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.

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.

RECOIL IMPLANTATION MÖSSBAUER EXPERIMENT ON 57Fe IN Ge IN THE PRESENCE OF AN EXTERNAL MAGNETIC FIELD

1980 ◽  
Vol 41 (C1) ◽  
pp. C1-445-C1-445
Author(s):  
G. Langouche ◽  
N. S. Dixon ◽  
L. Gettner ◽  
S. S. Hanna
Keyword(s):  
Magnetic Field ◽  
External Magnetic Field ◽  
Recoil Implantation
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