ESTIMATION OF THE DOMAIN CONTAINING ALL COMPACT INVARIANT SETS OF THE OPTICALLY INJECTED LASER SYSTEM

2007 ◽  
Vol 17 (11) ◽  
pp. 4213-4217 ◽  
Author(s):  
KONSTANTIN E. STARKOV
Keyword(s):  
Laser System ◽  
Invariant Sets ◽  
Localization Problem ◽  
Cylindrical Coordinates ◽  
First Order ◽  
Localization Set ◽  
Localization Sets ◽  
Extremum Conditions

In this paper we study the localization problem of compact invariant sets of the optically injected laser system by applying the first order extremum conditions and cylindrical coordinates. Our main results consist in finding localization sets of simple forms which can be easily computed. Special attention is focussed on the case where we obtain a compact localization set.

BOUNDS FOR THE SET CONTAINING ALL COMPACT INVARIANT SETS OF THE LINEARLY COUPLED LASER SYSTEM

2008 ◽  
Vol 18 (04) ◽  
pp. 1211-1217 ◽  
Author(s):  
KONSTANTIN E. STARKOV ◽  
KONSTANTIN K. STARKOV
Keyword(s):  
Circular Cylinder ◽  
Laser System ◽  
Invariant Sets ◽  
Localization Problem ◽  
First Order ◽  
Quadratic Surfaces ◽  
Localization Sets ◽  
Extremum Conditions

In this paper we study the localization problem of all compact invariant sets of the five-dimensional coupled laser system by applying the first-order extremum conditions. Our main results consist in finding a number of localization sets formed by frusta, a circular cylinder, two parabolic cylinders and some other quadratic surfaces. Parameters of these localization sets are computed explicitly.

BOUNDS FOR THE DOMAIN CONTAINING ALL COMPACT INVARIANT SETS OF THE SYSTEM MODELING DYNAMICS OF ACOUSTIC GRAVITY WAVES

2009 ◽  
Vol 19 (10) ◽  
pp. 3425-3432 ◽  
Author(s):  
KONSTANTIN E. STARKOV
Keyword(s):  
Gravity Waves ◽  
Lorenz System ◽  
System Modeling ◽  
Invariant Sets ◽  
Localization Problem ◽  
Acoustic Gravity Waves ◽  
First Order ◽  
The Lorenz System ◽  
Quadratic Surfaces ◽  
Extremum Conditions

In this paper, we study the localization problem of compact invariant sets of the system modeling the dynamics of acoustic gravity waves constructed by Stenflo (the Lorenz–Stenflo system). We discuss relations between compact localization domains and trapping domains with the help of extended invariance principle due to Rodrigues et al. Based on this analysis, we compute the trapping domain for the system modeling the dynamics of acoustic gravity waves. By using the first order extremum conditions and a comparison with localization results obtained earlier for the Lorenz system we find that all compact invariant sets of the Lorenz–Stenflo system are located in the intersection of one-parameter set of ellipsoids with a few domains bounded by some quadratic surfaces. Further, we derive polytopic bounds for the locus of compact invariant sets. Finally, we present the general formulae validating the application of rational localizing functions and use these formulae for the Lorenz–Stenflo system and for the Lorenz system.

LOCALIZING BOUNDS FOR COMPACT INVARIANT SETS OF NONLINEAR SYSTEMS POSSESSING FIRST INTEGRALS WITH APPLICATIONS TO HAMILTONIAN SYSTEMS

2010 ◽  
Vol 20 (05) ◽  
pp. 1477-1483 ◽  
Author(s):  
KONSTANTIN E. STARKOV
Keyword(s):  
Hamiltonian System ◽  
Nonlinear Systems ◽  
Hamiltonian Systems ◽  
Positive Definite ◽  
First Integrals ◽  
Invariant Sets ◽  
Yang Mills ◽  
First Order ◽  
Special Case ◽  
Extremum Conditions

In this paper, we study the localization problem of compact invariant sets of nonlinear systems possessing first integrals by using the first order extremum conditions and positive definite polynomials. In the case of natural polynomial Hamiltonian systems, our results include those in [Starkov, 2008] as a special case. This paper discusses the application to studies of the generalized Yang–Mills Hamiltonian system and the Hamiltonian system describing dynamics of hydrogenic atoms in external fields.

scholarly journals Global Dynamics of the Hastings-Powell System

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Luis N. Coria
Keyword(s):  
Numerical Simulations ◽  
Equilibrium Points ◽  
Heteroclinic Orbits ◽  
Invariant Sets ◽  
Chaotic Attractors ◽  
Global Dynamics ◽  
State Variables ◽  
System Parameters ◽  
First Order ◽  
Extremum Conditions

This paper studies the problem of bounding a domain that contains all compact invariant sets of the Hastings-Powell system. The results were obtained using the first-order extremum conditions and the iterative theorem to a biologically meaningful model. As a result, we calculate the bounds given by a tetrahedron with excisions, described by several inequalities of the state variables and system parameters. Therefore, a region is identified where all the system dynamics are located, that is, its compact invariant sets: equilibrium points, periodic-homoclinic-heteroclinic orbits, and chaotic attractors. It was also possible to formulate a nonexistence condition of the compact invariant sets. Additionally, numerical simulations provide examples of the calculated boundaries for the chaotic attractors or periodic orbits. The results provide insights regarding the global dynamics of the system.

BOUNDING A DOMAIN THAT CONTAINS ALL COMPACT INVARIANT SETS OF THE BLOCH SYSTEM

2009 ◽  
Vol 19 (03) ◽  
pp. 1037-1042 ◽  
Author(s):  
KONSTANTIN E. STARKOV
Keyword(s):  
Magnetic Field ◽  
External Magnetic Field ◽  
Periodic Orbits ◽  
First Integral ◽  
Invariant Set ◽  
Invariant Sets ◽  
Localization Problem ◽  
Cylindrical Coordinates ◽  
Specific Restriction

In this paper we consider the localization problem of compact invariant sets of the Bloch system describing dynamics of an ensemble of spins in an external magnetic field. Our main results are related to finding a domain containing all compact invariant sets of the Bloch system. This domain is described as an intersection of one-parameter set of balls with two half spaces. Further, we describe the location of periodic orbits respecting two circular paraboloids and one semipermeable plane. In addition, we find conditions under which the origin is the unique compact invariant set. Finally, taking the Bloch system in cylindrical coordinates we construct one first integral for some specific restriction imposed on its parameters and, we also establish conditions under which this system has no compact invariant sets.

LOCALIZATION OF COMPACT INVARIANT SETS OF NONLINEAR SYSTEMS WITH APPLICATIONS TO THE LANFORD SYSTEM

2006 ◽  
Vol 16 (11) ◽  
pp. 3249-3256 ◽  
Author(s):  
ALEXANDER P. KRISHCHENKO ◽  
KONSTANTIN E. STARKOV
Keyword(s):  
Nonlinear Systems ◽  
Iterative Algorithm ◽  
Periodic Orbits ◽  
Invariant Sets ◽  
Localization Problem ◽  
First Order ◽  
Extremum Condition

In this paper, we examine the localization problem of compact invariant sets of systems with the differentiable right-side. The localization procedure consists in applying the iterative algorithm based on the first order extremum condition originally proposed by one of authors for periodic orbits. Analysis of a location of compact invariant sets of the Lanford system is realized for all values of its bifurcational parameter.

scholarly journals An Ultra-High-SMSR External-Cavity Diode Laser with a Wide Tunable Range around 1550 nm

2019 ◽  
Vol 9 (20) ◽  
pp. 4390
Author(s):  
Yan Wang ◽  
Hao Wu ◽  
Chao Chen ◽  
Yinli Zhou ◽  
Yubing Wang ◽  
...  
Keyword(s):  
Diode Laser ◽  
Diffraction Efficiency ◽  
Laser System ◽  
External Cavity ◽  
External Cavity Diode Laser ◽  
External Cavity Laser ◽  
First Order ◽  
Mode Suppression ◽  
Tunable Range ◽  
Order Diffraction

In this paper, a widely tunable external cavity diode laser (ECDL) with an ultra-high side mode suppression ratio (SMSR) was fabricated. Three configurations were constructed to investigate the relationship between the grating features and the SMSR. When a 1200 grooves/mm grating with a first order diffraction efficiency of 91% is utilized in the external-cavity laser system, a maximum SMSR of 65 dB can be achieved. In addition, the tunable range reaches 209.9 nm. The results show that the laser performance can be improved by proper high grating groove number and first-order diffraction efficiency.

Improvement on Spherical Symmetry in Two-Dimensional Cylindrical Coordinates for a Class of Control Volume Lagrangian Schemes

2012 ◽  
Vol 11 (4) ◽  
pp. 1144-1168 ◽  
Author(s):  
Juan Cheng ◽  
Chi-Wang Shu
Keyword(s):  
Spherical Symmetry ◽  
Gas Dynamics ◽  
Control Volume ◽  
Two Dimensional ◽  
Cylindrical Coordinates ◽  
One Dimensional ◽  
First Order ◽  
Conservation Of Momentum ◽  
Numerical Fluxes ◽  
Geometric Conservation Law

AbstractIn, Maire developed a class of cell-centered Lagrangian schemes for solving Euler equations of compressible gas dynamics in cylindrical coordinates. These schemes use a node-based discretization of the numerical fluxes. The control volume version has several distinguished properties, including the conservation of mass, momentum and total energy and compatibility with the geometric conservation law (GCL). However it also has a limitation in that it cannot preserve spherical symmetry for one-dimensional spherical flow. An alternative is also given to use the first order area-weighted approach which can ensure spherical symmetry, at the price of sacrificing conservation of momentum. In this paper, we apply the methodology proposed in our recent work to the first order control volume scheme of Maire in to obtain the spherical symmetry property. The modified scheme can preserve one-dimensional spherical symmetry in a two-dimensional cylindrical geometry when computed on an equal-angle-zoned initial grid, and meanwhile it maintains its original good properties such as conservation and GCL. Several two-dimensional numerical examples in cylindrical coordinates are presented to demonstrate the good performance of the scheme in terms of symmetry, non-oscillation and robustness properties.

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