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.

Functional moduli of a smooth conjugacy of local $C^1$-diffeomorphisms and $C^1$-flows at a hyperbolic fixed point are introduced. Conditions for the smooth linearization, existence of iterative roots and embeddability in terms of these moduli are found

We study the dynamics of a family of planar vector fields that models certain populations of predators and their prey. This model is adapted from the standard Volterra–Lotka system by taking into account group defense, competition between prey and competition between predators. Also we initiate computer-assisted research on time-periodic perturbations, which model seasonal dependence.
We are interested in persistent features. For the planar autonomous model this amounts to structurally stable phase portraits. We focus on the attractors, where it turns out that multi-stability occurs. Further, we study the bifurcations between the various domains of structural stability. It is possible to fix the values of two of the parameters and study the bifurcations in terms of the remaining three. We find several codimension 3 bifurcations that form organizing centers for the global bifurcation set.
Studying the time-periodic system, our main interest is the chaotic dynamics. We plot several numerical examples of strange attractors.

We discuss the structure of asymptotic splitting formula for the lowest eigenvalues of multidimensional quantum double well problem. We show that the change of instanton by closed unstable trajectory of appropriate Hamiltonian system gives more natural and simpler preexponential factor (amplitude) in splitting formula. The projection of this trajectories onto configuration space are well know librations in classical mechanics.

In this paper, we study the family of Arneodo–Coullet–Tresser maps $F(x,y,z)=(ax-b(y-z)$, $bx+a(y-z)$, $cx-dxk+e z)$ where $a$, $b$, $c$, $d$, $e$ are real parameters with $bd \ne 0$ and $k>1$ is an integer. We find regions of parameters near anti-integrable limits and near singularities for which there exist hyperbolic invariant sets such that the restriction of $F$ to these sets is conjugate to the full shift on two or three symbols.

We study bifurcations of a three-dimensional diffeomorphism, $g_0$, that has a quadratic homoclinic tangency to a saddle-focus fixed point with multipliers $(\lambda e^{i\varphi}, \lambda e^{-i\varphi}, \gamma)$, where $0<\lambda<1<|\gamma|$ and $|\lambda^2 \gamma|=1$. We show that in a three-parameter family, $g_\varepsilon$, of diffeomorphisms close to $g_0$, there exist infinitely many open regions near $\varepsilon=0$ where the corresponding normal form of the first return map to a neighborhood of a homoclinic point is a three-dimensional Hénon-like map. This map possesses, in some parameter regions, a ''wild-hyperbolic'' Lorenz-type strange attractor. Thus, we show that this homoclinic bifurcation leads to a strange attractor. We also discuss the place that these three-dimensional Hénon maps occupy in the class of three-dimensional quadratic maps with constant Jacobian

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

The article is a survey on local and global structures (including classification results) of expanding attractors of diffeomorphisms $f : M \to M$ of a closed smooth manifold $M$. Beginning with the most familiar expanding attractors (Smale solenoid; DA-attractor; Plykin attractor; Robinson–Williams attractors), one reviews the Williams theory, Bothe's classification of one-dimensional solenoids in 3-manifolds, Grines–Plykin–Zhirov's classification of one-dimensional expanding attractors on surfaces, and Grines–Zhuzhoma's classification of codimension one expanding attractors of structurally stable diffeomorphisms. The main theorems are endowed with ideas of proof

We study bifurcations in dynamical systems with bounded random perturbations. Such systems, which arise quite naturally, have been nearly ignored in the literature, despite a rich body of work on systems with unbounded, usually normally distributed, noise. In systems with bounded random perturbations, new kinds of bifurcations that we call 'hard' may happen and in fact do occur in many situations when the unperturbed deterministic systems experience elementary, codimension-one bifurcations such as saddle-node and homoclinic bifurcations. A hard bifurcation is defined as discontinuous change in the density function or support of a stationary measure of the system.

The problem of coupled oscillations is considered in the case when a stable equilibrium of a globally averaged system passes through a resonance curve. Questions of persistence of invariant tori and transition to a self-synchronization are particularly discussed.

Upon a phenomenological consideration of possible modifications of gravity we introduce a bias operator $ρ_{DM} =\hat{K}ρ_{vis}$. We show that the empirical definition of a single bias function $K_{emp}(r,t)$ (i.e., of the kernel for the bias operator) allows to account for all the variety of Dark Matter halos in astrophysical systems. For every discrete source such a bias produces a specific correction to the Newton's potential $\phi=-GM(1/r+F(r,t))$ and therefore all DM effects can be explained as a modification of the gravity law. We also demonstrate that a specific choice of the bias $K \sim 1/r^2$ (which produces a logarithmic correction to the Newton's law $F \sim -\ln r$) shows quite a good qualitative agreement with the observed picture of the modern Universe

A definition of Zhukovski stability is introduced. A new research tool— a moving Poincaré section— is considered. With the help of this tool, generalizations of the Andronov–Vitt and Demidovich theorems are obtained.

For a smooth or real analytic Hamiltoniain vector field with two degrees of freedom we derive a local partial normal form of the vector field near a saddle equilibrium (two pairs of real eigenvalues $\pm \lambda_1$, $\pm \lambda_2$, $\lambda_1 > \lambda_2 > 0$). Only a resonance $\lambda_1 = n \lambda_2$ (if is present) influences on the normal form. This form allows one to get convenient almost linear estimates for solutions of the vector field using the Shilnikov's boundary value problem. Such technique is used when studying the orbit behavior near homoclinic orbits to saddle equilibria in a Hamiltonian system. The form obtained depends smoothly on parameters, if the vector field smoothly depends on parameters

The paper discusses a mechanism of excitation and control of two-frequency oscillations in the integrable Hamiltonian systems. It is close to the autoresonant technique for controlling the amplitude of nonlinear modes. Autoresonance is usually associated with single frequency mode excitations due to the synchronization and phase lock of various nonlinear modes with the driving force. Despite this we propose a model of multifrequency autoresonance which occur in completely integrable systems. This phenomenon is due to a number stable invariant tori governed by integrals of motion of the integrable system. The basic autoresonant effect of phase locking appears here as Whitham deformations of the invariant tori. This provides also a possibility to transfer a certain initial $n$-periodic motion to a given $m$-periodic motion as a final state.

The inverse shadowing property of a dynamical system means that, given a family of approximate trajectories, for any real trajectory we can find a close approximate trajectory from the given family. This property is of interest when we study dynamical systems numerically. In this paper, we describe some relations between the transversality of a heteroclinic trajectory of a diffeomorphism and the local inverse shadowing property for this heteroclinic trajectory.

Many effects of real turbulence can be observed in infinite-dimensional dynamical systems induced by certain classes of nonlinear boundary value problems for linear partial differential equations. The investigation of such infinite-dimensional dynamical systems leans upon one-dimensional maps theory, which allows one to understand mathematical mechanisms of the onset of complex structures in the solutions of the boundary value problems. We describe bifurcations in some infinite-dimensional systems, that result from bifurcations of one-dimensional maps and cause the relatively new mathematical phenomenon—ideal turbulence.

We consider a perturbation of the Anosov-type system, which leads to the appearance of a hierarchical set of islands-around-islands. We demonstrate by simulation that the boundaries of the islands are sticky to trajectories. This phenomenon leads to the distribution of Poincaré recurrences with power-like tails in contrast to the exponential distribution in the Anosov-type systems