SCIENTIFIC ABSTRACT TKACHEV, V.D. - TKACHEV, V.V.

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SCIENTIFIC ABSTRACT
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L 28001-66 ACC NRs AEbO3.2496 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 long-wave branch with a well- pronounced maximum. The narrow lines are due to phononleso radiative transitions within the centers, and the long-wave 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 Card 2/2 L 03C2-66 EIXT (m) 1EMF( t/_STT IJP(c) GGIJI) ACC NRz 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 p-type silicon-irradiated with fast electrons SOURCE: All BSSR. Doklady, v. 10, no. 2, 1966, 80-32 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 ed-kAth I MeV electrons at 25-300C and Mpe o rra at - integral doses of 1013-10 electrons cm . The change in the shape of the spectra of Lm-purity 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- Card -1, L0352-66 C NR& AP6019223 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 im-purity 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. SUB CODE: 20/ SUM DATE: 14Jun65/ ORIG REF: 005 /4t&/ Card L 2736Q=66 EWT(M)/EViP(t)/ET1 IJP(c) JD ACC NR- AP6011529 SOURCE CODE: UR/0250/6~/Mo/oo3/ol48/0150 AUTHORS: Sevchenk Tkachev V. D. 99~KQ o., A. K.; 1ki tk -YA-E. 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, 148-150 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 p-type silicon with resistivity 7 -- 10 ohm-cm, containing 8 X 1017 CM-3 oxygen. The irradiation was with 1-Mev electrons from n electrostatic generator at 80K and 30C. The spectral dependence of ~tbe stationary photoconductivity was plotted witb apparatus described by Cwd 1/2 L 27360-66 ACC NRi AP6011529 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 liquid-nitrogen 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 34820-6o 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, 374-376 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 p-type 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] SUB COIIE~, -20/ SUBM DATE: 09Mar66/ ORIG REF: 002/ ~ OTH AEF: 002/ ATD PRESS: Card J/ h. L_ 420.~7-!66 JD 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 gamma-quanta SOURCE: AN BSSR. Doklady, v. 10, no. 8, 1966, 550-552 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 n-type and p-type 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 1014-5 x 1017 kv/CM2. The specimens used were n- and p-type germanium single crystals with initial specific resistances of 48 ohm-cm and 6 ohmocm, respectively. The concentration of residual impurities in the electrically active state.for n-type Card L 42097-66 ACC NRs AP6029648 Material Was Up to 1012-1013 CM-3. Investigation of the electrical characteristics of specimens aft-er Lrr&*VWt1*WqthdW&if'the preseiice of EV + 0.01 ev, Ev + 0.008 ev, and Ev + 0.17 ev levels in p-type ger- manLum and E. - 0.20 ev level in n-typexrmanium. Irradiation of n-type crystals with doses up to 5 x 10 kv/cm2 did not change the conductivity &Lgn. At T - 80K the Fermi level in irradiated n-type specimens was located 0.16-0.22 ev from the bottom of the 'conductivity." one, while in p-type specimens it was found 0,20-0.26 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 these-centers 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 rW-dn 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 In-irviw4ated apacLmens for which the ratio Ac/o is greater, L-t 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 ATD PRESS$ . .. ... Cmd 2 / 2 af ACC NRi Ai,6o33157 SOURCE CODE: W0250/66/010/009/064VO643' 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, 641-643 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 n-type 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 3.0-4 mm Hg: The radiative recombination without additional illumination Card ACC NRt AP6033157 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 Card 2/2 0-407-67 N'?' AP602-t951 .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 gotudarstvcnny-y universfic-tj- TIT12: Energ.- spectrum of radiation damage in silicon sinrle crystals SOURCE: All SSS11. Doklady, v. 169, no. 3, 1966, 562-564 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 1-41'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 p-type crystals had a resistivity 0.03 -- 150 ohm-cm. 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 1/2 ard UDC: 537.312.5 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 radiation-daaged conduction and v 11 pairs. A complete energy-level r-stitial at0I je a; 3 figures' inte: . art. ha is presented. Or -oqmarGG[ -ORIG REP: 005 ./ OTI' REF: 002 SUB CODE: 20/ SUBM DATE: 2/241 SOURCE C _~T-UR/6-181r66[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, 3213-3217 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 25-6000C 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- Card 1/2 ACC NRt AF603696o 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 400-5000C. 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. 3:811-822 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*4:564-567 0 157. _ff -MIT (MMA 11:3 ) I Voronezhakiy gosudarstvennyy iiniversitet. Fredotavleno akademikom ;.S. Aleksandrovym. (Differential equations) 85226 14'-~L40 0 S/039/60/052/003/003/007 C ill/ 0 333 AUTHOESiTkachev, V. F. (Moscow), Tkachev. Vl. F. (Voronezh) TITLE: On Criteria for the Absence of Arbitrary and Multiple Limit Cycles PERIODICALs Matematicheskiy sbornik, 196o, Vol-52, NO,39 PP,811-822 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 Card 1/9 85226 S/039/60/052/003/003/007 C 111/ C 333 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. Card 2/9 85226 S/039/60/052/003/003/007 C Ill/ C 333 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 ay-2-> 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 (P-Q) > 0 M F~P-Q) h F Q-P) ~F(P-Q) - aF(P-Q),(=O ?X BY Card 3/ 9 85226 S/039/60/052/003/003/007 C 111/ C 333 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(P-KQ) Criteria Crit eria N - F KP q+ ax which gene whi-h gene- rall zetho- ralize tho o en- se of Ben- s f l dixon-Dula 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(PQ-PQ) =_ 0 PI = I ap + BA 0 Criterion ax ay of Bendixon Card 4/9 85226 S/039/60/052/003/003/007 C 111/ C 333 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 X-6 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 Card I/ a 85226 S/039/60/052/003/003/007 C 111/ C 333 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 Card 6/9 85226 S/039/60/052/003/003/007 C 111/ C 333 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 + -Z--M- 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); Card 7/ 9 85226 S/039/60/052/003/003/007 C 111/ C 333 On Criteria for the Absence of Arbitrary and Multiple Limit Cycles 2.) the expressions A D ggq i + " M ) and 97 a Y IM T-Y 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, 17-a ~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 Card 8/9 85226 S/039/60/052/00'7/003/007 C 111/ C 333 On Criteria for the Ab-74ence 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 Card 9/9 AUTHORs TKACHEV)VIF. 2.0-4-11/51 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,1957PV01-166,Nr-4,PP-564-567 (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 further-Lo x k98), y 104s641) be a closed integral curve of (1). Then, according-to Papuah.LRef.1_7 , in,a-sufficieatly small neighborhood of L (1) can be replaced by the equation . dn n) .. let (9)-n IfIf(s)-f !f' W-nf 2r'(s) (2) T8_ - F(Brrl) P F(s, 'to I +f V Card 1/3 1 on New Sufficient Conditions for the Stability, Selaistability 20-4-11/51 bind Instability of the Limit Cycle of the Equation & M PC x* Y-1 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 k-2[ exp [ Fn' d da n f 0 =0 -1 F1 d de < 0 t n In-0 2. 8emistabilityt (0>0) k - odd, On New Sufficient Conditions for the Stability, Semistability 20-4-11/51 Md Instability of the Limit Cycle of the Equation dy - p-(x'y~ dx I J-1 FI(e,n)da 0, p(j) xp dt] dis 0 f n in_o- j n fe If 'Fn' 0 0 0 in-o a k-1 F(k) exp F1 dt ds 0 J-2,39...lk-1 f n If a k - even. 0 a In-o 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.-281-300 Mr 162. (NIRA 15:4) (Stability) (Vibration) 'p"ONEV. -V - F a .I - I . I . I I "~ ' (.1' ' . b- COM-! Cil t'(-, lp-ci- of lh-d-& ntn-c` ot!-.n-:, lj'.,T,. i-at. 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.-811-822 N 160. (MIR-A 13:12) (Differential equations) 85226 S/039 6o/052/003/003/007 L4 0 0 C ill C 333 AUTHORS:Tkachev, V. F. (Moscow), Tkache . V1. F. (Voronezh) TITLEs On Criteria for the Absence of Arbitrar-y--a-n-T Multiple Limit Cycles PERIODICAL: Matematicheskiy sbornik, 1960, Vol-52, No.3, pp.811-822 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; o-J o; ~~t oJ Ez~ o, :5 o-1 ; [a o; !5. oJ or [S O;~~O_], then (1) possesses no limit cycles in G (the signs j~ 0 Card 1/9 85226 3/039/60/052/003/003/007 C 111/ C 333 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. Card 2/9 85226 S/039/60/052/003/003/007 C 111/ C 333 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 >N-O --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 P-Q) 2> 0 M = F(P-Q) 14= F(Q-P) ~F(P-Q) - '8 F (P- Q,) C x a Y Card 3/ 9 85226 S/039/60/052/003/003/007 C 111/ 0 333 On Criteria for the Absence of Arbitrary and Multiple Limit Cycles d 2(p2+q2)> 0 F M = F(P+KQ) N = P(-KP+Q) aF(R-KP) -. a F(P+KQ 3 x ay Ir(x, Y) ar bitrary function e F2(p2 +q2)> 0 M = F(P-KQ N = F(Q+K 'aF(Q+KP) - 3F(P-KQ) -dx 2y Criteria which gene- ralize tho- se of Ben- dixon-Dulac 2 M and N such that h ~_ 0 everywhere in G a F I (P Q-P Q) -~o M = _QYI N = PFI aF'P + ~Flq 0 Tx- ay Criterion of Dulac b F(PQ-Pq) =_ 0 PI = I -ap+ 21 < > 0 ax ay Criterion of Bendixon Card 4/9 85226 B/039/60/052/003/003/007 C 111/ C 333 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 a-K 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 85226 S/039/60/052/'003/003/007 C 111/ C 333 On Criteria for the Absence of Arbitrary and Multiple Limit Cycles 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 0--1 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 Card 6/9 85226 S/039/60/052/003/003/007 C 111/ 0 333 On Criteria for the Absence of Arbitrary and Multiple Limit Cycles 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 + -t-M 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); Card 7/ 9 85226 S/039 60/052/003/003/007 C 111~ C 333 On Criteria for the Absence of Arbitrary and Multiple Limit Cycles 1 NQ I + 9 M ) and 2.) the expressions A n (--& y WIZ 'Y-Y 'Z QNP 1F + qM 5-y -g ;Y-x 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 13-x -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 Card 8/9 85226 8/039/60/052/003/003/007 C III/ C 333 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 Card 9/9 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 woll-knc.,.mn theore% of Poiricare' for ~.-,ystems of second order is L--noralised ar, follows: Theorem 1: 11' in a cert"Lin dorti-ain 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 Ir-y - ~ / -, (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) '5-x 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 ~, 1-3 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