As we know, the second part of the Hilbert problem is to find the maximal number and relative locations of limit cycles of polynomial systems of degree n. Let H(n) denote this number, which is called the Hilbert number. Then the problem of finding H(n) is divided into two parts: find an upper and lower bounds of it. For the upper bound there are important works of Écalle [1990] and IIyashenko a...

* Correspondence: [email protected] School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471003, Henan, P. R. China Full list of author information is available at the end of the article Abstract For a class of fifth degree nilpotent system, the shortened expressions of the first eight quasi-Lyapunov constants are presented. It is shown that the orig...

We study the relative equilibria of the limit case of the planar Newtonian 4–body problem when three masses tend to zero, the so-called (1 + 3)–body problem. Depending on the values of the infinitesimal masses the number of relative equilibria varies from ten to fourteen. Always six of these relative equilibria are convex and the others are concave. Each convex relative equilibrium of the (1 + ...

Objective(s): Paraplegia is deterioration in motor or sensory function of the lower limbs that can occur after modification of a thoracoabdominal aortic aneurysm. The purpose of this survey was to determine the protective action of lutein on spinal cord ischemia-reperfusion (I-R) damage. Materials and Methods: Thirty-five male rats were distributed into five groups: intact, sham, dimethyl sulfo...

The bifurcation diagram of a model stochastic differential equation with delayed feedback is presented. We are motivated by recent research on stochastic effects in models of transcriptional gene regulation. We start from the normal form for a pitchfork bifurcation, and add multiplicative or parametric noise and linear delayed feedback. The latter is sufficient to originate a Hopf bifurcation i...

This paper presents an analysis on the bifurcation of limit cycles for a cubic Hamiltonian system with quintic perturbed terms using both qualitative analysis and numerical exploration. The perturbed terms considered here is in the form of R(x, y, λ) = S(x, y, λ) = mx+ny+kxy−λ, where m, n, k, and λ are all variable. The investigation is based on detection functions which are particularly effect...

Recently, the phase-flip bifurcation has been described as a fundamental transition in time-delay coupled, phase-synchronized nonlinear dynamical systems. The bifurcation is characterized by a change of the synchronized dynamics from being in-phase to antiphase, or vice versa; the phase-difference between the oscillators undergoes a jump of pi as a function of the coupling strength or the time ...

In this paper we classify the centers localized at the origin of coordinates, the cyclicity of their Hopf bifurcation and their isochronicity for the polynomial differential systems in R of degree d that in complex notation z = x+ iy can be written as ż = (λ+ i)z + (zz) d−5 2 ( Az4+jz1−j +Bzz + Cz2−jz3+j +Dz ) , where j is either 0 or 1, d is an arbitrary odd positive integer greater than or eq...

A fractional-order Leslie-Gower prey-predator-parasite system with delay is proposed in this article. The existence and uniqueness of the solutions, as well their non-negativity boundedness, are studied. Based on characteristic equations conditions stability Hopf bifurcation, local asymptotic each equilibrium point bifurcation interior investigated. Moreover, a Lyapunov function constructed to ...

percutaneous coronary intervention in the setting of coronary bifurcation lesions is associated with increased risk for complications. the goal of this manuscript is to review the currently available classifications for coronary artery bifurcation lesions with a guide to the use of specific coronary bifurcation techniques based on lesion characteristics