The bifurcation of limit cycles by perturbing a planar system which has a continuous family of cycles, i.e. periodic orbits, has been an intensively studied phenomenon; see for instance [13, 16, 2] and references therein. The simplest planar system having a continuous family of cycles is the linear center, and a special family of its perturbations is given by the generalized polynomial Liénard ...

Robust attracting heteroclinic cycles have been found in many models of dynamics with symmetries. In all previous examples, robust heteroclinic cycles appear between a number of symmetry broken equilibria. In this paper we examine the first example where there are robust attracting heteroclinic cycles that include the origin, ie a point with maximal symmetry. The example we study is for vector ...

The main modes of behavior of a food chain model, composed of logistic prey and Holling type II predator and superpredator, are discussed in this paper. The study is carried out through bifurcation analysis, alternating between a normal form approach and numerical continuation. The two-parameter bifurcation diagram of the model contains Hopf, fold and transcritical bifurcation curves of equilib...

We use blowup to study the regularization of codimension one two-fold singularities in planar piecewise smooth (PWS) dynamical systems. We focus on singular canards, pseudo-equlibria and limit cycles that can occur in the PWS system. Using the regularization of Sotomayor and Teixeira [30], we show rigorously how singular canards can persist and how the bifurcation of pseudo-equilibria is relate...

A new method with an efficient algorithm is developed for computing the Lyapunov constants of planar switching systems, and then applied to study bifurcation of limit cycles in a switching Bautin system. A complete classification on the conditions of a singular point being a center in this Bautin system is obtained. Further, an example of switching systems is constructed to show the existence o...

In this paper, we consider bifurcation of limit cycles in near-Hamiltonian systems. A new method is developed to study the analytical property of the Melnikov function near the origin for such systems. Based on the new method, a computationally efficient algorithm is established to systematically compute the coefficients of Melnikov function. Moreover, we consider the case that the Hamiltonian ...

The paper is concerned with the bifurcation of limit cycles in general quadratic perturbations of a quadratic reversible and non-Hamiltonian system, whose period annulus is bounded by an elliptic separatrix related to a singularity at infinity in the Poincaré disk. Attention goes to the number of limit cycles produced by the period annulus under perturbations. By using the appropriate Picard-Fu...

In this paper, we consider bifurcation of small limit cycles from Hopf-type singular points in Z5-equivariant planar vector fields of order 5. We apply normal form theory and the technique of solving coupled multivariate polynomial equations to prove that the maximal number of small limit cycles that such vector fields can have is 25. In addition, we show that no large limit cycles exist. Thus,...

This paper intends to explore bifurcation behavior of limit cycles for a cubic Hamiltonian system with quintic perturbed terms using both qualitative analysis and numerical exploration. To obtain the maximum number of limit cycles, a quintic perturbed function with the form of R(x, y, λ) = S(x, y, λ) = mx2 + ny2 + ky4 − λ is added to a cubic Hamiltonian system, where m, n, k and λ are all varia...