scholarly journals Bogdanov Map for Modelling a Phase-Conjugated Ring Resonator

Entropy ◽  
2019 ◽  
Vol 21 (4) ◽  
pp. 384
Author(s):  
Vicente Aboites ◽  
David Liceaga ◽  
Rider Jaimes-Reátegui ◽  
Juan Hugo García-López
Keyword(s):  
Phase Space ◽  
Dynamic Behavior ◽  
Periodic Orbits ◽  
Ring Resonator ◽  
Matrix Elements ◽  
Bifurcation Diagrams ◽  
Direct Dependence ◽  
Computer Calculations ◽  
Generating Matrix ◽  
Explicit Expressions

In this paper, we propose using paraxial matrix optics to describe a ring-phase conjugated resonator that includes an intracavity chaos-generating element; this allows the system to behave in phase space as a Bogdanov Map. Explicit expressions for the intracavity chaos-generating matrix elements were obtained. Furthermore, computer calculations for several parameter configurations were made; rich dynamic behavior among periodic orbits high periodicity and chaos were observed through bifurcation diagrams. These results confirm the direct dependence between the parameters present in the intracavity chaos-generating element.

Laser cavity with a Van der Pol dynamics

2021 ◽  
Vol 67 (1 Jan-Feb) ◽  
pp. 154
Author(s):  
M. Lozano ◽  
A. Kir’yanov ◽  
A. Pisarchik ◽  
V. Aboites
Keyword(s):  
Phase Diagrams ◽  
Optical Thickness ◽  
Laser Cavity ◽  
Matrix Approach ◽  
Matrix Elements ◽  
Van Der Pol ◽  
Computer Calculations ◽  
Dynamic Map ◽  
The Rich ◽  
Generating Matrix

In this article, a beam within a ring phase conjugated laser is described by means of a Van der Pol bidimensional dynamic map using an ABCD matrix approach. Explicit expressions for the intracavity chaos-generating matrix elements were obtained; furthermore, computer calculations for different values of Van der Pol map’s parameters were made. The rich dynamic behavior displays periodicity when the parameter ¹ (which determines the non-inearity term) takes values around zero. These results were observed in phase diagrams and in diagrams of the optical thickness of the intracavity element.

Conservative generalized bifurcation diagrams and phase space properties for oval-like billiards

2022 ◽  
Vol 155 ◽  
pp. 111707
Author(s):  
Diogo Ricardo da Costa ◽  
André Fujita ◽  
Antonio Marcos Batista ◽  
Matheus Rolim Sales ◽  
José Danilo Szezech Jr
Keyword(s):  
Phase Space ◽  
Bifurcation Diagrams

Various Types of Coexisting Attractors in a New 4D Autonomous Chaotic System

2017 ◽  
Vol 27 (09) ◽  
pp. 1750142 ◽  
Author(s):  
Qiang Lai ◽  
Akif Akgul ◽  
Xiao-Wen Zhao ◽  
Huiqin Pei
Keyword(s):  
Numerical Simulation ◽  
Dynamic Behavior ◽  
Chaotic System ◽  
Limit Cycles ◽  
Electronic Circuit ◽  
Strange Attractors ◽  
Bifurcation Diagrams ◽  
Coexisting Attractors ◽  
Signum Function ◽  
Multiple Coexisting Attractors

An unique 4D autonomous chaotic system with signum function term is proposed in this paper. The system has four unstable equilibria and various types of coexisting attractors appear. Four-wing and four-scroll strange attractors are observed in the system and they will be broken into two coexisting butterfly attractors and two coexisting double-scroll attractors with the variation of the parameters. Numerical simulation shows that the system has various types of multiple coexisting attractors including two butterfly attractors with four limit cycles, two double-scroll attractors with a limit cycle, four single-scroll strange attractors, four limit cycles with regard to different parameters and initial values. The coexistence of the attractors is determined by the bifurcation diagrams. The chaotic and hyperchaotic properties of the attractors are verified by the Lyapunov exponents. Moreover, we present an electronic circuit to experimentally realize the dynamic behavior of the 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.

Piecewise isometries, uniform distribution and 3log 2−π2/8

2011 ◽  
Vol 32 (6) ◽  
pp. 1862-1888 ◽  
Author(s):  
YITWAH CHEUNG ◽  
AREK GOETZ ◽  
ANTHONY QUAS
Keyword(s):  
Phase Space ◽  
Uniform Distribution ◽  
Periodic Orbits ◽  
Periodic Points ◽  
Full Measure ◽  
Infinite Measure ◽  
Important Family ◽  
Large Numbers ◽  
Piecewise Isometries ◽  
Measure Preserving Transformations

AbstractWe use analytic tools to study a simple family of piecewise isometries of the plane parameterized by an angle. In previous work, we showed the existence of large numbers of periodic points, each surrounded by a ‘periodic island’. We also proved conservativity of the systems as infinite measure-preserving transformations. In experiments it is observed that the periodic islands fill up a large part of the phase space and it has been asked whether the periodic islands form a set of full measure. In this paper we study the periodic islands around an important family of periodic orbits and demonstrate that for all angle parameters that are irrational multiples of π, the islands have asymptotic density in the plane of 3log 2−π2/8≈0.846.

ENERGY-LIKE FUNCTIONS FOR SOME DISSIPATIVE CHAOTIC SYSTEMS

2005 ◽  
Vol 15 (08) ◽  
pp. 2507-2521 ◽  
Author(s):  
C. SARASOLA ◽  
A. D'ANJOU ◽  
F. J. TORREALDEA ◽  
A. MOUJAHID
Keyword(s):  
Phase Space ◽  
Dynamic Behavior ◽  
Fourth Order ◽  
Chaotic Systems ◽  
Quantitative Measure ◽  
Energy Functions ◽  
Dissipation Of Energy ◽  
Rossler System ◽  
Order Polynomial ◽  
Fourth Order Polynomial

Functions of the phase space variables that can considered as possible energy functions for a given family of dissipative chaotic systems are discussed. This kind of functions are interesting due to their use as an energy-like quantitative measure to characterize different aspects of dynamic behavior of associated chaotic systems. We have calculated quadratic energy-like functions for the cases of Lorenz, Chen, Lü–Chen and Chua, and show the patterns of dissipation of energy on their respective attractors. We also show that in the case of the Rössler system at least a fourth-order polynomial is required to properly represent its energy.

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