Y U R I I A L E K S E E V I C H M I T R O P O L V S K I I

( O N HIS 5 0 t h B I R T H D A Y )

V . M . G l u s h k o v , V . S . K o r o l y u k , O . S . P a r a s y u k , a n d O . B . L y k o v a

On January 3, 1967, I_enin Pr ize Laureate and Academician of the Academy of Sciences of the Ukrai- nian SSR Yu. A. Mitropol 'ski i celebrated his fiftieth birthday.

Yurii Alekseevich Mitropol 'ski i was born in the town of Shishaki in Poltava Province. On completion of the seven-yea r school Mitropol 'ski i went to work in the factory; in 1938 he finished the ten-year school and entered Kiev University (KDU) in the Department of Mechanics and Mathematics.

His studies at the universi ty were interrupted by the Second World War. In 1942 he graduated from the Department of Physics and Mathematics at Kazakh University and was sent to the a r t i l l e ry school and subsequently to the front. During his mi l i ta ry service as commander of an ar t i l le ry reconnaissance platoon Mitropol 'ski i was awarded two Orders of the Red Star.

At the end of the war and demobilization, in 1946, Mitropol 'skii was engaged as a scientific co -worker at the Institute of Structural Mechanics (ISM) of the Academy of Sciences of the Ukrainian SSI:t, thus begin- ning his prolific scientific c a r e e r under the direction of the eminent Academician N.N. Bogolyubov.

Two years later, in 1948, Yurii Atekseevich successful ly defended his mas t e r s thesis, which was devoted to resonance effects in nonlinear osc i l la tory sys tems , and three years later, in 1951, he defended his doctoral d isser ta t ion on a vitM problem area in nonlinear mechanics and mathemat ical phy-sies, viz., nonstat ionary p rocesses in nonlinear osci l la tory sys tems.

Since 1951 MitropoI 'ski i has worked in the Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR, serving since 1953 as Chairman of the Department of Mathematical Physics.

In 1954 he was conferred a professorship.

Translated from Ukrainskii Matematicheskii Zhurnal, VoI. 19, No. 1, pp. 3-8, J anua ry -Februa ry , 1967.

In 1958 Mit ropol ' sk i i was elected a Corresponding Member of the Academy and in 1961 an Academician of the Academy of Sciences of the Ukrainian SSR.

In 1958 he was appointed Di rec to r of the Insti tute of Mathemat ics , se rv ing until the p resen t t ime as its d i rec tor .

Yurii Alekseevich has been elected seve ra l t imes to s e rve on the s taff of the Bureau of the Phys ico- ma themat i ca l Sciences Division of the Academy of Sciences of the Ukrainian SSR, f rom 1961 through 1963 as its d i rec to r ; s ince 1963 he has been Academic Sec re t a ry of the Division of Mathemat ics , Mechanics, and Cybernet ics and a m e m b e r of the P res id ium of the Academy of Sciences of the Ukrainian SSR.

Yurii Alekseevich Mit ropol ' sk i i is recognized as an outstanding scholar in the theory of osci l lat ions and nonlinear different ia l equations. His published works on the fundamental p rob lems of the theory of non- l inear osci l la t ions and nonlinear different ia l equations have r e p r e s e n t e d a m a j o r contribution to nonlinear mechanics and the conceptual theory of nonlinear different ia l equations.

He has published m o r e than a hundred scientif ic works, including six monographs , two of which (one in co-au thorsh ip with N.N. Bogolyubov) have been t rans la ted into English, French, German, Chinese, and Japanese .

Mi t ropo l ' sk i i ' s monographs , pa r t i cu la r ly Asymptot ic Methods in the Theory of Nonlinear Osci l la t ions (co-author N.N. Bogolyubov) and P rob l em s in the Asymptot ic Theory of Nonstat ionary Oscil lat ions, a r e widely known and a re used as r e f e r e n c e books by spec ia l i s t s in var ious branches of physics and engineering per ta in ing to the theory of osci l la t ions.

Yurii Alekseevich was awarded the Lenin P r i ze in 1965 for his outstanding achievements in the theory of nonlinear osci l la t ions and nonlinear different ial equations.

One of the core p rob lems , and one of the mos t t roub lesome , in the theory of osci l la t ions is the analys is of nonsta t ionary phenomena in s y s t e m s cha rac t e r i zed by var iab le p a r a m e t e r s . We have been confronted with the need to invest igate this p rob lem in the frequently encountered problem of the t ransi t ion of s y s t e m s through resonance , in the investigation of osci l la t ions in v a r i a b l e - m a s s va r i ab l e - s t i f fnes s sy s t ems , in p rob- l ems re la t ing to the vibrat ion of br idges and lift c ranes under the influence of moving loads and pulsat ing forces , in calculat ions of rocke t t r a j ec to r i e s in powered a rcs , in the analys is of r esonance effects during par t i c le acce le ra t ion in the synchrophasot ron, etc.

With the basic concept of the asymptot ic methods of Krylov and Bogolyubov as a foundation, Mi t ro- pol ' sk i i developed a method for the analysis of nonlinear s y s t e m s with slowly varying p a r a m e t e r s . The a lgor i thms that he formula ted pe rmi t the effect ive investigation of these p rob lems and the formulat ion of approx imate asympto t ic solutions of nonlinear different ia l equations with var iab le coefficients for the des - cr ipt ion of nonsta t ionary osc i l l a to ry p r o c e s s e s in s y s t e m s having e i ther one or many degrees of f reedom.

It is impor tan t to mention that p r io r to Mi t ropo l ' sk i i ' s work these p rob l ems could be solved m o r e or less r igo rous ly only in the mos t e l emen ta ry cases of l inear osc i l l a to ry s y s t e m s with one degree of f reedom.

The method developed by Yu. A. Mit ropol ' sk i i for the investigation of s y s t e m s with slowly varying p a r a m e t e r s enabled him to uncover a number of new and in teres t ing effects in nonlinear osc i l l a tory s y s t e m s with slowly varying p a r a m e t e r s , for example, in the passage of s y s t e m s through resonance: ampli tude pulling, ampli tude jumps and discontinuit ies, beats , etc.; the mutual influence of p a r a m e t r i c and ord inary re sonance during a nonsta t ionary p roce s s has also been studied.

A m a j o r contribution to the theory of osci l la t ions and the theory of nonlinear different ial equations was the method developed by Mit ropol ' sk i i for the analysis of one- f requency modes in osc i l l a to ry s y s t e m s with many deg rees of f reedom. This method has made it poss ib le in many instances to gain insight into the pa t te rn of osc i l l a to ry p r o c e s s e s in s y s t e m s with many degrees of f reedom. Mit ropol ' sk i i used it l a te r for an investigation of d i s t r i b u t e d - p a r a m e t e r s y s t e m s desc r ibed by nonlinear par t ia l d i f ferent ia l equations s i m i l a r to hyperbol ic equations. As we know, the one - f r equencymethod only pe rmi t s one to obtain an ap- p rox ima te r ep resen ta t ion for a specia l solution, viz., the t w o - p a r a m e t e r solution, r a the r than the genera l solution of the cor responding set of equations containing n a r b i t r a r y constants. By writ ing a lgor i thms for the formulat ion of the solutions of var ious c l a s s e s of nonlinear different ia l equations, Mit ropol ' sk i i was able to p rove a number of t heo rems set t ing for th s tabi l i ty c r i t e r i a for the invest igated t w o - p a r a m e t e r famil ies of pa r t i cu l a r solutions; these theorems were subsequently e labora ted into a sys t ema t i c theory of in tegral manifolds, which takes on specia l s ignif icance in the investigation of different ia l equations with both smal l and large p a r a m e t e r s .

The methods developed by Mit ropol ' sk i i for proving the existence theo rems for in tegral manifolds, as well as his effect ive techniques for their computation, have provided a m a j o r contribution to the theory of o rd inary different ia l equations. The method of integral manifolds is not only of g rea t theoret ica l importance, but of p rac t i ca l value as well; for example, it has been used to de te rmine the opt imal s t a t ionary s ta tes in complex dynamical sy s t ems .

In addition to its high theoret ical plane, a dist inguishing fea ture of Mi t ropo l ' sk i i ' s work is the leaning toward effect ive solutions of urgent p rac t i ca l p rob lems . His r e s e a r c h has s ingular values for its s teadfas t attention to the physical substance of the effects invest igated and for his awareness of how to d i sc lose and in te rp re t new effects .

The methods developed by Yurii Alekseevich have been used in studies devoted to v ibra t ions of turbine blading at va r i ab le driving f requencies , the investigation of control s y s t e m s with slowly varying p a r a m e t e r s , the invest igat ion of resonance effects in e lec t r i ca l networks, and in a g rea t many physica l p rob lems re la ted to p a r a m e t e r s slowly varying in space and t ime, as is a lmos t always the ease in many physical p rob lems (in the invest igat ion of r esonance and noise depletion of synchronous osci l la t ions in an acce le ra to r , in the invest igat ion of var ious cases of r esonance with Mow phase osci l lat ions, and in the analys is of a number of p rob lems in p l a s m a physics , p r i m a r i l y in the development of the theory of p l a sma s tabi l i ty in inhomogeneous and va r i ab le f ie lds) .

The asymptotic method of Yu. A. Mitropol'skii for the analysis of nonlinear oscillatory systems with slowly varying parameters is a powerful and deeply penetrating tool with enormous potentials for widespread future application in the problems of acoustics, fluid mechanics, gas dynamics, and other areas.

Yurii Alekseevich belongs to that category of scientists for whom there is no sharp distinction between pure and applied mathematics. The methods he has developed are marked by straightforwardness and ease of implementation, whereas the solutions of specific problems are stated in his papers in precise mathe-

matical terms and are always rigorously substantiated.

Yurii Alekseevich has a large following of disciples and students working toward the further develop- ment of asymptotic methods, as well as their application to the solution of practical problems. More than twenty masters and doctoral dissertations have been written and defended under his sponsorship.

Yurii Alekseevieh brilliantly combines his scientific work with scientific-administrative, educational, and communal activities.

For many years he has delivered lecture courses at Kiev University.

As Director of the Institute of Mathematics since 1958, Mitropol'skii has devoted tremendous effort and energy to the growth of the Institute.

Not to be overlooked either is the great attention that Yu. A. Mitropol'skii has given over the mathe- matical t ra ining of young people. On his init iative and with his par t ic ipat ion, in 1963 for the f i r s t t ime in the Soviet Union s u m m e r ma thema t i c s schools were instituted, where young students f rom s e v e r a l ci t ies throughout the Soviet Union rece ived the opportunity to attend lec tures by eminent spec ia l i s t s f rom various branches of ma thema t i c s .

Yurii Alekseevich Mit ropol ' sk i i is an outstanding scient is t , a m a j o r scientif ic organizer , a gifted educator of the young, and a highly cult ivated human being. For this he has won the r e spec t and admirat ion, not only of the m e m b e r s of the Insti tute of Mathemat ics , but also of the scientif ic communi ty at large.

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LIST OF THE PUBLISHED WORKS OF u MITROPOL'SKII

"Natural osci l la t ions of a nonlinear s y s t em with slowly varying p a r a m e t e r s , " Sb. Trudov ~M,~r No. 1_,, 107-114 (1948). "Invest igat ion of the natural osci l la t ions of a nonl inear sys t em s imi l a r to an exact ly in tegrable sys tem, with slowly varying p a r a m e t e r s , " Sb. Trudov ISM, No. 11, 99-106 (1948). "Invest igat ion of osci l la t ions in nonlinear s y s t e m s with many degrees of f reedom and slowly varying p a r a m e t e r s , " Ukrainsk. Matem. Zh., 1, No. 2, 85-98 (1949). "Application of symbol ic methods to the investigation of nonlinear s y s t e m s with slowly varying p a r a - m e t e r s , " Sb. Trudov ISM, No. 13, 99-111 (1949).

5. "Stationary oscillations in nonlinear systems with many degrees of freedom," Sb. Trudov ISM, No. 12, 228-233 (1950).

6. "Investigation of the oscillations of a nonlinear system with slowly varying parameters," Sb. Trudov ISM, No. 14, 134-144 (1950).

7. "Slow processes in nonlinear oscillatory systems with many degrees of freedom," Prikl. Matem. Mekhan., 14, No. 2, 130-170 (1950).

8. "Passage through resonance in a nonlinear oscillatory system with many degrees of freedom," Sb. Trudov ISM, No. 17, 47-50 (1952).

9. "Forced oscillations in nonlinear system on passing through resonance, Inzh. Shorn., 15, 89-98 (1953).

i0. "Oscillations in gyroscopic systems on passing through resonance," Ukrainsk. Matem. Zh., 5, No. 3, 333-349 (1953).

11. "Influence of elastic elements with a nonlinear characteristic on small oscillations in active gyroscopic systems," Nauk. Zap. KDU, No. 5, 107-114 (1954).

12. "Nonstationary oscillations in systems with many degrees of freedom," Ukrainsk. Matem. Zh., 6, No. 2, 176-189 (1954).

13. ,'Effect of a frequency-modulated sinusoidal force on a nonlinear vibrator," Ukrainsk. Matem. Zh., 6, No. 4, 442-447 (1954).

14. "Passage through a second-order resonance," Ukrainsk. Matem. Zh., 7, No.l, 121-123 (1955). 15. "Asymptotic methods of N.N. Bogolyubov and their continued development," Rev. Math. Pures et

Appl. (Academie de la RPR), 1, No. 3, 15-26 (1956). 16. ,Effect of variable-frequency external forces on nonlinear oscillatory systems," Bul. Inst. Politehnic

din Jasi, Serie Nova, 3 (7), Nos. l-2, 15-24 (1957). 17. "Intrinsic resonance in nonlinear oscillatory systems," Nauk. Zap. KDU, Matem. Zb. No. 9, 16,

No. 2, 53-61 (1957). 18. "Certain differential equations encountered in the theory of relaxation oscillations," Ukrainsk. Matem.

Zh., 9, No. 3, 296-309 (1957). 19. "Uns~ble processes in active relaxation oscillatory systems," Nauk. Zap. KDU, Matem. Zb. No. 9,

16, No. 2, 93-101 (1957). 20. "Asymptotic representations of the solutions of systems of nonlinear equations with variable coeffi-

cients," Scientific Progress Report of Kiev University for 1956 [in Ukrainian] (1957), pp. 504-506. 21. "Investigation of the integral manifold for a system of nonlinear equations with variable coefficients,"

Ukrainsk. Matem. Zh., 10, No. 3, 270-280 (1958). 22. "Stability of a one-parameter family of solutions of a system of equations with variable coefficients,"

Ukrainsk. Matem. Zh., 10, No. 4, 389-393 (1958). 23. "Equations similar to exactly integrable equations," Visn. KDU, Set. Astr., Matem. ta Mekhan.,

Edition 1, No. 1, 97-100 (1958). 24. "Order of e r r o r in the asymptotic integration of equations similar to exactly integrable equations,"

Visn. KDU, Ser. Astr., Matem. ta Mekhan., Edition 2, No.l, 3-6 (1958) 25. "Asymptotic methods in the theory of nonlinear oscillations," Proceedings of the Third All-Union

Mathematics Congress [in Russian[, Vol. 3 (1958), pp. 531-542. 26. Quelques questions de l'int~gration asymptotique des 6quations differentielles non-lin~ares [Problems

in the Asymptotic Integration of Nonlinear Differential Equations], Izd. AN UkrSSR (1958), pp. 1-17. 27. "Investigations of nonstationary oscillatory modes in systems with distributed parameters ," Visn.

KDU, Ser. Astr., Matem. ta Mekhan., Edition 1, No. 2, 3-17 (1959) (co-author B.I. Moseenkov). 28. "Certain equations similar to exactly integrabie equations," Collection: Automatic Control and Com-

puter Engineering [in Russian], No. 2 (1959), pp. 221-248. 29. "Periodic solutions of a system of nonlinear differential equations with nondifferentiable right mem-

bers ," Dokl. Akad. NaukSSSR, 128, No. 6, 1118-1121 (1959). 30. "Periodic solutions of systems of---almost-autonomous nonlinear differential equations," Dop. ANURSR,

No. 11, 1175-1178 (1959) (co-author O.B. Lykova). 31. "Periodic solutions of systems of nonlinear differential equations with nondifferentiable right mem-

bers ," Ukrainsk. Matem. Zh., 11, No.4, 366-379 (1959). 32. "Nonlinear equations with periodic coefficients," Visn. KDU, Ser. Astr., Matem. ta Mekhan., Edition

2, 3-12 (1959) (co-author O.B. Lykova). 33. "Periodic solutions of nonautonomous systems with the separation of an isolated generating solution,"

Dop. AN URSR, No. 1, 3-6 (1960) (co-author O.B. Lykova).

34. "Sur les multiplicities int6grales des syst6mes d'6quations diff6rentielles non-lin6ares ayant un petit param~tre [Integral manifold of systems of nonlinear differential equations having a small para- meter],' Ann. Matem. Pura ed Appl., Series IX, 49, 181-192, Bologna (1960),

35. MNajnowsze osiagniecia wdziedzinie mechaniki nielineowej [Recent advances in research on nonlinear mechanics]," Rozprawy Inzynierskie CXLVI, 8, No. 2, 125-135 (1960).

36. "Periodic solutions of nonlinear equations wit-h a small parameter, ~ Ukrainsk. Matem0 Zb., 12, No. 4, 391-401 (1960) (co-author O.B. Lykova).

37. "Nonlinear different ia l equations with per iodic coefficients and a r b i t r a r y var iab le p a r a m e t e r s , ~ Visn. KDU, Ser. Matem. ta Mekhan., Edition 2, No. 3, 3-10 (1960) (co -au tho r O.B. Lykova) .

38. "Per iod ic solutions of a s y s t em of nonlinear different ial equations with nondifferent iable right m e m - b e r s , " Bul. Inst. Poli tehnic din Jas i , Ser. Nova, 6 (10), Nos. 3-4, 7-12 (1960) (co-author O. B. Lykova) .

39. "Behavior of the asymptotic solution of a strongly nonlinear autonomous system," Dop. AN URSR, No. 7, 839-844 (19615 (co-author O.B. Lykova).

40. "Analytical methods in the theory of nonlinear oscillations, ~ Proceedings of the All-Union Congress on Theoretical and Applied Mechanics [in Russian], Izd. AN SSSR (1962), pp. 25-35 (co-author N. N. Bogolyubov).

41. "M6thodes analytiques de la th~orie des oscillations non-lin~aires [Analytical methods in the theory of nonlinear oscillations}," Proceedings of the Tenth International Congress of Applied Mechanics, Stresa-Amsterdam-New York (1960), pp. 9-25 (co-author N.N. Bogolyubov).

42. "The method of integral manifolds in the theory of nonlinear oscillations," International S~nposlum on Nonlinear Differential Equations and Nonlinear Mechanics, New York-London (19635, PP. 1-15.

43. "The method of integral manifolds in nonlinear mechanics," Proceedings of the International Sympo- sium on Nonlinear Oscillations [in Russianl, Vol. 1 (19635, pp. 93-154 (co-author N.N. Bogolyubov)~

44. English translation: "The method of integral manifolds in nonlinear mechanics," Contributions to Dif- ferential Equations, Vol. 2 (1963), pp. 123-196 (co-author N.N. Bogolyubov).

45. "Investigation of nonstationary oscillations in nonlinear systems," Proceedings of the International Symposium on Nonlinear Oscillations [in Russian], Vol. 3 (1963), pp. 241-274.

46. "Method of integral manifolds in the theory of differential equations, ' Proceedings of the Fourth All- Union Mathematics Congress [in Russian], Vol. 2 (19645, pp. 432-437 (co-author N.N. Bogolyubov).

47. "Integral manifold of nonlinear differential equations containing slow and fast motions," Ukrainsk. Matem. Zh., 16, No. 2, 157-163 (1964)(co-author O.B. Lykova).

48. "Investigation of the integral manifold for a system of nonlinear equations similar to equations with variable coefficients in Hilbert space," Ukrainsk. Matem. Zh., 16, No.3, 334-338 (1964).

49. "Structure of the general solution of nonlinear differential equations by a method ensuring 'accelerated' convergence," Ukrainsk. Matem. Zh., 16, No. 4, 475-501 (19645.

50. "Structure of the trajectories on toroidal manifolds," Dop. AN URSR, No. 8, 934-986 (1964) (co-author A.M. SamoilenkoS.

51. Investigation of the Behavior of the Solutions of Nonlinear Equations in the Neighborhood of the Equilib- rium Position [in Russian], Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR, Institute of Technical Information, Kiev (1964), pp. 3-44 (co-author O.B. Lykova).

52. "Investigation of quasi-periodic states in nonlinear oscillatory systems ~ [in Russian], Les vibrations fore6es dans les syst~mes non-lin6aires [Forced Vibrations in Nonlinear Systems], Paris (19655, pp. 181-192 (co-author N. N. Bogolyubov).

53. "Sur l'existence de solutions quasi-p~riodique d'un syst~me canonique troubl~ [Existence of quasi- periodic solutions of a perturbed canonical system], ~ Los vibrations forc@es dans les systemes non- lin~aires, Paris (1965), pp. 407-414 (co-author O.B. Lykova).

54. "Application of the asymptotic methods of nonlinear mechanics to the analysis of nonlinear oscillatory systems with distributed parameters" [in Russian], III Konferenziiber Nichtlineare Schwingungen [Third Conference on Nonlinear Oscillations] (Berlin, May 25-30, 1964), Vol. i, Akademie-Verlag, Berlin (1965), pp. 10-20.

55. "Investigation of the behavior of the solutions of nonlinear equations in the neighborhood of the equilib- rium position," Republic Joint Collection: Mathematical Physics [in Russianl, Kiev (1965), pp. 74-96 (co-author O.B. LykovaS.

56. "Integral manifold of a nonlinear system inHilbertspace," Ukrainsk. Matem. Zh., 17, No. 5, 43-53 (1965) (co-author O.B. Lykova) .

57. "Structure of the solutions of linear differential equations with quasi-periodic coefficients by the method of accelerated convergence, ~ Ukrainsk. Matem. Zh., 17, No.6, 42-59 (1965)(co-author A.M. Samoilenko).

58. ~Asyrnptotic methods of nonlinear mechanics in application to nonlinear equations with a delayed argument," Ukrainsk. Matem. Zh., 18, No. 3, 65-~84 (1965) (co-author V.I. Fodchuk).

Monographs

1. Nonstationary Processes in Nonlinear Oscillatory Systems [in Russian], Izd. AN UkrSSR (1955). 2. Chinese translation: NonstationaryProcesses in Nonlinear Oscillatory Systems, Izd. Nauka,

Peking (1958). 3. English translation: Nonstationary Processes in Nonlinear Oscillatory Systems, Air Technical Intel-

ligence Translation ATIC-270579 F-TS-9085/V (1961). 4. Japanese translation: Nonstationary Processes in Nonlinear Oscillatory Systems, Tokyo (1962). 5. Asymptotic Methods in the Theory of Nonlinear Oscillations [in Russian], Gostekhizdat (1955) (co-

author N.N. Bogolyubov). 6. Second edition, corrected and augmented [in Russian], Fizmatgiz, Moscow (1958). 7. Third edition, corrected and augmented [in Russian], Fizmatgiz, Moscow (1963). 8. Japanese translation: Asymptotic Methods in the Theory of Nonlinear Oscillations. 9. English translation: Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon and Breach,

Science Publishers, New York; Hindustan Publ. Corp., Delhi 6 (1961). 10. Chinese translation: Asymptotic Methods in the Theory of Nonlinear Oscillations, Peking (1962). 11. French translation: Les m~thodes asymptotiques en th~orie des oscillations nonlin6aires, Gauthier-

Villars, Paris (1962). 12. German translation: Asymptotische Methoden in der Theorie der nichtlinearen Schwingungen, Aca-

demie-Verlag, Berlin (1965). 13. Investigations of Oscillations in Distributed-Parameter Systems (Asymptotic Methods) [in Ukrainian],

Vid. KDU (1961) (co-author B.I. Moseenkov). 14. Problems in the Asymptotic Theory of Nonstationary Oscillations [in Russian], Izd. Nauka, Moscow

(1964). 15. Lectures on the Averaging Method in Nonlinear Mechanics [in Russian], Izd. AN UkrSSR, Kiev(1966).