scholarly journals Periodic Orbits of Third Kind in the Zonal J2 + J3 Problem

Symmetry ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 111
Author(s):  
M. de Bustos ◽  
Antonio Fernández ◽  
Miguel López ◽  
Raquel Martínez ◽  
Juan Vera
Keyword(s):  
Dynamical Systems ◽  
Phase Portrait ◽  
Periodic Orbits ◽  
Averaging Theory ◽  
The Third ◽  
Explicit Expressions

In this work, the periodic orbits’ phase portrait of the zonal J 2 + J 3 problem is studied. In particular, we center our attention on the periodic orbits of the third kind in the Poincaré sense using the averaging theory of dynamical systems. We find three families of polar periodic orbits and four families of inclined periodic orbits for which we are able to state their explicit expressions.

scholarly journals Third derivatives of the integrable part of an asteroid Hamiltonian

2007 ◽  
pp. 53-60 ◽  
Author(s):  
R. Pavlovic
Keyword(s):  
Third Order ◽  
Geometrical Condition ◽  
The Third ◽  
Derivatives Of ◽  
Quasi Convexity ◽  
Explicit Expressions

To apply the theorem of Nekhoroshev (1977) to asteroids, one first has to check whether a necessary geometrical condition is fulfilled: either convexity, or quasi-convexity, or only a 3-jet non-degeneracy. This requires computation of the derivatives of the integrable part of the corresponding Hamiltonian up to the third order over actions and a thorough analysis of their properties. In this paper we describe in detail the procedure of derivation and we give explicit expressions for the obtained derivatives. .

scholarly journals Numerical detection of unstable periodic orbits in continuous-time dynamical systems with chaotic behaviors

2007 ◽  
Vol 14 (5) ◽  
pp. 615-620 ◽  
Author(s):  
Y. Saiki
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbits ◽  
Infinite Number ◽  
Continuous Time ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Complex Phenomenon ◽  
Unstable Periodic Orbits ◽  
Newton Raphson ◽  
The Lorenz System

Abstract. An infinite number of unstable periodic orbits (UPOs) are embedded in a chaotic system which models some complex phenomenon. Several algorithms which extract UPOs numerically from continuous-time chaotic systems have been proposed. In this article the damped Newton-Raphson-Mees algorithm is reviewed, and some important techniques and remarks concerning the practical numerical computations are exemplified by employing the Lorenz system.

scholarly journals On the Periodic Structure of Parallel Dynamical Systems on Generalized Independent Boolean Functions

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1088 ◽  
Author(s):  
Juan A. Aledo ◽  
Ali Barzanouni ◽  
Ghazaleh Malekbala ◽  
Leila Sharifan ◽  
Jose C. Valverde
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Periodic Orbits ◽  
Periodic Structure ◽  
Boolean Functions ◽  
Directed Graphs ◽  
General Pattern ◽  
General Situation ◽  
Point System ◽  
Local Functions

In this paper, based on previous results on AND-OR parallel dynamical systems over directed graphs, we give a more general pattern of local functions that also provides fixed point systems. Moreover, by considering independent sets, this pattern is also generalized to get systems in which periodic orbits are only fixed points or 2-periodic orbits. The results obtained are also applicable to homogeneous systems. On the other hand, we study the periodic structure of parallel dynamical systems given by the composition of two parallel systems, which are conjugate under an invertible map in which the inverse is equal to the original map. This allows us to prove that the composition of any parallel system on a maxterm (or minterm) Boolean function and its conjugate one by means of the complement map is a fixed point system, when the associated graph is undirected. However, when the associated graph is directed, we demonstrate that the corresponding composition may have points of any period, even if we restrict ourselves to the simplest maxterm OR and the simplest minterm AND. In spite of this general situation, we prove that, when the associated digraph is acyclic, the composition of OR and AND is a fixed point system.

scholarly journals Periodic Orbits and Subharmonics of Dynamical Systems on Non-Compact Riemannian Manifolds

1996 ◽  
Vol 130 (1) ◽  
pp. 142-161 ◽  
Author(s):  
Silvia Cingolani ◽  
Elvira Mirenghi ◽  
Maria Tucci
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbits ◽  
Riemannian Manifolds

scholarly journals SYMMETRIC PERIODIC ORBITS NEAR HETEROCLINIC LOOPS AT INFINITY FOR A CLASS OF POLYNOMIAL VECTOR FIELDS

2006 ◽  
Vol 16 (11) ◽  
pp. 3401-3410 ◽  
Author(s):  
MONTSERRAT CORBERA ◽  
JAUME LLIBRE
Keyword(s):  
Phase Portrait ◽  
Periodic Orbits ◽  
Vector Fields ◽  
Phase Portraits ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Open Set ◽  
Symmetric Periodic Orbits

For polynomial vector fields in ℝ3, in general, it is very difficult to detect the existence of an open set of periodic orbits in their phase portraits. Here, we characterize a class of polynomial vector fields of arbitrary even degree having an open set of periodic orbits. The main two tools for proving this result are, first, the existence in the phase portrait of a symmetry with respect to a plane and, second, the existence of two symmetric heteroclinic loops.

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