Dmitry Turaev

This special issue is dedicated to the anniversaries of two famous Russian mathematicians, Sergey V.Gonchenko and Vladimir N.Belykh. Over the years, they have made a lasting impact in the theory of dynamical systems and applications. In this issue we have collected a series of papers by their friends and colleagues devoted to modern aspects and trends of the theory of dynamical chaos.

We show that the stable invariant foliation of codimension 1 near a zero-dimensional hyperbolic set of a $C^{\beta}$ map with $\beta>1$ is $C^{1+\varepsilon}$ with some $\varepsilon>0$. The result is applied to the restriction of higher regularity maps to normally hyperbolic manifolds. An application to the theory of the Newhouse phenomenon is discussed.

A complete description of dynamics in a neighborhood of a finite bunch of homoclinic loops to a saddle equilibrium state of a Hamiltonian system is given.

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.

A system consisting of a chaotic (billiard-like) oscillator coupled to a linear wave equation in the three-dimensional space is considered. It is shown that the chaotic behavior of the oscillator can cause the transfer of energy from a monochromatic wave to the oscillator, whose energy can grow without bound.

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).

We show that a generic area-preserving two-dimensional map with an elliptic periodic point is $C^\omega$-universal, i.e., its renormalized iterates are dense in the set of all real-analytic symplectic maps of a two-dimensional disk. The results naturally extend onto Hamiltonian and volume-preserving flows.

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.

In connection with the 70th birthday of Professor L.P. Shilnikov, an outstanding scientist and the leader of the famous Nizhny Novgorod Nonlinear Dynamics school, his colleagues and disciples organized the International Conference "Dynamics, Bifurcations and Chaos", which was held on January 31-February 4, 2005 in Nizhny Novgorod, Russia.
This special issue is a collection of research papers which were either contributed by participants of this conference or submitted in reply to a call for papers announced by Editorial Board of RCD in March 2005.

We consider a mode approximation model for the longitudinal dynamics of a multisection semiconductor laser which represents a slow-fast system of ordinary differential equations for the electromagnetic field and the carrier densities. Under the condition that the number of active sections $q$ coincides with the number of critical eigenvalues we introduce a normal form which admits to establish the existence of invariant tori. The case $q=2$ is investigated in more detail where we also derive conditions for the stability of the quasiperiodic regime

We study bifurcations leading to the appearance of elliptic orbits in the case of four-dimensional symplectic diffeomorphisms (and Hamiltonian flows with three degrees of freedom) with a homoclinic tangency to a saddle-focus periodic orbit.

Asymptotic normal forms for equilibria with a triplet of zero characteristic exponents in systems with $\mathbb{Z}_q$-symmetry are listed.

4D-Hamiltonian systems with discrete symmetries are studied. The symmetries under consideration are such that a system possesses two invariant sub-planes which intersect each other transversally at an equilibrium state. The equilibrium state is supposed to to be of saddle type; moreover, in each invariant sub-plane there are two homoclinic loops to the saddle. We establish the existence of stable and unstable invariant manifolds for the bouquet comprised by the four homoclinic trajectories at the Hamiltonian level corresponding to the saddle. These manifolds may intersect transversely along some orbit. We call such a trajectory a super-homoclinic one. We prove that the existence of a super-homoclinic orbit implies the existence of a countable set of multi-pulse homoclinic trajectories to the saddle.