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# Volume 11, Number 2, 2006 On the 70th birthday of L.P. Shilnikov

 Gonchenko S. V.,  Lerman L. M.,  Turaev D. V. Leonid Pavlovich Shilnikov. On his 70th birthday Abstract 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. Citation: Gonchenko S. V.,  Lerman L. M.,  Turaev D. V., Leonid Pavlovich Shilnikov. On his 70th birthday , Regular and Chaotic Dynamics, 2006, vol. 11, no. 2, pp. 139-140 DOI: 10.1070/RD2006v011n02ABEH000340
 Belitskii G. R.,  Tkachenko V. A. Moduli of local one-dimensional dynamical systems at a hyperbolic fixed point Abstract 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 Keywords: local diffeomorphisms, local flows, smooth conjugacy, hyperbolic fixed point, functional moduli Citation: Belitskii G. R.,  Tkachenko V. A., Moduli of local one-dimensional dynamical systems at a hyperbolic fixed point , Regular and Chaotic Dynamics, 2006, vol. 11, no. 2, pp. 141-154 DOI: 10.1070/RD2006v011n02ABEH000341
 Broer H. W.,  Naudot V.,  Roussarie R.,  Saleh K. A predator-prey model with non-monotonic response function Abstract 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. Keywords: predator-prey dynamics, organizing center, bi-furcation, strange attractor Citation: Broer H. W.,  Naudot V.,  Roussarie R.,  Saleh K., A predator-prey model with non-monotonic response function , Regular and Chaotic Dynamics, 2006, vol. 11, no. 2, pp. 155-165 DOI: 10.1070/RD2006v011n02ABEH000342
 Bruening J.,  Dobrokhotov S. Y.,  Semenov E. S. Unstable closed trajectories, librations and splitting of the lowest eigenvalues in quantum double well problem Abstract 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. Keywords: quantum double well problem, librations, splitting value, tunneling effect, normal forms, complex phase space Citation: Bruening J.,  Dobrokhotov S. Y.,  Semenov E. S., Unstable closed trajectories, librations and splitting of the lowest eigenvalues in quantum double well problem , Regular and Chaotic Dynamics, 2006, vol. 11, no. 2, pp. 167-180 DOI: 10.1070/RD2006v011n02ABEH000343
 Du B.,  Li M.,  Malkin M. I. Topological horseshoes for Arneodo–Coullet–Tresser maps Abstract 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. Keywords: topological horseshoe, full shift, polynomial maps, generalized Hénon maps, nonwandering set, inverse limit, topological entropy Citation: Du B.,  Li M.,  Malkin M. I., Topological horseshoes for Arneodo–Coullet–Tresser maps , Regular and Chaotic Dynamics, 2006, vol. 11, no. 2, pp. 181-190 DOI: 10.1070/RD2006v011n02ABEH000344
 Gonchenko S. V.,  Meiss J. D.,  Ovsyannikov I. I. Chaotic dynamics of three-dimensional Hénon maps that originate from a homoclinic bifurcation Abstract 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 Keywords: saddle-focus fixed point, three-dimensional quadratic map, homoclinic bifurcation, strange attractor Citation: Gonchenko S. V.,  Meiss J. D.,  Ovsyannikov I. I., Chaotic dynamics of three-dimensional Hénon maps that originate from a homoclinic bifurcation , Regular and Chaotic Dynamics, 2006, vol. 11, no. 2, pp. 191-212 DOI: 10.1070/RD2006v011n02ABEH000345
 Gonchenko S. V.,  Schneider K. R.,  Turaev D. V. Quasiperiodic regimes in multisection semiconductor lasers Abstract 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 Keywords: multisection semiconductor laser, averaging, mode approximation, invariant torus, normal form, stability Citation: Gonchenko S. V.,  Schneider K. R.,  Turaev D. V., Quasiperiodic regimes in multisection semiconductor lasers , Regular and Chaotic Dynamics, 2006, vol. 11, no. 2, pp. 213-224 DOI: 10.1070/RD2006v011n02ABEH000346
 Grines V. Z.,  Zhuzhoma V. S. Expanding attractors Abstract 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 Keywords: Axiom A diffeomorphisms, (codimension one) expanding attractors, structurally stable diffeomorphisms, hyperbolic automorphisms Citation: Grines V. Z.,  Zhuzhoma V. S., Expanding attractors , Regular and Chaotic Dynamics, 2006, vol. 11, no. 2, pp. 225-246 DOI:10.1070/RD2006v011n02ABEH000347
 Homburg A. J.,  Young T. Hard bifurcations in dynamical systems with bounded random perturbations Abstract 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. Keywords: bifurcations, random perturbations Citation: Homburg A. J.,  Young T., Hard bifurcations in dynamical systems with bounded random perturbations , Regular and Chaotic Dynamics, 2006, vol. 11, no. 2, pp. 247-258 DOI: 10.1070/RD2006v011n02ABEH000348
 Karabanov A. A.,  Morozov A. D. To the theory of coupled oscillations passing through the resonance Abstract 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. Keywords: near-integrable systems, coupled oscillations, resonances, self-synchronization Citation: Karabanov A. A.,  Morozov A. D., To the theory of coupled oscillations passing through the resonance , Regular and Chaotic Dynamics, 2006, vol. 11, no. 2, pp. 259-268 DOI: 10.1070/RD2006v011n02ABEH000349
 Kirillov A. A. Modification of gravity and Dark Matter Abstract 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 Keywords: galaxy formation, clusters, dark matter Citation: Kirillov A. A., Modification of gravity and Dark Matter , Regular and Chaotic Dynamics, 2006, vol. 11, no. 2, pp. 269-280 DOI: 10.1070/RD2006v011n02ABEH000350
 Leonov G. A. Generalization of the Andronov–Vitt theorem Abstract 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. Keywords: Orbital stability, Zhukovski stability, Characteristic exponent, Poincaré section Citation: Leonov G. A., Generalization of the Andronov–Vitt theorem , Regular and Chaotic Dynamics, 2006, vol. 11, no. 2, pp. 281-289 DOI: 10.1070/RD2006v011n02ABEH000351
 Lerman L. M. Partial normal form near a saddle of a Hamiltonian system Abstract 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 Keywords: Hamiltonian, saddle, normal form, symplectic transformation, invariant manifold Citation: Lerman L. M., Partial normal form near a saddle of a Hamiltonian system , Regular and Chaotic Dynamics, 2006, vol. 11, no. 2, pp. 291-297 DOI: 10.1070/RD2006v011n02ABEH000352
 Novokshenov V. Y. Adiabatic deformations of integrable two-frequency Hamiltonians Abstract 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. Keywords: integrable Hamiltonian system, Lax pairs, perturbation theory, adiabatic invariants, autoresonance, Whitham equations Citation: Novokshenov V. Y., Adiabatic deformations of integrable two-frequency Hamiltonians , Regular and Chaotic Dynamics, 2006, vol. 11, no. 2, pp. 299-310 DOI: 10.1070/RD2006v011n02ABEH000353
 Pilyugin S. Y. Transversality and local inverse shadowing Abstract 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. Keywords: shadowing, heteroclinic trajectory, transversality Citation: Pilyugin S. Y., Transversality and local inverse shadowing , Regular and Chaotic Dynamics, 2006, vol. 11, no. 2, pp. 311-318 DOI: 10.1070/RD2006v011n02ABEH000354
 Sharkovsky A. N.,  Romanenko E. Y.,  Fedorenko V. V. One-dimensional bifurcations in some infinite-dimensional dynamical systems and ideal turbulence Abstract 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. Keywords: dynamical system, boundary value problem, difference equation, one-dimensional map, bifurcation, ideal turbulence, fractal, random process Citation: Sharkovsky A. N.,  Romanenko E. Y.,  Fedorenko V. V., One-dimensional bifurcations in some infinite-dimensional dynamical systems and ideal turbulence , Regular and Chaotic Dynamics, 2006, vol. 11, no. 2, pp. 319-328 DOI:10.1070/RD2006v011n02ABEH000355
 Zaslavsky G. M.,  Edelman M. Stickiness of trajectories in a perturbed Anosov system Abstract 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 Keywords: Anosov systems, sticky trajectories, Poincaré recurrences Citation: Zaslavsky G. M.,  Edelman M., Stickiness of trajectories in a perturbed Anosov system , Regular and Chaotic Dynamics, 2006, vol. 11, no. 2, pp. 329-336 DOI: 10.1070/RD2006v011n02ABEH000356

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