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.     

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.     

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 cyclicity of weight-homogeneous centers

Let W be a weight-homogeneous planar polynomial differential system with a center. We find an upper bound of the number of limit cycles which bifurcate from the period annulus of W under a generic polynomial perturbation. We apply this result to a particular family of planar polynomial systems having a nilpotent center without meromorphic first integral.     

On the Number of Limit Cycles in Small Perturbations of a Piecewise Linear Hamiltonian System with a Heteroclinic Loop

In this paper, the authors consider limit cycle bifurcations for a kind of nonsmooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with a center at the origin and a heteroclinic loop around the origin. When the degree of perturbing polynomial terms is n(n ≥ 1), it is obtained that n limit cycles can appear near the origin and the heteroclinic loop respectively by using the first Melnikov function of piecewise near-Hamiltonian systems, and that there are at most n + [(n+1)/2] limit cycles bifurcating from the periodic annulus between the center and the heteroclinic loop up to the first order in ε. Especially, for n = 1, 2, 3 and 4, a precise result on the maximal number of zeros of the first Melnikov function is derived.     

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.     

The cyclicity of the elliptic segment loops of the reversible quadratic Hamiltonian systems under quadratic perturbations

Denote by Q^H and Q^R the Hamiltonian class and reversible class of quadratic integrable systems. There are several topological types for systems belong to Q^H∩Q^R. One of them is the case that the corresponding system has two heteroclinic loops, sharing one saddle-connection, which is a line segment, and the other part of the loops is an ellipse. In this paper we prove that the maximal number of limit cycles, which bifurcate from the loops with respect to quadratic perturbations in a conic neighborhood of the direction transversal to reversible systems (called in reversible direction), is two. We also give the corresponding bifurcation diagram.     

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.     

Limit cycles and invariant cylinders for a class of continuous and discontinuous vector field in dimention 2n

The subject of this paper concerns with the bifurcation of limit cycles and invariant cylinders from a global center of a linear differential system in dimension 2n perturbed inside a class of continuous and discontinuous piecewise linear differential systems. Our main results show that at most one limit cycle and at most one invariant cylinder can bifurcate using the expansion of the displacement function up to first order with respect to a small parameter. This upper bound is reached. For proving these results we use the averaging theory in a form where the differentiability of the system is not needed.     

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.     

Limit cycles near generalized homoclinic and double homoclinic loops in piecewise smooth systems

In this paper, we study the bifurcation of limit cycles in piecewise smooth systems by perturbing a piecewise Hamiltonian system with a generalized homoclinic or generalized double homoclinic loop. We first obtain the form of the expansion of the first Melnikov function. Then by using the first coefficients in the expansion, we give some new results on the number of limit cycles bifurcated from a periodic annulus near the generalized (double) homoclinic loop. As applications, we study the number of limit cycles of a piecewise near-Hamiltonian systems with a generalized homoclinic loop and a central symmetric piecewise smooth system with a generalized double homoclinic loop.     

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.     

On the number of zeros of Abelian integrals for a polynomial Hamiltonian irregular at infinity

Up to now, most of the results on the tangential Hilbert 16th problem have been concerned with the Hamiltonian regular at infinity, i.e., its principal homogeneous part is a product of the pairwise different linear forms. In this paper, we study a polynomial Hamiltonian which is not regular at infinity. It is shown that the space of Abelian integral for this Hamiltonian is finitely generated as a R[h] module by several basic integrals which satisfy the Picard-Fuchs system of linear differential equations. Applying the bound meandering principle, an upper bound for the number of complex isolated zeros of Abelian integrals is obtained on a positive distance from critical locus. This result is a partial solution of tangential Hilbert 16th problem for this Hamiltonian. As a consequence, we get an upper bound of the number of limit cycles produced by the period annulus of the non-Hamiltonian integrable quadratic systems whose almost all orbits are algebraic curves of degree k+n, under polynomial perturbation of arbitrary degree.     

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.     

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.     

Bifurcation of limit cycles from a 4-dimensional center in 1:n resonance

We study the bifurcation of limit cycles from the periodic orbits of a linear differential system in R⁴ in resonance 1:n perturbed inside a class of piecewise linear differential systems, which appear in a natural way in control theory. Our main result shows that at most 1 limit cycle can bifurcate using expansion of the displacement function up to first order with respect to a small parameter. This upper bound is reached. For proving this result we use the averaging theory in a form where the differentiability of the system is not needed.     

Bifurcation of periodic orbits emanated from a vertex in discontinuous planar systems

We discuss bifurcation of periodic orbits in discontinuous planar systems with discontinuities on finitely many straight lines intersecting at the origin and the unperturbed system has either a limit cycle or an annulus of periodic orbits. Assume that the unperturbed periodic orbits cross every switching line transversally exactly once. For the first case we give a condition for the persistence of the limit cycle. For the second case, we obtain the expression of the first order Melnikov function and establish sufficient conditions on the number of limit cycles bifurcate from the periodic annulus. Then we generalize our results to systems with discontinuities on finitely many smooth curves. As an application, we present a piecewise cubic system with 4 switching lines and show that the maximum number of limit cycles bifurcate from the periodic annulus can be affected by the position of the switching lines.     

On Hopf bifurcation in non-smooth planar systems

As we know, for non-smooth planar systems there are foci of three different types, called focus-focus (FF), focus-parabolic (FP) and parabolic-parabolic (PP) type respectively. The Poincaré map with its analytical property and the problem of Hopf bifurcation have been studied in Coll et al. (2001) [3] and Filippov (1988) [6] for general systems and in Zou et al. (2006) [13] for piecewise linear systems. In this paper we also study the problem of Hopf bifurcation for non-smooth planar systems, obtaining new results. More precisely, we prove that one or two limit cycles can be produced from an elementary focus of the least order (order 1 for foci of FF or FP type and order 2 for foci of PP type) (Theorem 2.3), different from the case of smooth systems. For piecewise linear systems we prove that 2 limit cycles can appear near a focus of either FF, FP or PP type (Theorem 3.3).     

NEW FAMILIES OF CENTERS AND LIMIT CYCLES FOR POLYNOMIAL DIFFERENTIAL SYSTEMS WITH HOMOGENEOUS NONLINEARITIES

We consider the class of polynomial differential equations x = -y+Pn(x,y), y = x + Qn(x, y), where Pn and Qn are homogeneous polynomials of degree n. Inside this class we identify a new subclass of systems having a center at the origin. We show that this subclass contains at least two subfamilies of isochro-nous centers. By using a method different from the classical ones, we study the limit cycles that bifurcate from the periodic orbits of such centers when we perturb them inside the class of all polynomial differential systems of the above form. In particular, we present a function whose simple zeros correspond to the limit cycles vvhich bifurcate from the periodic orbits of Hamiltonian systems.