle bifurcations of some Liénard systems

In this paper we first give some general theorems on the limit cycle bifurcation for near-Hamiltonian systems near a double homoclinic loop or a center as a preliminary. Then we use these theorems to study some polynomial Liénard systems with perturbations and give new lower bounds for the maximal number of limit cycles of these systems.     

Homoclinic,heteroclinic and periodic orbits of singularly perturbed systems

The main aims of this paper are to study the persistence of homoclinic and heteroclinic orbits of the reduced systems on normally hyperbolic critical manifolds, and also the limit cycle bifurcations either from the homoclinic loop of the reduced systems or from a family of periodic orbits of the layer systems. For the persistence of homoclinic and heteroclinic orbits, and the limit cycles bifurcating from a homolinic loop of the reduced systems, we provide a new and readily detectable method to characterize them compared with the usual Melnikov method when the reduced system forms a generalized rotated vector field. To determine the limit cycles bifurcating from the families of periodic orbits of the layer systems, we apply the averaging methods.We also provide two four-dimensional singularly perturbed differential systems, which have either heteroclinic or homoclinic orbits located on the slow manifolds and also three limit cycles bifurcating from the periodic orbits of the layer system.     

On the number of limit cycles in double homoclinic bifurcations

LetL be a double homoclinic loop of a Hamiltonian system on the plane. We obtain a condition under whichL generates at most two large limit cycles by perturbations. We also give conditions for the existence of at most five or six limit cycles which appear nearL under perturbations.     

Cyclicity of planar homoclinic loops and quadratic integrable systems

A general method for a homoclinic loop of planar Hamiltonian systems to bifurcate two or three limit cycles under perturbations is established. Certain conditions are given under which the cyclicity of a homoclinic loop equals 1 or 2. As an application to quadratic systems, it is proved that the cyclicity of homoclinic loops of quadratic integrable and non-Hamiltonian systems equals 2 except for one case. Project supported by the National Natural Science Foundation of China.     

Limit cycle bifurcations of some Liénard systems with a cuspidal loop and a homoclinic loop

In this paper, we study the number of limit cycles of some polynomial Liénard systems with a cuspidal loop and a homoclinic loop, and obtain some new results on the lower bound of the maximal number of limit cycles for these systems.     

The Maximum Number of Zeros of Functions with Parameters and Application to Differential Equations

In this paper, we first study the problem of finding the maximum number of zeros of functions with parameters and then apply the results obtained to smooth or piecewise smooth planar autonomous systems and scalar periodic equations to study the number of limit cycles or periodic solutions, improving some fundamental results both on the maximum number of limit cycles bifurcating from an elementary focus of order $k$ or a limit cycle of multiplicity $k$, or from a period annulus, and on the maximum number of periodic solutions for scalar periodic smooth or piecewise smooth equations as well.     

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.     

The Number of Limit Cycles in a Class of Piecewise Polynomial Systems

In this paper, we pay attention to the number of limit cycles for a class of piecewise smooth near-Hamiltonian systems. By using the expression of the first order Melnikov function and some known results about Chebyshev systems, we study upper bound of the number of limit cycles in Hopf bifurcation and Poincar\''{e} bifurcation respectively.     

Limit cycle bifurcations by perturbing a cuspidal loop in a Hamiltonian system

In this paper, we first study the analytical property of the first Melnikov function for general Hamiltonian systems exhibiting a cuspidal loop and obtain its expansion at the Hamiltonian value corresponding to the loop. Then by using the first coefficients of the expansion we give some conditions for the perturbed system to have 4, 5 or 6 limit cycles in a neighborhood of loop. As an application of our main results, we consider some polynomial Lienard systems and find 4, 5 or 6 limit cycles.     

Bifurcations of limit cycles in a perturbed quintic Hamiltonian system with six double homoclinic loops

This paper concerns with the number and distributions of limit cycles of a quintic subject to a seven-degree perturbation. With the aid of numeric integral computation provided by Mathematica 4.1, at least 45 limit cycles are found in the above system by applying the method of double homoclinic loops bifurcation, Hopf bifurcation and qualitative analysis. The four configurations of 45 limit cycles of the system are also shown. The results obtained are useful to the study of the weakened 16th Hilbert Problem.     

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].