In this paper we find an upper bound for the maximum number of limit cycles bifurcating from the periodic orbits of any planar polynomial quasi-homogeneous center, which can be obtained using first order averaging method. This result improves the upper bounds given in [7].

We study the maximum number of limit cycles that bifurcate from the periodic solutions of the family of isochronous cubic polynomial centers ẋ = y(−1+2αx+2βx), ẏ = x+α(y−x)+ 2βxy, α ∈ R, β < 0, when it is perturbed inside the classes of all continuous and discontinuous cubic polynomial differential systems. We obtain that the maximum number of limit cycles which can be obtained by the averaging...

In this article we study Hopf bifurcations and small amplitude limit cycles in a family of quadratic systems in the three dimensional space called Rucklidge systems. Bifurcation analysis at the equilibria of Rucklidge system is pushed forward toward the calculation of the second Lyapunov coefficient, which makes possible the determination of the Lyapunov and higher order structural stability.

Using the method of qualitative analysis we show that five perturbed cubic Hamiltonian systems have the same distribution of limit cycles and have 11 limit cycles for some parameters. The accurate location of each limit cycle is given by numerical exploration. In other words, we demonstrate the existence of 11 limit cycles and their distribution in five perturbed systems in two ways, the result...

We present a control method which makes the trajectories of a switched system converge to a stable limit cycle lying in a desired region of equilibrium. The method is illustrated on the boost DC-DC converter example. We also point out in this example the sensitivity of limit cycles to parameter variations by showing how the limit cycle evolves in presence of small perturbations of some system p...

In this paper, we carry out a computational study using the spectral decomposition of the fluctuations of a two-pathogen epidemic model around its deterministic attractor, i.e., steady state or limit cycle, to examine the role of partial vaccination and between-host pathogen interaction on early pathogen replacement during seasonal epidemics of influenza and respiratory syncytial virus.

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. Mor...

The Bogdanov-Takens system has at most one limit cycle and, in the parameter space, it exists between a Hopf and a saddle-loop bifurcation curves. The aim of this paper is to prove the Perko’s conjectures about some analytic properties of the saddle-loop bifurcation curve. Moreover, we provide sharp piecewise algebraic upper and lower bounds for this curve.

Recently Braga and Mello conjectured that for a given n ∈ N there is a piecewise linear system with two zones in the plane with exactly n limit cycles. In this paper we prove a result from which the conjecture is an immediate consequence. Several explicit examples are given where location and stability of limit cycles are provided.

We study asymptotics of parameter estimates in conditional heteroscedastic models. The estimators considered are those obtained by minimizing certain functionals and those obtained by solving estimation equations. We establish consistency and derive asymptotic limit laws of the estimators. Condition under which the limit law is normal is studied. Further, bootstrap for these estimators is discu...