Let a system of differential equations possess a saddle periodic orbit such that every orbit in its unstable manifold is homoclinic, i.e. the unstable manifold is a subset of the (global) stable manifold. We study several bifurcation cases of the breakdown of such a homoclinic connection that causes the blue sky catastrophe, as well as the onset of complex dynamics. The birth of an invariant torus and a Klein bottle is also described.

We derive sufficient conditions for the existence of an invariant set in an absorbing region homeomorphic to the product of a multidimensional torus and a ball. This set consists of low dimensional tori labeled by symbolic sequences. It may appear as a result of the breakdown of an attracting multidimensional torus. Trajectories on the set manifest chaotic behavior for some angular coordinates and may behave regularly for others, i.e. the dynamics on the set is of the chimera state type.

К вопросу о сценариях возникновения хаоса у трехмерных отображений А. С

On 7 October 2010 Romen Vasil’evich Plykin, a leading expert on the qualitative theory of dynamical systems, passed away. He was born on 5 July 1935, in Tambov. His father, the Bulgarian political émigré Christo Pakov, was a well-known aviator, the commander of the aviation unit of Glavsevmorput’. During World War II he commanded an air squadron at the front, but was arrested in 1943. His family, who had been evacuated, received no information about him and believed him to be dead. Some time later Romen’s mother married again, and he got the family name and patronymic after his stepfather. After being freed from a Soviet forced-labour camp in the GULAG system, Pakov could not find his family and returned to Bulgaria. It was not until 1958 that the father and the son resumed relations. In 1953 Plykin graduated from a high school in Urgench (in Uzbekistan) and enrolled in the Faculty of Physics and Mathematics of the Central Asia State University, from which he graduated in 1958 with the qualification of a mathematician. After that he taught at the Tashkent Military School and Tashkent University. In 1967 he moved to Kalinin (now Tver’), where he was a docent in the Department of Numerical Mathematics of the Polytechnic Institute, and in 1969 he settled in Obninsk, where till the end of life he remained a professor at the Obninsk Branch of the Moscow Institute for Physics and Engineering (the Obninsk Institute for Atomic Power Engineering since 1985). The first published mathematics papers of Plykin are concerned with general topology. On their basis he wrote his Ph.D. thesis, “Metrization and completeness over semifields”, which he defended in 1964 (his advisor was M.Ya. Antonovskii). In this period his acquaintance with Professor Yu.M. Smirnov of Moscow State University had a strong influence on Plykin. He regarded Smirnov as one of his teachers and maintained close relations with him until Smirnov’s death.

On December 17, 2009 we all raised our glasses to celebrate the 75th anniversary of Leonid Pavlovich Shilnikov, our dear friend, mentor and fellow researcher, creator of the homoclinic bifurcation theory for high-dimensional dynamical systems. His works greatly influenced the overall development of mathematical theory of dynamical systems, as well as nonlinear dynamics in general. Shilnikov’s findings have became the classics, and been included in the most textand reference books which are used worldwide by mathematics students and nonlinear dynamists to study the qualitative theory of dynamical systems and chaos. The elegance and completeness of his results let them reach to “the heart of the matter,” and provide applied researchers with in-depth mathematical understanding of outcomes of natural experiments. No doubt that this popularity is due the status of “a living classic” that Professor Shilnikov has attained over several decades for his continuous hard works on the bifurcation theory of multi-dimensional dynamical systems, mathematical chaos theory and theory of strange attractors. Out of his so many fundamental achievements, here we will brief on a few, the key ones in our view. We will begin with his works on the theory of global bifurcations in multi-dimensional dynamical systems, the works which had built the foundation for the theory. The basics of bifurcations for systems in a plane had originally been discovered and studied by A. A.Andronov and E. A. Leontovich as early as in the 1930s. Among them, of special, for this story, interest are two nonlocal bifurcations that occur in the system with a homoclinic loop of either a saddle or a saddle-node equilibrium state. In the late 1950s and early 1960s L. P. Shilnikov studied highdimensional versions of these bifurcations, and he identified the cases, for which the breakdown of the homoclinic loop would lead to the emergence of a singe periodic trajectory. This research direction was held yet along the traditional lines drawn by the Andronov School in Gorky.

We review bifurcations of homoclinic tangencies leading to Hénon-like maps of various kinds.

We study dynamics and bifurcations of three-dimensional diffeomorphisms with nontransverse heteroclinic cycles. We show that bifurcations under consideration lead to the birth of wild-hyperbolic Lorenz attractors. These attractors can be viewed as periodically perturbed classical Lorenz attractors, however, they allow for the existence of homoclinic tangencies and, hence, wild hyperbolic sets.

The phenomenon of the generic coexistence of infinitely many periodic orbits with different numbers of positive Lyapunov exponents is analysed. Bifurcations of periodic orbits near a homoclinic tangency are studied. Criteria for the coexistence of infinitely many stable periodic orbits and for the coexistence of infinitely many stable invariant tori are given.

Пусть C r -гладкий, r 5, двумерный диффеоморфизм f имеет негрубый гетероклинический контур, содержащий несколько седловых периодических и гетероклинических траекторий, причем среди последних есть негрубые, в точках которых инвариантные многообразия соответствующих сёдел пересекаются нетрансверсально.Предположим, что контур содержит по крайней мере две такие седловые периодические траектории, что седловая величина (модуль произведения мультипликаторов) одной из них меньше 1, а другойбольше 1. Тогда, как показано в работе, в любой окрестности, в C r -топологии, диффеоморфизма f в пространстве C r -гладких диффеоморфизмов существуют области (области Ньюхауса с гетероклиническими касаниями), в которых плотны диффеоморфизмы, имеющие одновременно счетное множество устойчивых и неустойчивых замкнутых инвариантных кривых.Для случая трехмерных потоков этот результат означает существование областей Ньюхауса, в которых плотны потоки со счетным множеством устойчивых и неустойчивых двумерных инвариантных торов.Ключевые слова: негрубый

We study the semilocal dynamics of two-dimensional symplectic diffeomorphisms with homoclinic tangencies. Conditions for the existence of infinitely many generic elliptic periodic orbits of successive periods starting with some integer are found. Bibliography: 14 titles.

A few mathematical problems arising in the classical synchronization theory are discussed; especially those relating to complex dynamics. The roots of the theory originate in the pioneering experiments by van der Pol and van der Mark, followed by the theoretical studies done by Cartwright and Littlewood. Today we focus specifically on the problem on a periodically forced stable limit cycle emerging from a homoclinic loop to a saddle point. Its analysis allows us to single out the regions of simple and complex dynamics, as well as to yield a comprehensive description of bifurcational phenomena in the two-parameter case. Of a particular value among ones is the global bifurcation of a saddle-node periodic orbit. For this bifurcation, we

Structurally Stable Systems Bifurcations of Dynamical Systems The Behavior of Dynamical Systems on Stability Boundaries of Equilibrium States The Behavior of Dynamical Systems on Stability Boundaries of Periodic Trajectories Local Bifurcations on the Route Over Stability Boundaries Global Bifurcations at the Disappearance of a Saddle-Node Equilibrium States and Periodic Orbits Bifurcations of Homoclinic Loops of Saddle Equilibrium States Safe and Dangerous Boundaries.