LEONID PAVLOVICH SHILNIKOV Memorial Volume

LEONID PAVLOVICH SHILNIKOV Memorial Volume This special issue of the journal presents a selection of the proceeding papers of the conference “Dynamics, Bifurcations and Strange Attractors” dedicated to the memory of Leonid Pavlovich Shilnikov (1934–2011) to commemorate his contribution to the theory of dynamical systems and bifurcations. The conference was held at Nizhny Novgorod State University, Russia, on July 1–5, 2013. The conference was attended by 155 partic- ipants from all over the world, who contributed to the threefocal topics of the conference: Bi- furcations and strange attractors; Dynamical systems with additional structures (Hamil- tonian, time-reversible, etc.); Applications of dynamical systems. The topics were chosen to concur with the pivotal contributions by L.P.Shilnikov to the fields. The speakers pre- sented on their current research and outlined future directions in the theory and frontier applications. The organizers of the conference are grate- ful to its sponsors: Russian Foundation of Basic Research, D. Zimin’s Russian Charitable Foundation “Dynasty”, R&D company Mera- NN, and K. V. Kirsenko (Russia), as well as Office of Naval Research, London (UK). Leonid Pavlovich Shilnikov was a creator of the theory of homoclinic bifurcations and afounderofthetheoryofstrangeattractors. Several decades of his continuous, pioneering work on global bifurcations and strange attractors helped to turn Chaos Theory into amathematicalmarvel.Thebeautifulrigor and ultimate transparency of Shilnikov’s findings have allowed other researchers to use his mathematically sound theorems and develop methods for solving other problems, theoretical and applied, including the understanding of complex outcomes of natural experiments. His works continue to propel forward the further development of the qualitative theory of dynamical systems, and nonlinear dynamics in a whole. Every recent text- or reference book used worldwide by mathematics students and nonlinear dynamists to study dynamical systems includes Shilnikov’s discoveries and theorems. They laid the foundations of the advanced theory of dynamical systems and chaos needed for education of new generations of researchers to understand the complexity of our nonlinear world. His contributions to the theory are pivotal and often treated as scientific folklore, i. e. without acknowledging his original papers. In the opening paper of this volume we start presenting our best attempt to compile a detailed and scientifically and chronologically accurate account of L. P.Shilnikov scientific heritage. Leonid Pavlovich has built the world famous Shilnikov School of dynamical systems and bifurcations in the city of Nizhny Novgorod. For many years he had run the laboratory at the Department of Differential Equations at the Research Institute for Applied Math- ematics and Cybernetics, which was headed by E.A.Leontovich- Andronova. Since 1984 he became the Head of the Department, which soon transformed into an informal science club for former students and minded colleagues. A weekly, two-hour long seminar led by Leonid was always a center of vivid and passionate discussions.To give a presentation at that seminar was considered both an honor and an invaluable experience. In addition to dynamical systems, the scope of the seminar was of a broad range, from algebra to theoretical physics. Shilnikov was a true educator, keen to bringnew ideas and findings in the dynamical system theory to researchers in other fields. He co-organized and delivered lectures at a large number of meetings and workshops on nonlinear dynamics. Leonid Pavlovich became a “global attractor” for colleagues and research fellows from mathematics to physics, biology, neuroscience, chemistry and engineering, and he stayed that way always. Many researchers acknowledge Shilnikov’s ideas and charisma that have vastly influenced their own development, both professional and personal. For nearly four decades Leonid Shilnikov lectured at his alma-matter — Gorky and later Nizhny Novgorod State University. His courses werevery popular with students of mathematics and physics concentrations. Leonid was able to demonstrate the beauty of mathematics to students, which is why the students who had keen interest in sciences, highly valued his lectures and seminars. He was a magnet for strong, motivated students. Anyone who stepped into his attraction basin, sensed the extraordinary atmosphere of the true scientific environment. L.P.Shilnikov was one of the celebrated and influential figures in the theory of dynamical systems. He possessed the pivotal qualities of the exceptional researcher and human being — integrity, ethics, and scientific bravery. He did science his way, by creating new directions and original ap- proaches. Leonid was a wise and caring mentor for his PhD students and scholars at his department, who became active and independent researchers: N.K.Gavrilov, V.S.Afraimovich, A.D.Morozov, L. M. Lerman, Ya. L. Umanskii, V. S. Grines, G. M. Polotovskii, N. V. Roschin, L. A. Belyakov, V. V. Bykov, V. I. Lukyanov, A. N. Bautin, S. V. Gonchenko, E. V. Zhuzhoma, M. I. Malkin, I. M. Ov- syannikov, V. S. Biragov, A. L. Shilnikov, D. V. Turaev, M. V. Shashkov, O. V. Sten’kin, I. V. Belykh, V. S. Gonchenko. Leonid was the leader of the Nizhny Novgorod mathematical community. He helped found the Nizhny Novgorod Mathematical Society, and was its first president. He never missed its meetings, unless he was traveling. L.P.Shilnikov published nearly two hundred research papers and co-authored a few books. Among those, Meth- ods of Qualitative Theory in Nonlinear Dynamics (Parts 1 and 2) with A.L.Shilnikov, D.V.Turaev, and L.Chua (1998, 2001), and its recent translations in Russian (2003, 2009) and Chinese (2010), as well as Bifurcation Theory with V.I.Arnold, V.S.Afraimovich, Yu.S.Ilyashenko, in Russian (1986) and English (1994). In the last papers he returned to his favorite subject — the Lorenz attractor and its multi-dimensional generalizations. Professor Shilnikov’s scientific achievements were ac- knowledged by several awards, including A. M. Lyapunov Award of Russian Academy of Science (1998), M. A. Lavrentiev Award of National Academy of Science of Ukraine (2005), and Professorship of Alexander von Humboldt Foundation of Germany in 2002. He served on editorial boards of many peer-reviewed journals. He was akeynotespeakeratlargenumberofconferencesand workshops held in Russia and all over the world, and was invited to visit and speak at leading research universities in USA, Belgium, France, Israel, Germany, Italy, China. For many years, Leonid’s group had collaborated with the Nobel Prize awardee I.R.Prigogine and his colleagues at Universitélibre de Bruxelles that began with the acclaimed conference “Homoclinic Chaos” (Brussels, 1991) honoring Shilnikov’s contributions to mathematics and nonlinear dynamics. Back at his university years Leonid Pavlovich met his future wife Lyudmila Ivanovna, and they lived together for 55 years. He had been her faithful companion and friend, and for their children and grandchildren remained the considerate and fair family head of an utmost respect and an unquestioned role model. He was a book collector, loved and knew history, particularly history of science. He was an aficionado of football, and summertime fishing on the Volga River was his passion. For each and every of us, Leonid Pavlovich Shilnikov will always remain Teacher, extraordinary Expert and Visionary in science. On behalf of all of the Organizers of the L. P. Shilnikov memorial conference: V. S. Afraimovich, S. V. Gonchenko, L. M. Lerman, A. L. Shilnikov and D. V. Turaev. ISSN 1560-3547, Regular and Chaotic Dynamics, 2014, Vol. 19, No. 4, pp. 435–460. c Pleiades Publishing, Ltd., 2014. ⃝ Scientific Heritage of L. P. Shilnikov Valentin S. Afraimovich 1*,SergeyV.Gonchenko2**,LevM.Lerman2***, Andrey L. Shilnikov2, 3****,andDmitryV.Turaev4***** 1Universidad Autónoma de San Luis Potos´ı, Av. Karakorum 1470, Lomas 4a. San Luis Potos´ı, 78210, México 2Lobachevsky State University of Nizhny Novgorod, pr. Gagarina 23, Nizhny Novgorod, 603950 Russia 3Neuroscience Institute and Department of Mathematics and Statistics, Georgia State University, Atlanta 30303, USA 4Imperial College, SW7 2 AZ London, UK Received July 1, 2014; accepted July 11, 2014 Abstract—This is the first part of a review of the scientific works of L.P. Shilnikov. We group his papers according to 7 major research topics: bifurcations of homoclinic loops; the loop of a saddle-focus and spiral chaos; Poincare homoclinics to periodic orbits and invariant tori, homoclinic in noautonous and infinite-dimensional systems; Homoclinic tangency; Saddle- node bifurcation — quasiperiodicity-to-chaos transition, blue-sky catastrophe; Lorenz attractor; Hamiltonian dynamics. The first two topics are covered in this part. The review will be continued in the further issues of the journal. MSC2010 numbers: 37-01, 37-02, 01A65, 37C29, 37D45 DOI: 10.1134/S1560354714040017 Keywords: Homoclinic chaos, global bifurcations, spiral chaos, strange attractor, saddle-focus, homoclinic loop, saddle-node, saddle-saddle, Lorenz attractor, hyperbolic set Our dear friend, mentor and fellow researcher, Leonid Pavlovich Shilnikov was a creator of the theory of global bifurcations of high-dimensional systems and one of the founders of the mathematical theory of dynamical chaos. He built a profound research school in the city of Nizhny Novgorod (Gorky formerly) – the Shilnikov School that continues to this day. His works

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