Andronov School of Nonlinear Oscillations

Andronov's school began to take shape in 1931, when Alexander Alexandrovich himself, together with his wife E.A. Leontovich, moved from Moscow to Nizhny Novgorod. By the time of the move, A.A. Andronov was an established scientist. Even then, he introduced a number of new concepts into science, including self-oscillations, concepts of the roughness of the system, the bifurcation value of the parameter, the phase portrait, and so on. This is a long-lived school in which a unified scientific program has been actively developed by several generations of scientists. In my report, I will touch upon the scientific direction of the school, which is associated with rough (structurally stable) dynamic systems. The simplest of them - "Morse-Smale systems" got their name after the publication of S. Smale's work "On gradient dynamical system // Ann. Math. 74, 1961, P.199-206". He introduced a class of flows on manifolds of arbitrary dimension that copy the properties of coarse flows on the plane described in 1937 by A. Andronov and L. Pontryagin. For the introduced streams With . Smale proved the validity of inequalities similar to Morse inequalities for non-degenerate functions, after which such flows were called Morse-Smale flows. S. Smale considered it extremely important to study such flows, since he assumed that, by analogy with coarse flows on the plane, Morse-Smale flows exhaust the class of structurally stable flows on manifolds and are dense in the set all threads. Fortunately, it turned out that the multidimensional structurally stable world is much wider, and the Morse-Smale systems represent only its regular part - structurally stable systems with a non-wandering set consisting of a finite number of orbits. Due to the close connection of Morse-Smale systems with the supported manifold, various topological objects, including wild ones, are realized as invariant subsets of such systems. This leads to a wide variety of Morse-Smale systems (especially on multidimensional manifolds) and, accordingly, complicates their topological classification.