Bifurcation of limit cycles and the cusp of ordern

The author first investigates the limit cycles bifurcating from a center for general two dimensional systems, and then proves the conjecture that any unfolding of the cusp of ordern has at mostn−1 limit cycles. Supported by the Chinese National Natural Science Foundation.     

Bifurcations and distribution of limit cycles which appear from two nests of periodic orbits

In this paper, we study the distribution and simultaneous bifurcation of limit cycles bifurcated from the two periodic annuli of the holomorphic differential equation , after a small polynomial perturbation. We first show that, under small perturbations of the form , where is a polynomial of degree 2m−1 in which the power of z is odd and the power of is even, the only possible distribution of limit cycles is (u,u) for all values of u=0,1,2,…,m−3. Hence, the sharp upper bound for the number of limit cycles bifurcated from each two period annuli of is m−3, for m≥4. Then we consider a perturbation of the form , where is a polynomial of degree m in which the power of z is odd and obtain the upper bound m−5, for m≥6. Moreover, we show that the distribution (u,v) of limit cycles is possible for 0≤u≤m−5, 0≤v≤m−5 with u+v≤m−2 and m≥9.     

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 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.     

The number of limit cycles bifurcating from the periodic orbits of an isochronous center

This paper concerns the bifurcation of limit cycles for a quartic system with an isochronous center. By using the averaging theory, it shows that under any small quintic homogeneous perturbations, at most 14 limit cycles birfucate from the period annulus of the considered system.     

On the number of limit cycles bifurcating from a non-global degenerated center

We give an upper bound for the number of zeros of an Abelian integral. This integral controls the number of limit cycles that bifurcate, by a polynomial perturbation of arbitrary degree n, from the periodic orbits of the integrable system , where H is the quasi-homogeneous Hamiltonian H(x,y)=x2k/(2k)+y²/2. The tools used in our proofs are the Argument Principle applied to a suitable complex extension of the Abelian integral and some techniques in real analysis.     

Abelian integrals and limit cycles

The paper deals with generic perturbations from a Hamiltonian planar vector field and more precisely with the number and bifurcation pattern of the limit cycles. In this paper we show that near a 2-saddle cycle, the number of limit cycles produced in unfoldings with one unbroken connection, can exceed the number of zeros of the related Abelian integral, even if the latter represents a stable elementary catastrophe. We however also show that in general, finite codimension of the Abelian integral leads to a finite upper bound on the local cyclicity. In the treatment, we introduce the notion of simple asymptotic scale deformation.     

The bifurcation of homoclinic orbits from two heteroclinic orbits-a topological approach

Conditions are found for a homoclinic orbit to bifurcate from a heteroclinic loop for autonomous ordinary differential equations. The results axe applied to prove the existence of traveling wave solutions of the FitzHugh-Nagumo equations and a system of reaction diffusion equations which arise as a model for a two step combustion process.     

Uniqueness of limit cycles for polynomial first-order differential equations

We study the uniqueness of limit cycles (periodic solutions that are isolated in the set of periodic solutions) in the scalar ODE in terms of {i_k}, {j_k}, {n_k}. Our main result characterizes, under some additional hypotheses, the exponents {i_k}, {j_k}, {n_k}, such that for any choice of the equation has at most one limit cycle. The obtained results have direct application to rigid planar vector fields, thus, planar systems of the form x^′=y+xR(x,y), y^′=−x+yR(x,y), where . Concretely, when the set has at least three elements (or exactly one) and another technical condition is satisfied, we characterize the exponents {i_k}, {j_k} such that the origin of the rigid system is a center for any choice of and also when there are no limit cycles surrounding the origin for any choice of .     

Exact number of limit cycles for a family of rigid systems

For a given family of planar differential equations it is a very difficult problem to determine an upper bound for the number of its limit cycles. Even when this upper bound is one it is not always an easy problem to distinguish between the case of zero and one limit cycle. This note mainly deals with this second problem for a family of systems with a homogeneous nonlinear part. While the condition that allows us to separate the existence and the nonexistence of limit cycles can be described, it is very intricate.

     

Bifurcations of limit cycles in equivariant quintic planar vector fields

In this paper, we obtain 23 limit cycles for a _Z3$Z_3$-equivariant near-Hamiltonian system of degree 5 which is the perturbation of a _Z6$Z_6$-equivariant quintic Hamiltonian system. The configuration of these limit cycles is new and different from the configuration obtained by H.S.Y. Chan, K.W. Chung and J. Li, where the unperturbed system is a _Z3$Z_3$-equivariant quintic Hamiltonian system. Our unperturbed system is different from the unperturbed systems studied by Y. Wu and M. Han. The limit cycles are obtained by Poincaré–Pontryagin theorem and Poincaré–Bendixson theorem.     

The center-focus problem and bifurcation of limit cycles in a class of 7th-degree polynomial systems

By computing singular point values, the center conditions are established for a class of 7th-degree planar polynomial systems with 15 parameters. It is proved that such systems can have 13 small-amplitude limit cycles in the neighborhood of the origin. To the best of our knowledge, this is the first example of a 7th-degree system having non-homogeneous nonlinearities with thirteen limit cycles bifurcated from a fine focus.     

On the stability of limit cycles for planar differential systems

We consider a planar differential system , , where P and Q are C¹ functions in some open set U⊆R², and . Let γ be a periodic orbit of the system in U. Let f(x,y):U⊆R²→R be a C¹ function such that

     

Limit cycles for rigid cubic systems

We consider planar cubic systems with a unique rest point of center-focus type and constant angular velocity. For such systems we obtain an affine classification in three families, and, for two of them, their corresponding phase portraits on the Poincaré sphere. We also prove that for two of these families there is uniqueness of limit cycle. With respect the third family, we give the bifurcation diagram and phase portraits on the Poincaré sphere of a one-parameter sub-family exhibiting at least two limit cycles.     

Exact bound on the number of limit cycles arising from a periodic annulus bounded by a symmetric heteroclinic loop

In this paper, the bound on the number of limit cycles by Poincare bifurcation in a small perturbation of some seventh-degree Hamiltonian system is concerned. The lower and upper bounds on the number of limit cycles have been obtained in two previous works, however, the sharp bound is still unknown. We will employ some new techniques to determine which is the exact bound between $3$ and $4$. The asymptotic expansions are used to determine the four vertexes of a tetrahedron, and the sharp bound can be reached when the parameters belong to this tetrahedron.     

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].     

Uniqueness of limit cycles in predator-prey system: the role of weight functions

We consider a Gause type model of interactions between predator and prey populations. Using the ideas of Cheng and Liou we give a sufficient condition for uniqueness of the limit cycle, which is more general than their condition. That is, we include a kind of weight function in the condition. It was motivated by a result due to Hwang, where the prey isocline plays a role of weight function. Moreover, we show that the interval where the condition from Hwang's result is to be fulfilled can be narrowed.     

Bifurcation of limit cycles from the global center of a class of integrable non-Hamilton systems

In this paper, we consider the bifurcation of limit cycles for system $\dot{x}=-y(x^2+a^2)^m,~\dot{y}=x(x^2+a^2)^m$ under perturbations of polynomials with degree n, where $a\neq0$, $m\in \mathbb{N}$. By using the averaging method of first order, we bound the number of limit cycles that can bifurcate from periodic orbits of the center of the unperturbed system. Particularly, if $m=2, n=5$, the sharp bound is 5.