The 3-dimensional cored and logarithm potencials: Periodic orits

Agraïments: The first author is partially supported by CNPq grant 201802/2012-0.We study analytically families of periodic orbits for the cored and logarithmic Hamiltonians H(x, y, z, px, py, pz) = (p2x +p2y +p2z/q)/2+ (1+x2 +(y2 +z2)/q2)1/2, and H(x, y, z, px, py, pz) = (p2x +p2y +p2z/q)/2+ (log(1+x2 +(y2 + z2)/q2))/2, with 3 degrees of freedom, which are relevant in the analysis of the galactic dynamics. First, after introducing a scale transformation in the coordinates and momenta with a parameter ε, we show that both systems give essentially the same set of equations of motion up to first order in ε. Then the conditions for finding families of periodic orbits, using the averaging theory up to first order in ε, apply equally to both systems in every energy level H = h > 0. The averaging method used proves the existence of at most three periodic orbits, for ε small enough, and gives an analytic approximation for the initial conditions of these periodic orbits

Limit cycles in uniform isochronous centers of discontinuous differential systems with four zones

We apply the averaging theory of first order for discontinuous differential systems to study the bifurcation of limit cycles from the periodic orbits of the uniform isochronous center of the differential systems ẋ = -y+x, y = x + xy, and ẋ = -y + xy, y = x + xy, when they are perturbed inside the class of all discontinuous quadratic and cubic polynomials differential systems with four zones separately by the axes of coordinates, respectively. Using averaging theory of first order the maximum number of limit cycles that we can obtain is twice the maximum number of limit cycles obtained in a previous work for discontinuous quadratic differential systems perturbing the same uniform isochronous quadratic center at origin perturbed with two zones separately by a straight line, and 5 more limit cycles than those achieved in a prior result for discontinuous cubic differential systems with the same uniform isochronous cubic center at the origin perturbed with two zones separately by a straight line. Comparing our results with those obtained perturbing the mentioned centers by the continuous quadratic and cubic differential systems we obtain 8 and 9 more limit cycles respectively

Periodic solutions of the perturbed symmetric Euler top

We study the problem of persistence of $T$-periodic solutions of the celebrated symmetric Euler top when subjected to a small $T$-periodic stimulus. All solutions of the unperturbed system are periodic (of different periods, including continua of equilibria). In the case that the perturbation depends also on the three components of the angular momentum (the unknowns of the system) we provide bifurcation functions whose simple zeros correspond to $T$-periodic solutions of the perturbed system

A Note on Forced Oscillations in Differential Equations with Jumping Nonlinearities

Agraïments: The first author is supported by a grant of the Romanian National Authority for Scientific Research, CNCS UEFISCDI, project number PN-II-ID-PCE-2011- 3-0094. The third author is partially supported by RFBR Grant 13-01-00347. We thank the referees for useful comments which improved our note.The goal of this paper is to study bifurcations of asymptotically stable 2-periodic solutions in the forced asymmetric oscillator u c u u a u^ =1 t by means of a Lipschitz generalization of the second Bogolubov's theorem due to the authors. The small parameter >0 is introduced in such a way that any solution of the system corresponding to =0 is 2-periodic. We show that exactly one of these solutions whose amplitude is ^2 c^2 generates a branch of 2-periodic solutions when >0 increases. The solutions of this branch are asymptotically stable provided that c>0