SCIENTIFIC ABSTRACT TKACHEV, V.D.  TKACHEV, V.V.
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SCIENTIFIC ABSTRACT
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L 2800166
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served bands of the temperature dependences of the intensities of the individual
components with the published data leads to the conclusion that the observed effect
is an optic analog of the Mossbauer effect in five different recombinations centers
of silicon. Each band has a narrow line adjacent to a longwave branch with a well
pronounced maximum. The narrow lines are due to phononleso radiative transitions
within the centers, and the longwave components are due to radiative transitions
with emission of acoustic phonons. The appearance of new bands during annealing is
evidence of the complex nature of the centers. The authors thank P# S. SolovIvev
for help with the preparation for the experiment. Orig. art. has: 1 figure and
1 table. [021
SUB CODB:_ 20/ SUBM DATE: 04.0at65/ ORIG MPS 004/ OM MW. OOIJ ATD PRES
#;z
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AP6019223 SOURCE CODE: UR/0250/66/olO/002/0080/0082
AUTHOR: Sevchenko, A. N.; Tkachev, V. D.
'~ORG: Belorussian State University im. V. 1. Lenin (Belorusskiy gosudarstvannyy univer"!
sitet) 1
TITLE: Photoconductivity Spectra of ptype siliconirradiated with fast electrons
SOURCE: All BSSR. Doklady, v. 10, no. 2, 1966, 8032
TOPIC TAG3: photoconductivity, silicon, impurity center, irradiation effect, irradia
tion damage, Fermi level
ABSTRACT: The object of the work was to study t~e kinetics of formation and stability
of radiation damage in t i di edkAth I MeV electrons at 25300C and
Mpe o rra at 
integral doses of 101310 electrons cm . The change in the shape of the spectra of
Lmpurity photoconductivity was studied as a function of the irradiation dose. The ef
fo~tiveness of the formation of radiation damage was found to depend on the imr)urity
composition of the crystals. The shaloe of the spectrum of the photoconductivity signal
donends on the position of the Fermi level in the forbidden zone. By raising or lo"Ter
ing the Fermi level, one can eliminate certain centers from the photoconductivity pro
cesses, which confirms the authors' earlier hypothesis that the structure of the spec
tra is related to volume defects, since the change in the position of the Fermi level
reflects the conditions in the volume of the crystal. The observed change in the SD.OC
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Ara following the cessation of irradiation is attributed not only to different stabil
,ities of the radiation defects$ but also to the occurrence of redistribution Of point
radiation defects among the various impurity atoms. In conclusion, the authors expres
their sincere appreciation to V. S. Vavilov for discussing the work and his many usefxi
comments. Orig. art. has: 2 figures.
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ACC NR AP6011529 SOURCE CODE: UR/0250/6~/Mo/oo3/ol48/0150
AUTHORS: Sevchenk Tkachev V. D. 99~KQ
o., A. K.; 1ki tk YAE. F..;
Yukbnevich
ORG: Belorussian State University Im. V. I. Lenin LBelorusskiy
gosudarstvennyy universitet
TITLE: Investigation of the influence of heat treatment on the photo
electric properties of silicon with radiation dgj~A~ In its structure
SOURCE: AN BSSR. Doklady, v.'110, no. 3, 1966, 148150
TOPIC TAGS: silicon, sinSle crystal, crystal structure, radiation
damage, photoelectric property, beat effect, photoconductivity, fine
structure., crystal defect
ABSTRACT: The purpose of the investigation was to study the temperature
stability of different radiation damages which are produced in single
crystal silicon when irradiated witb energy particles. The Initial
material was ptype silicon with resistivity 7  10 ohmcm, containing
8 X 1017 CM3 oxygen. The irradiation was with 1Mev electrons from
n electrostatic generator at 80K and 30C. The spectral dependence of
~tbe stationary photoconductivity was plotted witb apparatus described by
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A. F. Plotnikov et al. (PTE, no. 3, 183, 1962). The results show that
samples whose photoconductivity spectrum displayed no structure shortly
after the cessation of'the irradiation, acquired a pronoitneed structure
after prolonged storage at liquidnitrogen temperature. This is attrib
uted to diffusion of the vacancy pairs resulting from the electron bom
bardment. An'increase in the temperature and longer storage following
the bombardment causes the point defects due to the bombardment to becom
annealed. The results are interpreted and reconciled with the level
scheme of the defects. A quantitative interpretation of the phenomenon
is made difficult by the presence of different types of structure defecti
which can become transformed Into each other during annealing. Orig.
art. has:.2 figures.
SUB CODE: 20/ SUBM DATE.:L4Jun65/ ORIG REP: 004/
L 348206o nT ( m,
ACC NRt AP6021921 SOURCE CODE: uR/0250/66/010/006/0374/U376
AUTHOR: Sev henko,.A. N.; Stel'makh, V. F.; Tkachev, V. D. 73
ORG: Belorussian State Universl~y tm. V. 1. Lenin (Belorusskiy gosudarstvennyy
universitet)
TITLE: Photoelectric properties of 8alliu arsenide*containing structure defects due
to radiation 3,7
SOURCE: AN BSSR. Doklady, v. 10, no. 6, 374376
TOPIC TAGS: gallium arsenide, radiation effect, photoresistance, photoconductivity,
resistivity , photoelectric property, fast neutron, neutron irradiation
ABSTRACT* The energy spectrum of local levels in n and ptype gallium arsenide
'single crystals irradiated with fast neutronslas investigated by studying the
structure of photoconductivity spectra beyond the absorption edge. Spectral de
pendencies of photoconductivity were recorded at temperatures of 300 and 80K using
samples with a resistivity up to 1012 ohm. The specific.resistivity of the irradiaced~
saWes was found to depend markedly on the density of neutron beams: at fluxes of
101 neutrons/CM2 ed
the resistivity increased slowly; at higher densities it increas
rapidly, showing a tendency toward saturation at 1017 neutrons/cm2. It was conclured
that the irradiation of gallium arsenide produces a great number of stable combina
tions of point defects and residual chemical impurities. Orig. art. has: 2 figures
~ZL]
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AVX NR1 AP6029648 SOURCE CODE: UR/0250/66/010/008/0530/0552
AUTHORs Sevchenko. A, N.g Tkachev. V._P.; Urenev, V. 1,
ORG: Belorussian State University im. V. I. Lenin (Belorueskly
g9addarstvennyy universitet)
TITLEt Photoconductivity spectra of germanium single crystals irradi
ated with gammaquanta
SOURCE: AN BSSR. Doklady, v. 10, no. 8, 1966, 550552
TOPIC TAGSt germanium single crystal, gamma irradiation, electron
energy levelp impurity level
ABSTRACTs An investigation was made of the system of energy levels
which appears in ntype and ptype germanium irradiated with gamma/191
quanta from Co6o at room temperature. Particular attention was gLven'to
a Study Of the stable centers which introduce deep energy levels into
the forbidden zone. Photoconductivity spectra were taken in a range
from 1 to 5 V for temperatures from 300 to BOK before and after Lrradi
ation with integrated fluxes of 10145 x 1017 kv/CM2. The specimens
used were n and ptype germanium single crystals with initial specific
resistances of 48 ohmcm and 6 ohmocm, respectively. The concentration
of residual impurities in the electrically active state.for ntype
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Material Was Up to 10121013 CM3. Investigation of the electrical
characteristics of specimens after Lrr&*VWt1*WqthdW&if'the preseiice of
EV + 0.01 ev, Ev + 0.008 ev, and Ev + 0.17 ev levels in ptype ger
manLum and E.  0.20 ev level in ntypexrmanium. Irradiation of
ntype crystals with doses up to 5 x 10 kv/cm2 did not change the
conductivity &Lgn. At T  80K the Fermi level in irradiated ntype
specimens was located 0.160.22 ev from the bottom of the 'conductivity."
one, while in ptype specimens it was found 0,200.26 ev from the top
the valence zone. The presence of deep centers in both irradiated
N
and nonLrradLated crystals indicates that these centers are not geh
erated due to the irradiation but are only displaygd.as thq.rA*O'ult of
Lt. The transition of electrons between the tonbs and thesecenters
in irradiated crystals can lead to a significant ctri*W6 in zhir';ib~can
T1.4
tration of free carriers. Such a chailge in condon rWdn s.'06osible'
q "~ted conce tr n,:.or a er#
8ince after irradiation the a uil1hr n
becomes very low as a result of their capture by
which introduces shallower energy levels. Therefore Inirviw4ated
apacLmens for which the ratio Ac/o is greater, Lt to 'to detect
deep energy levels which belong to residual imperfactibaii *'_U.the crystal
lattice, OrLg. art. hast 1 figure. (JAI
SUB COM 20/ SUBM DATEi 1SApr66/ ORIG. kEV: 002/ OTH. .AO2
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AUMOR: Sevchenkop A. N. (Academician AN BSSR); loomako, V. M.; Tkachev, V. D.
ORG: Belorussian State University im. V. 1. Lenin (Belormuss gosudarstvenrVy uni
versitet)
TITLE: Temperature and optical quenching of radiative recombination of gallium ar
senide
SOURCE: AN BSSR. Doklady, v. 10, no. 9, 1966, 641643
TOPIC TAGS: gallium arsenide, luminescence quenching.. radiative recombination, tem
perature dependence, absorption band, Raman spectrum
ABSTRACT: The purpose of the investigation was to determine the influence of temper
ature on the spectral distribution of the radiative recombination of electrolumines
cent diodes obtained from ntype GaAs by diffusion of beryllium, and the influence of
constant external illumination in the intrinsic absorption band (% < 0.83 nm) on the
intensity of the recombination band. The carrier density i~a the irfftial GaAs was
8 x, 1017  6 x 1018 cm73. The optical resonators were made from the crystal by
cleavage. The Raman spectra were investigated with apparatus based on the IKS;12
spectrometer. The radiation receiver was a germanium photodiode or a cooled PbS
photoresistance. The constant illumination was with the aid of an incandescent lamp
and a number of filters The tests were made at temperatures 80  30K in a cryostat
evacuated to 3.04 mm Hg: The radiative recombination without additional illumination
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had three peaks, at 1.02.. 1.26., and 1.44 ev. Additional illumination reduced greatly!
the 1.26 ev peak and both shifted and attenuated the two other bands. The radiative
recombination also decreased with increasing temperature. The results are related to
the formation of effective centers for electron adhesion. The authors thank M. T.
Iappo, and V. S. Veliyev for help with the experiments. Orig. art. has: 2 figures.
SUB CODE: 2D/ SUBM DATE: 2LMay66/ ORIG REF: 004/ OTH REF: 001
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N'?' AP602t951 .1;0URCE CON1,;: uii/ou.,u/66/iGg/00~/o56"n/o56)t
AU111011: Sevc*acnko, A. N. (Academician AN 13SSR); Tkachcv. V, D.; Lul
_,a;ov, P. F.
OhG: Belorussian State University im. V. 1. Lenin (I'cloru.&;kiy gotudarstvcnnyy
universfictj
TIT12: Energ. spectrum of radiation damage in silicon sinrle crystals
SOURCE: All SSS11. Doklady, v. 169, no. 3, 1966, 562564
TOPIC TAGS: silicon semiconductor, semiconductor band structure, crystal lattice
defect, irradiation damage, impurity level
ABSTRACT: The purpose of the investigation was to determine the cncrr,,r levels thAt
appear when the crystal is irradiated with different integral fluxes of 141'ev clec
trons (1013
 1018 el/cm2), fast reactor neutrons (1012  1019 ncut/cm2), and y quanta,
from Co6o (1015  1019 qu/cm2). The initial n and ptype crystals had a resistivity
0.03  150 ohmcm. and a low concentration of impurities capable of producing deep
levels in the forbidden band. The investigation consisted of measuring the tempera
ture dependence of the conductivity and of the Hall coefficient, and the spectral dis
tribution of the photoconductivity signal. The electrical measurements identified thei
i
principal donor and acceptor centers, while the photoelectric measurements identified
the levels due to point defects and some of the chemical impurities. The resulto
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C
~ Oqhjj _67_~
(3
A.CC NR, AP6027951 ous set of local levels near the
give an almost continu varied separat.LOn of the vacancy
int defects to the silicon
show that the PO alence bands this being due scheme of radiationdaaged
conduction and v 11 pairs. A complete energylevel
rstitial at0I je a; 3 figures'
inte: . art. ha
is presented. Or oqmarGG[ ORIG REP: 005 ./ OTI' REF: 002
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2/241
SOURCE C _~TUR/6181r66[6 217
4Cd___N_4_t_Ai:P6o_ 696o A, /V) 3h
AUTHORS Yukhnevich, A. V.; Tkachev, V. D.; Bortnik# M, Ve
ORG: Belorussian State University im. V. I. Lening Minsk (Bolorusakiy gosudarstvenn:
universitet)
TITLE: Annealing of bands of impurity recombination radiation in silicon irradiated
'With gamma quanta
SOURCES Fizika tvardogo tela, v. 81 no. lit 1966, 32133217
TOPIC TAGS: recombination radiation, radiative recombinationp semiconductor carrier,
gamma irradiation
ABSTRACT: The isochronous annealing of infrared radiation bands arising in silicon
from the radiative recombination of excess carriers across the levels of radiation de
fects was studied. In the 256000C range, the successive appearance and disappear
ance of various bands was observed, indicating a complex character of the rearrange
ment of defects during annealing. The results obtained show an important role of
oxygen in the formation of recombination centers in silicon upon irradiation with
gamma quanta. On the other hand, this recombination radiation is a good indicator of
low oxygen concentrationsy and can be used to determine the latter. Thusp recambina
tion radiation can be used as a means of studying the radiation defects of silicon and
processes of their rearrangement during heat treatment, Nine different "radiating"
radiation defects were observedg and the kinetics of their annealing ahowed the atruc
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ture of stable radiation defects to be complex. Oxygen atoms are an integral part of
most of the radiation defects responsible for the observed bands of impurity recombi
.nation radiation. Phosphorus atoms participate in the formation of centers radiating
.,D and E bands, and boron atoms take part in the formation of centers radiating F and
,bands. The maJority recombination centers (determining the lifetime of excess
`;:dairiers) are annealed at 4005000C. They are also linked to oxygen and are centers'
.0
f'nonradiative recombination. The intensity and energy distribution of the various
."~.bands of recombination radiation of silicon containing radiation defects and subjecte
.,..to heat treatment permit an analysis of the content of chemical impurities in the ini,
tial single crystals. Both active (boron, phosphorus) and inactive impurities (oxy,
gen) can thus be analyzed. Authors thank Z. M. Afanaslyev and Pe 3'.Soloylyev for
their systematic assistance in the course of the wcpariments. Orig: art. has# .1
figureand I table.
SUB CODE'S 20/ SUBM DAT&I 21Mar66/ ORIG REFS 006/ M IUWI 008
TKAGHEIIJ II.F.
4 7
~N"ew theorem cn,
Uap. mat. L
(TXUUV, V.F. (Moskva) , TKUHEV, V1.F. (Voronezh)
Criterial for the absence of arbitrary and multiple limit c7cles.
Hat. sbor. 52 no. 3:811822 N 160. (MIRA 13:12)
(Differential eauations)
10 sufficient conditions for stlibility, semistabilit7 and Instabilitjr
of, the limit cycle of the !&)L ELLII~ ovation, Dokl, AS SSSR 116
no*4:564567 0 157. _ff MIT (MMA 11:3 )
I Voronezhakiy gosudarstvennyy iiniversitet. Fredotavleno akademikom
;.S. Aleksandrovym.
(Differential equations)
85226
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AUTHOESiTkachev, V. F. (Moscow), Tkachev. Vl. F. (Voronezh)
TITLE: On Criteria for the Absence of Arbitrary and Multiple Limit
Cycles
PERIODICALs Matematicheskiy sbornik, 196o, Vol52, NO,39 PP,811822
TEXTs The author considers the system
(1) ok P(Y,Y), ly Q(X,Y).
dt dt
General theorem (Theorem 1)z Let the system (1) be given in a simply
connected domain G; let P and q be continuou AJIf there are functions
N(x,y), M(x,y) continuous in G, the partial derivativesvof which are
continuous in G and which possess the property that for the functions
h(xly) = PM + QN and k(x,y) N M in G there holds one of
x y
the following systemB of signs
D~ 0; 01) Es 03 ; C= 0; ~t OJ E~ 0 o Ek o 02
or[:S O;~tO_], then (1) possesses no limit cycles in G (the signs:~, 0
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On Criteria for the Absence of Arbitrary and Multiple Limit Cycles
and ~~ 0 mean that the function is  0 at most on single curves and
otherwise < 0 or > 0).
Under the additional assumptions that P(X,Y), Q(x,y) are continuous and
continuously differentiable in G and that (.whereever it is necessary)
N(x,y), M(x,y) are twice continuously differentiable, the author
collects in a scheme the most essential conditions (doubly framed)
under which (1) possesses no limit cycles in G. Criteria based on
the inverse signs (Dt o; !s: o.3, E!~: o; im o 1 ) of the functions
h(x,y) and k(x,y) must be used in the Green formula with regard to the
sign.
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On Criteria for the Absence of Arbitrary and Multiple Limit Cycles
Scheme
r
M
X,Y)
h(x,y) N x,y)
~ k(x,y) Remarks
F(x,y) >0
0
FPM + FQN :50
=0 0
aN ahl >=O
ax ay2> 0 General
criterion
Mand N such that h > 0 everywhere in G
a 2 2 2
F (P +Q >0 M = FP
N = FQ _aFQ  ')FPSjo
ax by
b 2(p2+q2 2
F > 0 M F/P+Q?
'N F~ P+Q) 3F(P+Q)  aF(P+Q)z.FO
ax ay
c 2 2
P (PQ) > 0 M F~PQ)
h F QP) ~F(PQ)  aF(PQ),(=O
?X BY
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On Criteria for the Absence of Arbitrary and Multiple Limit Cycles
d 2 2 2
p (P +Q )>0 M = F(PAQ) ~P(~KP) 3P(P+KQ)$j0
 Z(xpy) ar
"" v
'
N  P(KP+Q) a x dy
a
bitrary
i tr ry
~b
function
f ct' on
e 2 2 2
F (P +Q )> o
_K
M  P
~P
Q~ *aF(Q+KP)  aF(PKQ) Criteria
Crit eria
N  F
KP
q+ ax which gene
whih gene
rall zetho
ralize tho
o en
se of Ben
s f
l
dixonDula
2 M and N such that h 0 everywhere in G
a FFI(PQ_PQM M ~ QFI apip + DFIQ
_
0 Criterion
N = PF' x ay of Dulac
b F(PQPQ) =_ 0 PI = I ap + BA 0 Criterion
ax ay of Bendixon
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On Criteria for the Absence of Arbitrary and Multiple Limit Cycles
3 M and N such that k 0 everywhere in G
aK
U 
Criterion of
1
.4? 3K + pq 9 K 0 x
T 2K 'iK the type of'
ax ay  ~K r4 x by 6x by 0 H. Poincare
N .
ZY
b P aK a.! > 0 1 2K 2K 0 Somewhat gene
?x Y
X6 Y ralized Poin
care Criterion
4 M and N such that k~~ 0 everywhere in G
F(P dK + qN 0
ax
a (3 F &M K

N  aK a mono
T M =,~x T 
x '~:eDy
y (M ~_X) 0
n,tll,,r:
Po on ~a,xing
34n 1,
5x
5Y
1
1
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On Criteria for the Absence of Arbitrary and Multiple Limit Cycles
Other cases
a 0
2PqK 0 M  FKQ 'D CP C Q 0
0
C FK
0 N = FKP ~D x 'D yz > 0
b 2
2 0
P M ~ PF Fq + ; PP 0
_
Q
0 1
I N = QF 3 x
y
A cycle C 4S denoted as Multiple limit cycle if for it
j~p
+
dt vanishes, where 1 is the length of
0 and the integration is carried out along the limit cycle in the
direction of increasing t.
Theorem 2: Let P(X,y), Q(Y,y) in (1) be continuous functions with
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On Criteria for the Absence of Arbitrary and Multiple Limit Cycles
continuous partial derivatives in a simply connected domain G. If
there are functions N(X,Y) > 0, M(x,y) continuous and continuously
differentiable in G such that it is everywhere in G
'a NQ + ZM ITP + ~M NQ0 0)
h ONP +
ox GY
and that h vanishes identically in no partial domain of G, then in G
there are no multiple limit cycles of (!).~
In theorem 3 the author gives four further criteria for the absence
of multiple limit cycles. A generalized form of the first of these
criteria is given in
theorem 31 : Let P(XY), Q(x,y) be twice continuously differentiable
in the simply connected domain G., Lot twice continuously differentiab
le functionn N(X,.Y)> 0 and M(x,y) exist such that
1.) the curves N(x,y) P(x,y) = 0 and N(x,y) Q(x,y) = 0 are representab
le as monotone functions y = f(x) and x = T(y);
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On Criteria for the Absence of Arbitrary and Multiple Limit Cycles
2.) the expressions A D ggq i + " M ) and
97 a Y IM TY
1~ A
B IF +
()y (2~Np
N 'a x
are of constant and equal sign and both expressions do not
simultaneously vanish identically;
'~ N 1 + ('~ M
3.) the signs of A and LP, 17a ~7y (or of B and
C~ NP 1 + MM) are in a sufficiently small neighborhood of the
, ~ X 9 Tox
curve x = ~(y) (y = f(x))identical in the lower (upper) part of the
plane under (bLbove) the curve or, however,
NQ 1 a M 16 lip 1 CC) M
5 y iFQ )y dx TIT + Tjx
vanish identically, while in the other parts of the curves
X =~f(y) or y ~ f(x) those expressions have equal signs opposite to
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On Criteria for the Ab74ence of Arbitrary and Multiple Limit Cycles
the sign of A or vanish identically.
Then (1) possesses no multiiple limit cycles in G.
There are 3 fieurils, und 6 references' 3 Soviet, 2 French and 1
American.
SUBMITTED: March 6, 1959
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AUTHORs TKACHEV)VIF. 2.0411/51
TITLE: On New Sufficient Conditions for the.Stability, Semistabilitj
dy P(x,v)
and.Instability of the Limit Cycle of the Equation dx . Q~xy)_
(0 novykh dostatochnykh usloviyakh ustoychivosti, poluustoychivosti
i neustoychivo,sti predellnogo tsikla uravneniya dy . P(x.y)
dx 0.(X,Y)
PERIODICAM Doklad; Akademii Nauk$M,1957PV01166,Nr4,PP564567 (USSR)
ABSTRACTs Given the differential equation
(1) dy f(x,y),
dx
x9y
where f(xpy) _and. P(]C,Y) and q(x,y) have partial
x
derivatives of sufficiently high order. Let furtherLo x k98),
y 104s641) be a closed integral curve of (1). Then,
accordingto Papuah.LRef.1_7 , in,asufficieatly small neighborhood
of L (1) can be replaced by the equation
. dn n) .. let (9)n IfIf(s)f !f' Wnf 2r'(s)
(2) T8_  F(Brrl) P F(s,
'to I +f V
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on New Sufficient Conditions for the Stability, Selaistability 20411/51
bind Instability of the Limit Cycle of the Equation & M PC x* Y1
dx Q(x,y)
where 9 is the arc length and n is the length of the normal.
Theorem: The following conditions are sufficient for
1. stability (instability) of the cycles
CaTd 2/ 3
Pl(a,n)de; 0,
n inno
f
0
1 8
F" exp [f Fn' dtjds =of...,
f n ~=o j
0 0 0
1 a
(k)
I F exp
f n If
0 0
a t3 k2[
exp [ Fn' d da
n f
0 =0
1
F1 d de < 0
t
n
In0
2. 8emistabilityt
(0>0) k  odd,
On New Sufficient Conditions for the Stability, Semistability 20411/51
Md Instability of the Limit Cycle of the Equation dy  p(x'y~
dx
I
J1
FI(e,n)da 0, p(j) xp dt] dis 0
f n in_o j n fe If 'Fn'
0 0 0 ino
a k1
F(k) exp F1 dt ds 0 J2,39...lk1
f n If a k  even.
0 a Ino
ASSOCTATTONt Voronezh SUte University (Voronezhsldy goatidarstvenMy universitet)
PRESENTED BT:P, 8, Aleksandrovs Academician, A~ril 27, 1957
S UBMITM r Aprn 180, 195V ' ''
AVAILABUs Library of Congress
Card 3/3
TKACREV, V.F. (14oskva)
Necessary and sufficient conditions of stability., semistability,
and instability of a limited cycle and some of their applications.
Mut.sbor. 56 no.3.281300 Mr 162. (NIRA 15:4)
(Stability) (Vibration)
'p"ONEV. V  F
a
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"~ ' (.1' ' . b COM! Cil t'(, lpci of lhd&
ntnc` ot!.n:, lj'.,T,. iat. nav,I: 16
207 c',r', 76j. ,  
. I , (Different~,,1. Oal""~, 14:7.0)
GHIGIRINETS, A.A.) TKAGIIEVY V.P.
Multiple machini%, of supports. Mashinostroitell no. 4:310 kP 161.
(Killing machines) (MITUI 14:4)
TKACHEV, V.F. (Moskva); TKA.CRW, VI.F. (Voronezh)
Criterial for the absence of arbitrary and MUltiDle liMit Cycles.
Mat. abor. 52 no. 3.811822 N 160. (MIRA 13:12)
(Differential equations)
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AUTHORS:Tkachev, V. F. (Moscow), Tkache . V1. F. (Voronezh)
TITLEs On Criteria for the Absence of ArbitraryanT Multiple Limit
Cycles
PERIODICAL: Matematicheskiy sbornik, 1960, Vol52, No.3, pp.811822
TEXTs The author considers the system
(1) P(x,y), dy = Q(x,y).
dt dt
General theorem (Theorem 1); Let the system (1) be given in a simply
connected domain G; let P and 0. be continuous.UIf there are functions
N(x,y), M(x,y) continuous in G, the partial derivativesvof which are
continuous in G and which possess the property that for the functions
h(x,y) = PM + QN and k(x,y) N 1~ M in G there holds one of
x y
the following systems of signs
D: o; ol, 65 o; oJ o; ~~t oJ Ez~ o, :5 o1 ; [a o; !5. oJ
or [S O;~~O_], then (1) possesses no limit cycles in G (the signs j~ 0
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On Criteria for the Absence of Arbitrary and Multiple Limit Cycles
and ~~ 0 mean that the function is = 0 at most on single curves and
otherwise < 0 or > 0).
Under the additional assumptions that P(X,Y), Q(x,y) are continuous and
continuously differentiable in G and that (whereever it is necessary)
N(x,y), M(x,y) are twice continuously differentiable, the author
collects in a scheme the most essential conditions (doubly framed)
under which (1) possesses no limit cycles in G. Criteria based on
the inverse signs (Dt 0; ~S 03, E!!~L 0; ]~: 0 j ) of the functions
h(x,y) and k(x,y) must be used in the Green formula with regard to the
sign.
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On Criteria for the Absence of Arbitrary and Multiple Limit Cycles
Scheme
M
X,Y)
h(x,y) N x,y)
~ k(x,y) Remarks
F(x,y) >0
"0
FPM + FQN SO
~O
0
aN  am >NO
ax a Y.:S;,. 0 General
criterion
M and V such that h > 0 everywhere in G
a 2 2 2
F (P +Q >0 M = FP
N = FQ 'aFQ  'aFP,z,,

ax i!Ty
b 2(P2+Q2 2
F ;;10 1&= F~P+Q~
V = F P+Q aF(P+C0_  aF(P+Q,)
~x ay __O
F 2 PQ) 2> 0 M = F(PQ)
14= F(QP) ~F(PQ)  '8 F (P Q,) C
x a Y
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d 2(p2+q2)> 0
F M = F(P+KQ)
N = P(KP+Q) aF(RKP) . a F(P+KQ
3 x ay Ir(x, Y) ar
bitrary
function
e F2(p2 +q2)> 0 M = F(PKQ
N = F(Q+K 'aF(Q+KP)  3F(PKQ)
dx 2y Criteria
which gene
ralize tho
se of Ben
dixonDulac
2 M and N such that h ~_ 0 everywhere in G
a
F I (P QP Q) ~o M = _QYI
N = PFI aF'P + ~Flq
0
Tx ay Criterion
of Dulac
b F(PQPq) =_ 0 PI = I ap+ 21 < > 0
ax ay Criterion
of Bendixon
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On Criteria for the Absence of Arbitrary and Multiple Limit Cycles
3 M and N such that k 0 everywhere in G
i aK
M = 
Criterion of
a FP 3K + FQ UK 0 1
 a x 2K !ZFK
0 the type of'
ax ay K
D '~x ZY 'ax By H. Poincare

N
Z
y
b JP a K + Q aK 0 1 32 K 2K 0
 Somewhat gene
ralized Poin
J ~x y ,~Xay rzxzy
care Criterion
4 M and N such that k;_? 0 everywhere in G
P(P ~K + QN)= !~ 0
a ax Q F &M _ ta2 K
3K
N  Ty a mono M =Vx Tx s Xay
,~) (M_ IjK)j 0
tqnq.j~i2reasing 1 57 0y
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Other cases
M = FKQ 0
'DCP 2CRi~;
a
2PQK < 0
1
0

1 C FK
0 N = FKP 6 x
0 yz > 0
b 2
2 0 M ~ PF LFq FP 0
_
P Q
0 1
N
x a y 01
P
A cycle C is denoted as multiple limit cyc e 1 or it
t (LP + ~6A) dt vanishes, where 1 is the length of
0 a x 'DY
0 and the integration is carrIed out along the limit cycle in the
direction of increasing t.
Theorem 2: Let P(X,Y), Q(x,y) in (1) be continuous functions with
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continuous partial derivatives in a simply connected domain 0. If
there are functions N(Xty) > 0, M(x,y) continuous and continuously
differentiable in G such that it is everywhere in G
h = r0 NP + IER + tM ITP + '~M NQ ~2 0 If 0)
ox !a y 1Z x G y
and that h vanishes identically in no partial domain of G, then in G
there are no multiple limit cycles of ().
In theorem 3 the author Gives four further criteria for the absence
of multiple limit cycles. A generalized form of the first of these
criteria is given in
theorem 3' : Let P(X,Y), Q.(x,y) be twice continuously differentiable
in the simply connected domain G. Let twice continuously differentiab
le functions N(X,,Y)> 0 and M(x,y) exist such that
1.) the curves N(x,y) P(x,y) = 0 and N(x,y) q(x,y) = 0 are representab
le as monotone functions y = f(x) and x = T(y);
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1
NQ I + 9 M ) and
2.) the expressions A n (& y WIZ 'YY
'Z QNP 1F + qM
5y g ;Yx
are of constant and equal sign and both expressions do not
simultaneously vanish identically;
N
3.) the signs of A and ~P I + J~ M (or of B and
y Tq ~T
1) NP '1_ + ~M) are in a sufficiently small neighborhood of the
13x NP IDX
curve x = ~(y) (y = f(x))identical in the lower (upper) part of the
plane under (4bove) the curve or, however,
NQ 1 M and Cd NP 1 M
D y NQ 'r ?y 9T 'ff I" UY
vanish identically, while in the other parts of the curves
x = (f(y) or y = f(x) these expressions have equal signs opposite to
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On Criteria for tirip Ab3enc!e of Arbitrary and Multiple Limit Cycles
the sign of A or vanish identically.
Then (1) possesses no mult1ple limit cycles in G,
There are 3 fiGures, and 6 references: 3 Soviet, 2 French and 1
American.
SUBMITTED: March 6, i9~.~9
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j/0411/61/016/005/00'V005
Cill/C444
AUTHOR:
T I T L H. Poincare' on the
of Li.AL (:'y(;LC3 5'..'1d oth,,t re:;ults
PERIODICAL: U wak.i i m ~'. t c;~,at i c; i ~., sk. i k I i I I z ~ u k, , v . 160' 5'
205  207
TZXT: A wollknc.,.mn theore% of Poiricare' for ~.,ystems of second
order is Lnoralised ar, follows:
Theorem 1: 11' in a cert"Lin dortiain G of the :ipace (X1, x2" ... xn)
there existo a continuous function N(X,, x 21 ... I xn) ,,,rith continu
ous partial derivatives such that
~jN N
h(xl, x N P + P 11 P
2'**.' xn x1 1 1~ x22 x n
has a constant sign and does not vanish in G, then the system
dx.
Card 1/3 dt1 Pi(x1' x2"'*' xn) (i  1, 21 ... , n) (2)
S/04 61/016/005/005/005
A generali5ation of the theorem C111YC444
togral curve9 in G.
tIlly clotloa in
does not kdjoirl,llg it is provedt dx Y), (4)
If the system d2x  + N (X, Y) fit
Theorem 2 (X, Y) Y) dy
Iry
 ~ / , (x , Y) 71 T
lip 2, are
where al I functions i (xj Y)v gi(x, Y) derivativesl'is defined in
continuous and possess continuous partial
a simply connected domain G9 and if there
+ 1 2 Y > 0 < 0)
h(x, Y) 11(x, Y) 0 0) ,
r7 ( f t f
(g, + 92 UY 1 2
h(x, Y) '5x possess any limit cycles in G,
then (4) does not system
Theorem 3S the + 9, (X, + Y), (5)
'Y + A (X, Y do
d3x rn, y) + n, (X, Y) LdI2 ~_Y + Y),
dt dx
~, 13 fill/ 92 (T, Y) di
~ILII ??I, Or" Y) + 11, (X, Y) /2
Card 2/3 dO dt'
S/042/61/01 6/00 5/00 5/00 5
A generalisation of the theorem.,. C111/C444
where all m, n, f, g, 1 are continuous, possessing continuous partial
derivatives, is defined in a simply connected domain G, and if there
0 (n, (x,
CI(ml(x, Y) + M2(x, Y)) Y) + n2(x' Y))  0
0 y 0 x
~D(ml(x` Y) + m2(x, Y)) D(nl(x, y) + P2(x, Y))
ID   '~>O)~   0)
h(x, y)  11(x' Y) + 12(x' Y) > 0 0