Limit cycle bifurcations from a nilpotent focus or center of planar systems

In this paper, we study the analytical property of the Poincare return map and the generalized focal values of an analytical planar system with a nilpotent focus or center. Then we use the focal values and the map to study the number of limit cycles of this kind of systems with parameters, and obtain some new results on the lower and upper bounds of the maximal number of limit cycles near the nilpotent focus or center.Comment: This paper was submitted to Journal of Mathematical Analysis and Application

Limit Cycle Bifurcations from Centers of Symmetric Hamiltonian Systems Perturbing by Cubic Polynomials

In this paper, we consider some cubic near-Hamiltonian systems obtained from perturbing the symmetric cubic Hamiltonian system with two symmetric singular points by cubic polynomials. First, following Han [2012] we develop a method to study the analytical property of the Melnikov function near the origin for near-Hamiltonian system having the origin as its elementary center or nilpotent center. Based on the method, a computationally efficient algorithm is established to systematically compute the coefficients of Melnikov function. Then, we consider the symmetric singular points and present the conditions for one of them to be elementary center or nilpotent center. Under the condition for the singular point to be a center, we obtain the normal form of the Hamiltonian systems near the center. Moreover, perturbing the symmetric cubic Hamiltonian systems by cubic polynomials, we consider limit cycles bifurcating from the center using the algorithm to compute the coefficients of Melnikov function. Finally, perturbing the symmetric hamiltonian system by symmetric cubic polynomials, we consider the number of limit cycles near one of the symmetric centers of the symmetric near-Hamiltonian system, which is same to that of another center

Invariants and reversibility in polynomial systems of ODEs

We first investigate the interconnection of invariants of certain group actions and time-reversibility of a class of two-dimensional polynomial systems with $1:-1$ resonant singularity at the origin. The time-reversibility is related to the Sibirsky subvariety of the center (integrability) variety of systems admitting a local analytic first integral near the origin. We propose a new algorithm to obtain a generating set for the Sibirsky ideal of such polynomial systems and investigate some algebraic properties of this ideal. Then, we discuss a generalization of the concept of time-reversibility in the three-dimensional case considering the systems with $1:\zeta:\zeta^2$ resonant singularity at the origin (where $\zeta$ is a primitive cubic root of unity) and study a connection of such reversibility with the invariants of some group actions in the space of parameters of the system

Environmental stress and the effects of mutation

The electronic version of this article is the complete one and can be found online at http://jbiol.com/content/2/2/12Mutations are the ultimate fuel for evolution, but most mutations have a negative effect on fitness. It has been widely accepted that these deleterious fitness effects are, on average, magnified in stressful environments. Recent results suggest that the effects of deleterious mutations can, instead, sometimes be ameliorated in stressful environments.The Spanish Consejo Superior de Investigaciones Científicas (CSIC) funds S.F.E., and J.A.G.M.dV. is funded by a fellowship from the Netherlands Organization of Scientific Research (NWO).Peer reviewe