Leonid Pavlovich Shil ’ nikov ( obituary )

A remarkable mathematician, one of the most prominent specialists in the theory of dynamical systems and bifurcation theory, a laureate of the Lyapunov Prize of the Russian Academy of Sciences and of the Lavrent’ev Prize of the National Academy of Sciences of Ukraine, a Humboldt Professor, Head of the Department of Differential Equations of the Research Institute of Applied Mathematics and Cybernetics of Nizhnii Novgorod University, Professor Leonid Pavlovich Shil’nikov passed away on 26 December 2011. Leonid Pavlovich was born on 17 December 1934 in the town of Kotel’nich of Kirov Province, in a family of workers. After graduating from secondary school in 1952 he entered the Physics-Mathematics Faculty of Gor’kii (now Nizhnii Novgorod) State University, from which he graduated in 1957. After completing his post-graduate studies (1957–1960), in 1962 he defended his Ph.D. dissertation, “Birth of periodic motions from singular trajectories”, where he generalized to the multidimensional case the non-local bifurcations that had been studied for systems on the plane by A. A. Andronov and E.A. Leontovich. Soon after that he began studying the theory of systems with complex (chaotic) dynamics, which was just then coming into existence. By that time, the example of Smale’s horseshoe had already appeared (1961), and D. V. Anosov’s note on geodesic flows had been published (1962), where the notion of hyperbolicity was formulated and its primary significance was revealed. In developing the results of his dissertation Shil’nikov discovered (Dokl. Akad. Nauk SSSR, 1965) that infinitely many Smale horseshoes, and hence a complex dynamics, exist in a neighbourhood of a homoclinic loop of a saddle-focus. Today we know that a Shil’nikov loop determines the chaotic dynamics of a wide range of models in various areas of science. But at that time the result that a complex dynamics can arise near such a simple and familiar object as a separatrix loop was totally unexpected. Shil’nikov realized that the study of homoclinic bifurcations gave him a unique opportunity for studying complex dynamics of multidimensional systems, and this became a central subject of his scientific activities.