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.

LOCALIZATION OF COMPACT INVARIANT SETS OF DISCRETE-TIME NONLINEAR SYSTEMS

2011 ◽  
Vol 21 (07) ◽  
pp. 2057-2065 ◽  
Author(s):  
ANATOLY N. KANATNIKOV ◽  
ALEXANDER P. KRISHCHENKO
Keyword(s):  
Nonlinear Systems ◽  
Iterative Algorithm ◽  
Discrete Time ◽  
Invariant Sets ◽  
Localization Problem ◽  
Extremum Condition

In this paper, we examine the localization problem of compact invariant sets of discrete-time nonlinear systems. The localization procedure consists in applying the iterative algorithm based on the extremum condition. An analysis of a location of compact invariant sets of the Henon system is realized for all values of its parameters.

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.

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.

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.

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.

scholarly journals Limit Cycles of a Class of Perturbed Differential Systems via the First-Order Averaging Method

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Amor Menaceur ◽  
Salah Mahmoud Boulaaras ◽  
Amar Makhlouf ◽  
Karthikeyan Rajagobal ◽  
Mohamed Abdalla
Keyword(s):  
Hamiltonian System ◽  
Periodic Orbits ◽  
Limit Cycles ◽  
Averaging Method ◽  
Differential Systems ◽  
First Order ◽  
Polynomial Differential Systems

By means of the averaging method of the first order, we introduce the maximum number of limit cycles which can be bifurcated from the periodic orbits of a Hamiltonian system. Besides, the perturbation has been used for a particular class of the polynomial differential systems.

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