Some questions of the theory of differential equations with a small parameter (Q1091531)

For the system (1) \(\epsilon\dot x=f(x,y)\), \(\dot y=g(x,y)\) where x is a quick and y a slow vector the following problem is regarded: tends the solution (2) \(x=\phi (t,\epsilon)\), \(y=\psi (t,\epsilon)\) of the system (1) to a definite limit? The studying of the behaviour of quick vector led to the consideration of the system (3) \({\dot \epsilon}=f(x,y)\) where y is a constant parameter. Particular attention is given to the asymptotic calculation of relaxational oscillations - periodic solutions of the system (1) whose trajectories contain sections of slow changes of phase variables and sections of quick motions. The formulation and solution of new problems, which cover and explain the most subtle asymptotic phenomena connected with transients, are regarded.