Lev Lerman

Scientific heritage of L.P. Shilnikov

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; Saddlenode 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.

Breakdown of Symmetry in Reversible Systems

We review results on local bifurcations of codimension 1 in reversible systems (flows and diffeomorphisms) which lead to the birth of attractor-repeller pairs from symmetric equilibria (for flows) or periodic points (for diffeomorphisms).

Separatrix splitting at a Hamiltonian 02iω bifurcation

We study the splitting of a separatrix in a generic unfolding of a degenerate equilibrium in a Hamiltonian system with two degrees of freedom. We assume that the unperturbed fixed point has two purely imaginary eigenvalues and a non-semisimple double zero one. It is well known that a one-parameter unfolding of the corresponding Hamiltonian can be described by an integrable normal form. The normal form has a normally elliptic invariant manifold of dimension two. On this manifold, the truncated normal form has a separatrix loop. This loop shrinks to a point when the unfolding parameter vanishes. Unlike the normal form, in the original system the stable and unstable separatrices of the equilibrium do not coincide in general. The splitting of this loop is exponentially small compared to the small parameter. This phenomenon implies nonexistence of single-round homoclinic orbits and divergence of series in normal form theory. We derive an asymptotic expression for the separatrix splitting. We also discuss relations with the behavior of analytic continuation of the system in a complex neighborhood of the equilibrium.

Symmetric Homoclinic Orbits at the Periodic Hamiltonian Hopf Bifurcation

We prove the existence of symmetric homoclinic orbits to a saddle-focus symmetric periodic orbit that appears in a generic family of reversible three degrees of freedom Hamiltonian system due to periodic Hamiltonian Hopf bifurcation, if some coefficient A of the normal form of the fourth order is positive. If this coefficient is negative, then for the opposite side of the bifurcation parameter value, we prove the existence of symmetric homoclinic orbits to saddle invariant 2-tori.

Editorial — Leonid Pavlovich Shilnikov

This special issue presents a selection of papers from the conference “Dynamics, Bifurcations and Strange Attractors” dedicated to the memory of Leonid Pavlovich Shilnikov (1934–2011) to commemorate his contributions to the theory of dynamical systems and bifurcations. The conference was held at the Lobachevsky State University of Nizhny Novgorod, Russia, on 1–5 July 2013. The conference was attended by 155 participants from all over the world, who contributed to the three focal topics: bifurcations and strange attractors; dynamical systems with additional structures (Hamiltonian, time-reversible, etc.); applications of dynamical systems. The topics were chosen in confluence with pivotal contributions by L. P. Shilnikov to the fields. The speakers presented their current research and outlined future directions in both theory and frontier applications. The organizers of the conference are grateful 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 (USA) and its officers, Drs. M. Harper (UK) and M. Shlesinger (USA). We thank the Editors-in-Chief of the International Journal of Bifurcations and Chaos: Ron Chen and Leon Chua for having the proceedings published here. L. P. Shilnikov served on the Editorial Board of the journal from the time it was founded. Our dear friend, mentor and fellow researcher, L. P. Shilnikov conceptualized the theory of global bifurcations of high-dimensional systems and was 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 greatly influenced the overall development of the mathematical theory of dynamical systems as well as nonlinear dynamics, in general. Shilnikov’s findings have been included in most textand reference books, and 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 “the heart of the matter,” and provide applied researchers with an in-depth mathematical understanding of the outcomes of natural experiments. The popularity and appreciation are due to the “living classic” status attained by Professor Shilnikov over several decades of his life through continuous hard work on bifurcation theory of multidimensional dynamical systems, mathematical chaos theory and theory of strange attractors. L. P. Shilnikov was born in Kotelnich, Kirov region of Russia on December 17, 1934. After graduating from a local high school in 1952, he became a student in the Department of Physics and Mathematics at Gorky State University. After graduation in 1957, he continued his PhD studies at the same university. He defended his PhD thesis “On birth of stable periodic orbits from singular trajectories” in 1962, it focused on the multidimensional generalization of basic homoclinic bifurcations, which were originally discovered and studied for systems on a plane by A. A. Andronov and E. A. Leontovich in the early 1930s.

A Saddle in a Corner—A Model of Collinear Triatomic Chemical Reactions

A geometrical model which captures the main ingredients governing atom-diatom collinear chemical reactions is proposed. This model is neither near-integrable nor hyperbolic, yet it is amenable to analysis using a combination of the recently developed tools for studying systems with steep potentials and the study of the phase space structure near a center-saddle equilibrium. The nontrivial dependence of the reaction rates on parameters, initial conditions, and energy is thus qualitatively explained. Conditions under which the phase space transition state theory assumptions are satisfied and conditions under which they fail are derived.

Dynamic systems and soliton states of completely integrable field equations

The dynamics of completely integrable field equations (as sine-Gordon, nonlinear Schroedinger, etc.) are studied by methods of the qualitative theory of dynamic systems. This permits one to predict the existence of important new solutions. e.g., solitons of the nonzero vacuum. These solutions are constructed by various method, and explicit formulas have been derived for them.