In this paper, we show that generic planar quadratic Hamiltonian systems with third degree polynomial perturbation can have eight small-amplitude limit cycles around a center. We use higher-order focus value computation to prove this result, which is equivalent to the computation of higher-order Melnikov functions. Previous results have shown, based on first-order and higher-order Melnikov func...

In this Letter, we investigate a class of Hamiltonians which, in addition to the usual center-of-mass momentum conservation, also have center-of-mass position conservation. We find that, regardless of the particle statistics, the energy spectrum is at least q-fold degenerate when the filling factor is p/q, where and are coprime integers. Interestingly, the simplest Hamiltonian respecting this t...

It is shown that the recently found non-uniqueness of the third post-Newtonian binary point-mass ADM-Hamiltonian is related to the non-uniqueness at the third post-Newtonian approximation of the applied ADM-coordinate conditions. PACS number(s): 04.25.Nx, 04.30.Db, 97.60.Jd, 97.60.Lf In a recent paper [1] the authors reported on the non-uniqueness of the 3rd post-Newtonian ADM-Hamiltonian for b...

In this paper, we study the global dynamics of continuous piecewise quadratic Hamiltonian systems separated by straight line [Formula: see text], where these kinds have a nilpotent center at which comes from combination two cusps both systems. By Poincaré compactification classify phase portraits We must mention that it is extremely rare to find works studying center-focus problem in smooth wit...

We consider a real analytic, two degrees of freedom Hamiltonian system possessing a homoclinic orbit to a saddle-center equilibrium p (two nonzero real and two nonzero imaginary eigenvalues). We take a two-parameter unfolding for such the system and show that in nonresonance case there are countable sets of multiround homoclinic orbits to p. We also find families of periodic orbits, accumulatin...

We study the existence of homoclic solutions for reversible Hamiltonian systems taking the family of differential equations u + au′′ − u + f(u, b) = 0 as a model. Here f is an analytic function and a, b real parameters. These equations are important in several physical situations such as solitons and in the existence of “finite energy” stationary states of partial differential equations. We red...