BOGOLYUBOV 58 



Bogolyubov was successful in extending the methods of the 

 turbulence theory to general nonconservative systems and in 

 developing new asymptotic methods in the theory of nonlinear 

 oscillations. These asymptotic methods, grounded in mathe- 

 matics, not only permitted a solution in the first approximation" 

 (as, for instance, does the Van der Pohl method) but also in 

 higher degrees of approximation and could be applied to the 

 study of both periodic and quasi-periodic oscillatory processes. 

 These methods were simple for practical use embodying a high- 

 ly effective principle of equivalent linearization, the symbolic 

 method, etc. 



A number of investigations by Bogolyubov in nonlinear me- 

 chanics deal with the rigorous foundation of asymptotic methods, 

 the estimation of error over a finite interval, the determination 

 of correspondence of some properties of precise and approxi- 

 mate solutions over an infinite interval, and the proof of some 

 existence and stability theorems of quasi -periodic solutions. 

 Interesting and elegant theorems were proven in the investi- 

 gation of stationary oscillatory processes. Making use of the 

 Poincare-Lyapunov theory, as well as of the Poincare-Danzhua 

 theory of trajectories on a tore, he was successful in investi- 

 gating the nature of a precise stationary solution in the vicinity 

 of an approximate solution. In the theoretical field of nonlinear 

 mechanics he also investigated the abstract theory of dynamic 

 systems. He made a full investigation of the structure of the 

 invariant dimensions of a compact dynamic system. A study 

 was made of the existence and the basic properties of ergodic 

 numbers emerging in the phase space of a dynamic system, 

 corresponding physically to a stationary oscillation science. 



In his first works in theoretical physics, which were related 

 to asymptotic methods, Bogolyubov examined problems dealing 

 with the influence of a random force on a harmonic oscillator, 

 and the establishment of statistical balance in a system con- 

 nected to a thermostat. 



A number of his investigations deal with questions in sta- 

 tistical mechanics of classical systems. Here, he has de- 

 veloped a method of distribution functions, the essence of which 

 lies in the development of analytical calculation methods which 

 give probability distribution function of the particle complexes 

 in the examined system. On the basis of Gibbs' distribution, he 

 arrived at a method for constructing a system of equations for 

 these functions, and indicated methods of their solution for 

 various cases. Extending the technique of distribution functions 

 to the case of unbalanced processes, Bogolyubov approaches 



