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.

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.

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.

COLOR MAP OF LYAPUNOV EXPONENTS OF INVARIANT SETS

1999 ◽  
Vol 09 (07) ◽  
pp. 1459-1463 ◽  
Author(s):  
MARCO MONTI ◽  
WILLIAM B. PARDO ◽  
JONATHAN A. WALKENSTEIN ◽  
EPAMINONDAS ROSA ◽  
CELSO GREBOGI
Keyword(s):  
Lyapunov Exponent ◽  
Parameter Space ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
Invariant Sets ◽  
Largest Lyapunov Exponent ◽  
Specific Behavior ◽  
Practical Applications ◽  
The Lorenz System ◽  
Different Levels

The largest Lyapunov exponent of the Lorenz system is used as a measure of chaotic behavior to construct parameter space color maps. Each color in these maps corresponds to different values of the Lyapunov exponent and indicates, in parameter space, the locations of different levels of chaos for the Lorenz system. Practical applications of these maps include moving in parameter space from place to place without leaving a region of specific behavior of the 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.

Distinguishing Lorenz and Chen Systems Based Upon Hamiltonian Energy Theory

2017 ◽  
Vol 27 (02) ◽  
pp. 1750024 ◽  
Author(s):  
Shijian Cang ◽  
Aiguo Wu ◽  
Zenghui Wang ◽  
Zengqiang Chen
Keyword(s):  
Energy Function ◽  
Lorenz System ◽  
Three Dimensional ◽  
Type System ◽  
Chen System ◽  
First Order ◽  
Dimensional Phase Space ◽  
Ellipsoidal Surface ◽  
Generalized Hamiltonian Realization ◽  
The Lorenz System

Solving the linear first-order Partial Differential Equations (PDEs) derived from the unified Lorenz system, it is found that there is a unified Hamiltonian (energy function) for the Lorenz and Chen systems, and the unified energy function shows a hyperboloid of one sheet for the Lorenz system and an ellipsoidal surface for the Chen system in three-dimensional phase space, which can be used to explain that the Lorenz system is not equivalent to the Chen system. Using the unified energy function, we obtain two generalized Hamiltonian realizations of these two chaotic systems, respectively. Moreover, the energy function and generalized Hamiltonian realization of the Lü system and a four-dimensional hyperchaotic Lorenz-type system are also discussed.

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.

PERIODIC ORBITS OF THE LORENZ SYSTEM THROUGH PERTURBATION THEORY

2001 ◽  
Vol 11 (10) ◽  
pp. 2559-2566 ◽  
Author(s):  
J. PALACIÁN ◽  
P. YANGUAS
Keyword(s):  
Periodic Orbits ◽  
Lorenz System ◽  
Linear Part ◽  
Three Dimensional ◽  
Invariant Sets ◽  
Dimensional System ◽  
The Matrix ◽  
Lie Transformations ◽  
The Lorenz System ◽  
The Stability

Different transformations are applied to the Lorenz system with the aim of reducing the initial three-dimensional system into others of dimension two. The symmetries of the linear part of the system are determined by calculating the matrices which commute with the matrix associated to the linear part. These symmetries are extended to the whole system up to an adequate order by using Lie transformations. After the reduction, we formulate the resulting systems using the invariants associated to each reduction. At this step, we calculate for each reduced system the equilibria and their stability. They are in correspondence with the periodic orbits and invariant sets of the initial system, the stability being the same.

Localization of compact invariant sets of the Lorenz system

2006 ◽  
Vol 353 (5) ◽  
pp. 383-388 ◽  
Author(s):  
Alexander P. Krishchenko ◽  
Konstantin E. Starkov
Keyword(s):  
Lorenz System ◽  
Invariant Sets ◽  
The Lorenz System

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.

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