Sharp Bound of the Number of Zeros for a Liénard System with a Heteroclinic Loop

In the presented paper, the Abelian integral I h of a Liénard system is investigated, with a heteroclinic loop passing through a nilpotent saddle. By using a new algebraic criterion, we try to find the least upper bound of the number of limit cycles bifurcating from periodic annulus.

Higher Codimension Bifurcation Analysis of Predator–Prey Systems with Nonmonotonic Functional Responses

In this paper, we study the dynamic behaviors of a predator–prey system with a general form of nonmonotonic functional response. Through analysis, it is found that the system exists in extinction equilibrium, boundary equilibrium and two positive equilibria, one or no positive equilibrium. Furthermore, the conditions are given such that the boundary equilibrium is a saddle, node or a saddle-node point of codimension 1, 2 or 3. Then, some conditions are obtained so that the unique positive equilibrium of the system is a cusp point of codimension 2, 3 and higher or a saddle-node one of codimension 1 or 3, or a nilpotent saddle-node of codimension 4. When there are two positive equilibria in the system, the equilibrium with a large number of preys is a saddle point. For the other one, the system may undergo Hopf bifurcation. To verify our conclusion, we consider the functional response function in the literature [ Zhu et al., 2002 ; Xiao & Ruan, 2001 ]. Finally, we give a brief discussion and numerical simulation.

Global Phase Portraits of ℤ2-Symmetric Planar Polynomial Hamiltonian Systems of Degree Three with a Nilpotent Saddle at the Origin

We characterize the phase portraits in the Poincaré disk of all planar polynomial Hamiltonian systems of degree three with a nilpotent saddle at the origin and [Formula: see text]-symmetric with [Formula: see text].

Poincaré Bifurcation of Limit Cycles from a Liénard System with a Homoclinic Loop Passing through a Nilpotent Saddle

In present paper, the number of zeros of the Abelian integral is studied, which is for some perturbed Hamiltonian system of degree 6. We prove the generating elements of the Abelian integral from a Chebyshev system of accuracy of 3; therefore there are at most 6 zeros of the Abelian integral.

On the Number of Limit Cycles Bifurcated from a Near-Hamiltonian System with a Double Homoclinic Loop of Cuspidal Type Surrounded by a Heteroclinic Loop

In this paper, we study the number of bifurcated limit cycles from some polynomial systems with a double homoclinic loop passing through a nilpotent saddle surrounded by a heteroclinic loop, and obtain some new results on the lower bound of the maximal number of limit cycles for these systems. In particular, we study the bifurcation of limit cycles in the following system: [Formula: see text] where [Formula: see text] is a polynomial of degree [Formula: see text].

On the Number of Limit Cycles Bifurcated from Some Hamiltonian Systems with a Double Homoclinic Loop and a Heteroclinic Loop

In this paper, we study the number of bifurcated limit cycles from near-Hamiltonian systems where the corresponding Hamiltonian system has a double homoclinic loop passing through a hyperbolic saddle surrounded by a heteroclinic loop with a hyperbolic saddle and a nilpotent saddle, and obtain some new results on the lower bound of the maximal number of limit cycles for these systems. In particular, we study the bifurcation of limit cycles of the following system [Formula: see text] as an application of our results, where [Formula: see text] is a polynomial of degree five.

On the Distribution of Limit Cycles in a Liénard System with a Nilpotent Center and a Nilpotent Saddle

In this work, we study the Abelian integral [Formula: see text] corresponding to the following Liénard system, [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] are real bounded parameters. By using the expansion of [Formula: see text] and a new algebraic criterion developed in [Grau et al., 2011], it will be shown that the sharp upper bound of the maximal number of isolated zeros of [Formula: see text] is 4. Hence, the above system can have at most four limit cycles bifurcating from the corresponding period annulus. Moreover, the configuration (distribution) of the limit cycles is also determined. The results obtained are new for this kind of Liénard system.