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1

Periodic orbits around the triangular points with prolate primaries

Abd El Motelp Nihad, Radwan Mohamed

Artificial Satellites : Journal of Planetary Geodesy

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2023

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Vol. 58, No. 1

1--13

EN

Periodic orbits play a fundamental role in the study and deep understanding of the behavior of dynamical systems. In the current work, we investigated the periodic orbits around the triangular libration points of the restricted three-body problem. The equations of motion of the restricted problem are presented when both primaries are prolate triaxial. Periodic orbits around the triangular points are obtained and then illustrated graphically for some selected initial conditions and for the entire domain of the mass ratio µ, as well. The eccentricities of the periodic orbits are obtained and then represented graphically. It is observed that the periodic orbits about the triangular stationary points are elliptical, and the frequencies of short and long orbits of the periodic motion are influenced by the shape of the primary bodies. Furthermore, we found that the perturbing forces influence the period, the orientation, and the eccentricities of the short and long periodic orbits.

2

Stabilization of polynomial systems in ℝ3 via homogeneous feedback

Jerbi Hamadi, Kharrat Thouraya, Mabrouk Fehmi

Journal of Applied Analysis

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2022

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Vol. 28, nr 2

189--197

EN

In this paper, we study the stabilization problem of a class of polynomial systems of odd degree in dimension three. The constructed stabilizing feedback is homogeneous and guarantee the homogeneity of the closed loop system.mynotered In the end of the paper, we show the efficiency of such a study in the local stabilization of nonlinear systems affine in control.

3

Periodic solutions for periodic second-ordervdifferential equations with variable potentials

Llibre J., Makhlouf A.

Journal of Applied Analysis

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2018

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Vol. 24, nr 2

127--137

EN

We provide sufficient conditions for the existence of periodic solutions of the second-order differentia equation with variable potentials −(px’)’(t) − r(t)p(t)x’(t) + q(t)x(t) = f(t, x(t)), where the functions p(t) > 0, q(t), r(t) and f(t, x) are C2 and T-periodic in the variable t.

4

Planar nonautonomous polynomial equations V. The Abel equation

Wilczyński P.

Opuscula Mathematica

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2013

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Vol. 33, no. 1

175-189

EN

We give a full description of the dynamics of the Abel equation [formula] for some special complex valued ƒ. We also prove the existence of at least three periodic solutions for equations of the form [formula] for odd n ≥ 5.

5

Planar nonautonomous polynomial equations IV. Nonholomorphic case

Wilczyński P.

Opuscula Mathematica

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2012

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Vol. 32, no. 2

357-375

EN

We give a few sufficient conditions for the existence of periodic solutions of the equation [formula] where n > r and aj 's, ck's are complex valued. We prove the existence of one up to two periodic solutions.

6

Periodic solutions of periodic retarded functional differential equations

Pawłowski M.

Bulletin of the Polish Academy of Sciences. Mathematics

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2004

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Vol. 52, nr 4

353-363

EN

The paper presents a geometric method of finding periodic solutions of retarded functional differential equations (REDE) x'(t) = f(t,x1), where f is T-periodic in t. We construct a pair of subsets of R x R^n called a T-periodic block and compute its Lefschetz number. If it is nonzero, then there exists a T-periodic solution.

7

Optimal periodic orbits for non-recurrent rational functions

Urbański M.

Bulletin of the Polish Academy of Sciences. Mathematics

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2003

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Vol. 51, no 2

175-184

EN

We prove that each non-parabolic periodic orbit contained in the omega-limit set of a measure-recurrent, optimal orbit for a continuous function defined on the Julia set of a non-recurrent rational function is also optimal. As a by-product, we prove in the next section appropriate versions of shadowing and closing lemmas for non-recurrent rational functions.