Bifurcations of Equilibria and Periodic Orbits in n-Dimensional Dynamical Systems

2008 ◽  
pp. 151-194
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbits

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 Periodic orbits of discrete and continuous dynamical systems via Poincaré-Miranda theorem

2020 ◽  
Vol 25 (2) ◽  
pp. 651-670
Author(s):  
Armengol Gasull ◽  
◽  
Víctor Mañosa ◽  
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbits

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 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 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

INFINITE DIMENSIONAL KRAWCZYK OPERATOR FOR FINDING PERIODIC ORBITS OF DISCRETE DYNAMICAL SYSTEMS

2007 ◽  
Vol 17 (12) ◽  
pp. 4261-4272 ◽  
Author(s):  
ZBIGNIEW GALIAS ◽  
PIOTR ZGLICZYŃSKI
Keyword(s):  
Dynamical Systems ◽  
Fixed Points ◽  
Periodic Orbits ◽  
Discrete Dynamical Systems ◽  
Krawczyk Operator ◽  
Infinite Dimensional ◽  
Dispersal Model ◽  
Period 2 ◽  
Existence Of Zeros

In this work, we introduce the Krawczyk operator for infinite dimensional maps. We prove two properties of this operator related to the existence of zeros of the map. We also show how the Krawczyk operator can be used to prove the existence of periodic orbits of infinite dimensional discrete dynamical systems and for finding all periodic orbits with a given period enclosed in a specified region. As an example, we consider the Kot–Schaffer growth-dispersal model, for which we find all fixed points and period-2 orbits enclosed in the region containing the attractor observed numerically.

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