Volume 15, Number 23
Volume 15, Number 23, 2010
On the 75th birthday of Professor L.P. Shilnikov
Wittig A., Berz M., Grote J., Makino K., Newhouse S.
Rigorous and accurate enclosure of invariant manifolds on surfaces
Abstract
Knowledge about stable and unstable manifolds of hyperbolic fixed points of certain maps is desirable in many fields of research, both in pure mathematics as well as in applications, ranging from forced oscillations to celestial mechanics and space mission design. We present a technique to find highly accurate polynomial approximations of local invariant manifolds for sufficiently smooth planar maps and rigorously enclose them with sharp interval remainder bounds using Taylor model techniques. Iteratively, significant portions of the global manifold tangle can be enclosed with high accuracy. Numerical examples are provided.

Afraimovich V. S., Bunimovich L. A., Moreno S. V.
Dynamical networks: continuous time and general discrete time models
Abstract
Dynamical networks are characterized by 1) their topology (structure of the graph of interactions among the elements of a network); 2) the interactions between the elements of the network; 3) the intrinsic (local) dynamics of the elements of the network. A general approach to studying the commulative effect of all these three factors on the evolution of networks of a very general type has been developed in [1]. Besides, in this paper there were obtained sufficient conditions for a global stability (generalized strong synchronization) of networks with an arbitrary topology and the dynamics which is a composition (action of one after another) of a local dynamics of the elements of a network and of the interactions between these elements. Here we extend the results of [1] on global stability (generalized strong synchronization) to the case of a general dynamics in discrete time dynamical networks and to general dynamical networks with continuous time.

Belykh I., Jalil S., Shilnikov A. L.
Burstduration mechanism of inphase bursting in inhibitory networks
Abstract
We study the emergence of inphase and antiphase synchronized rhythms in bursting networks of Hodgkin–Huxley–type neurons connected by inhibitory synapses.We show that when the state of the individual neuron composing the network is close to the transition from bursting into tonic spiking, the appearance of the network’s synchronous rhythms becomes sensitive to small changes in parameters and synaptic coupling strengths. This burstingspiking transition is associated with codimensionone bifurcations of a saddlenode limit cycle with homoclinic orbits, first described and studied by Leonid Pavlovich Shilnikov. By this paper, we pay tribute to his pioneering results and emphasize their importance for understanding the cooperative behavior of bursting neurons. We describe the burstduration mechanism of inphase synchronized bursting in a network with strong repulsive connections, induced by weak inhibition. Through the stability analysis, we also reveal the dual property of fast reciprocal inhibition to establish in and antiphase synchronized bursting.

Gelfreich V. G., Turaev D. V.
Universal dynamics in a neighborhood of a generic elliptic periodic point
Abstract
We show that a generic areapreserving twodimensional map with an elliptic periodic point is $C^\omega$universal, i.e., its renormalized iterates are dense in the set of all realanalytic symplectic maps of a twodimensional disk. The results naturally extend onto Hamiltonian and volumepreserving flows.

Gonchenko S. V., Li M.
Shilnikov’s crossmap method and hyperbolic dynamics of threedimensional Hénonlike maps
Abstract
We study the hyperbolic dynamics of threedimensional quadratic maps with constant Jacobian the inverse of which are again quadratic maps (the socalled 3D Hénon maps). We consider two classes of such maps having applications to the nonlinear dynamics and find certain sufficient conditions under which the maps possess hyperbolic nonwandering sets topologically conjugating to the Smale horseshoe. We apply the socalled Shilnikov’s crossmap for proving the existence of the horseshoes and show the existence of horseshoes of various types: (2,1) and (1,2)horseshoes (where the first (second) index denotes the dimension of stable (unstable) manifolds of horseshoe orbits) as well as horseshoes of saddle and saddlefocus types.

Grines V. Z., Pochinka O. V.
Energy functions for dynamical systems
Abstract
The paper contains exposition of results devoted to the existence of an energy functions for dynamical systems.

Lerman L. M.
Breaking hyperbolicity for smooth symplectic toral diffeomorphisms
Abstract
We study a smooth symplectic 2parameter unfolding of an almost hyperbolic diffeomorphism on twodimensional torus. This diffeomorphism has a fixed point of the type of the degenerate saddle. In the parameter plane there is a bifurcation curve corresponding to the transition from the degenerate saddle into a saddle and parabolic fixed point, separatrices of these latter points form a channel on the torus. We prove that a saddle period2 point exists for all parameter values close to the codimension two point whose separatrices intersect transversely the boundary curves of the channel that implies the existence of a quadratic homoclinic tangency for this period2 point which occurs along a sequence of parameter values accumulating at the codimension 2 point. This leads to the break of stable/unstable foliations existing for almost hyperbolic diffeomorphism. Using the results of Franks [1] on $\pi_1$diffeomorphisms, we discuss the possibility to get an invariant Cantor set of a positive measure being nonuniformly hyperbolic.

Li M., Malkin M. I.
Approximation of entropy on hyperbolic sets for onedimensional maps and their multidimensional perturbations
Abstract
We consider piecewise monotone (not necessarily, strictly) piecewise $C^2$ maps on the interval with positive topological entropy. For such a map $f$ we prove that its topological entropy $h_{top}(f)$ can be approximated (with any required accuracy) by restriction on a compact strictly $f$invariant hyperbolic set disjoint from some neighborhood of prescribed set consisting of periodic attractors, nonhyperbolic intervals and endpoints of monotonicity intervals. By using this result we are able to generalize main theorem from [1] on chaotic behavior of multidimensional perturbations of solutions for difference equations which depend on two variables at nonperturbed value of parameter.

Delshams A., Gutierrez P., Koltsova O. Y., Pacha J. R.
Transverse intersections between invariant manifolds of doubly hyperbolic invariant tori, via the Poincaré–Mel’nikov method
Abstract
We consider a perturbation of an integrable Hamiltonian system having an equilibrium point of elliptic–hyperbolic type, having a homoclinic orbit. More precisely, we consider an $(n+2)$degreeoffreedom near integrable Hamiltonian with $n$ centers and 2 saddles, and assume that the homoclinic orbit is preserved under the perturbation. On the center manifold near the equilibrium, there is a Cantorian family of hyperbolic KAM tori, and we study the homoclinic intersections between the stable and unstable manifolds associated to such tori. We establish that, in general, the manifolds intersect along transverse homoclinic orbits. In a more concrete model, such homoclinic orbits can be detected, in a first approximation, from nondegenerate critical points of a Mel’nikov potential. We provide bounds for the number of transverse homoclinic orbits using that, in general, the potential will be a Morse function (which gives a lower bound) and can be approximated by a trigonometric polynomial (which gives an upper bound).

Gardini L., Tramontana F.
Snapback repellers in nonsmooth functions
Abstract
In this work we consider the homoclinic bifurcations of expanding periodic points. After Marotto, when homoclinic orbits to expanding periodic points exist, the points are called snapbackrepellers. Several proofs of the existence of chaotic sets associated with such homoclinic orbits have been given in the last three decades. Here we propose a more general formulation of Marotto’s theorem, relaxing the assumption of smoothness, considering a generic piecewise smooth function, continuous or discontinuous. An example with a twodimensional smooth map is given and one with a twodimensional piecewise linear discontinuous map.

Mira C.
Noninvertible maps and their embedding into higher dimensional invertible maps
Abstract
The first part is devoted to a presentation of specific features of noninvertible maps with respect to the invertible ones. When embedded into a threedimensional invertible map, the specific dynamical features of a plane noninvertible map are the germ of the threedimensional dynamics, at least for sufficiently small absolute values of the embedding parameter. The form of the paper, as well as its contents, is approached from a non abstract point of view, in an elementary form from a simple class of examples.

Anishchenko V. S., Astakhov S. V., Vadivasova T. E.
Diagnostics of the degree of noise influence on a nonlinear system using relative metric entropy
Abstract
In this paper we summarize and substantiate the relative metric entropy approach introduced in our previous papers [1,2]. Using this approach we study the mixing influence of noise on both regular and chaotic systems. We show that the synchronization phenomenon as well as stochastic resonance decrease, the degree of mixing is caused by white Gaussian noise.

Belykh V. N., Pankratova E. V.
Chaotic dynamics of two Van der Pol–Duffing oscillators with Huygens coupling
Abstract
We consider a system of two coupled Van der PolDuffing oscillators with Huygens coupling as an appropriate model of two mechanical oscillators connected to a movable platform via a spring. We examine the complicated dynamics of the system and study its multistable behavior. In particular, we reveal the coexistence of several chaotic regimes and study the structure of the associated riddled basins.

Dobrokhotov S. Y., Minenkov D. S.
On various averaging methods for a nonlinear oscillator with slow timedependent potential and a nonconservative perturbation
Abstract
The main aim of the paper is to compare various averaging methods for constructing asymptotic solutions of the Cauchy problem for the onedimensional anharmonic oscillator with potential $V(x, \tau)$ depending on the slow time $\tau=\varepsilon t$ and with a small nonconservative term $\varepsilon g(\dot x, x, \tau)$, $\varepsilon \ll 1$. This problem was discussed in numerous papers, and in some sense the present paper looks like a "methodological" one. Nevertheless, it seems that we present the definitive result in a form useful for many nonlinear problems as well. Namely, it is well known that the leading term of the asymptotic solution can be represented in the form $X(\frac{S(\tau)+\varepsilon \phi(\tau))}{\varepsilon}, I(\tau), \tau)$, where the phase $S$, the "slow" parameter $I$, and the socalled phase shift $\phi$ are found from the system of "averaged" equations. The pragmatic result is that one can take into account the phase shift $\phi$ by considering it as a part of $S$ and by simultaneously changing the initial data for the equation for $I$ in an appropriate way.

Gelfreich V. G., Gelfreikh N.
Unique normal forms for area preserving maps near a fixed point with neutral multipliers
Abstract
We study normal forms for families of areapreserving maps which have a fixed point with neutral multipliers $\pm 1$ at $\varepsilon = 0$. Our study covers both the orientationpreserving and orientationreversing cases. In these cases Birkhoff normal forms do not provide a substantial simplification of the system. In the paper we prove that the Takens normal form vector field can be substantially simplified. We also show that if certain nondegeneracy conditions are satisfied no further simplification is generically possible since the constructed normal forms are unique. In particular, we provide a full system of formal invariants with respect to formal coordinate changes.

Grigorieva E. V., Kashchenko S. A.
Dynamics of spikes in delay coupled semiconductor lasers
Abstract
We derive the discrete maps to describe the dynamics of coupled laser diodes. The maps allow us to find analytically regions of parameters and initial conditions in the functional phase space that correspond to spiking with stable (or nearly stable) phase shift. The method developed is promising for further discussion of controlled switching between periodic states by an impulse injection signal.

Ilyashenko Y. S.
Thick attractors of step skew products
Abstract
A diffeomorphism is said to have a thick attractor provided that its Milnor attractor has positive but not full Lebesgue measure. We prove that there exists an open set in the space of boundary preserving step skew products with a fiber [0,1], such that any map in this set has a thick attractor.

Plykin R. V., Klinshpont N. E.
Strange attractors. Topologic, geometric and algebraic aspects
Abstract
This article is a review on research work of two authors on hyperbolic and Lorenz like strange attractors. In 1971 R. Plykin received a letter from two young scientists of Warwick University David Chillingworth and Anthony Manning with pointing out his errors in the article entitled "The topology of basic sets for Smale diffeomorphisms". The main error in that manuscript was the statement about nonexistence of one dimensional hyperbolic attractor of the diffeomorphism of twosphere. The first part of this report corrects previous errors and carries information about geometry and topology of hyperbolic strange attractors.
The second part of the report contains some results obtained by N. Klinshpont on the problem of topological classification of Lorenz type attractors and their generalizations. An investigation of stochastic properties of differentiable dynamical systems often results in study other limiting formations, the isolation of which is a difficult problem which requires the mobilization of not only analytical and geometrical methods but also substantial computational resources. In the geometrical approach, which this investigation follows, manifolds are studied together with dynamical systems on them in which the diversity of structures of attractors occurs and the main difficulties are the classification problems. 
Kuznetsov S. P.
Example of blue sky catastrophe accompanied by a birth of Smale–Williams attractor
Abstract
A model system is proposed, which manifests a blue sky catastrophe giving rise to a hyperbolic attractor of Smale–Williams type in accordance with theory of Shilnikov and Turaev. Some essential features of the transition are demonstrated in computations, including Bernoullitype discretestep evolution of the angular variable, inverse square root dependence of the first return time on the bifurcation parameter, certain type of dependence of Lyapunov exponents on control parameter for the differential equations and for the Poincaré map.

Leonov G. A., Kuznetsova O. A.
Lyapunov quantities and limit cycles of twodimensional dynamical systems. Analytical methods and symbolic computation
Abstract
In the present work the methods of computation of Lyapunov quantities and localization of limit cycles are demonstrated. These methods are applied to investigation of quadratic systems with small and large limit cycles. The expressions for the first five Lyapunov quantities for general Lienard system are obtained. By the transformation of quadratic system to Lienard system and the method of asymptotical integration, quadratic systems with large limit cycles are investigated. The domain of parameters of quadratic systems, for which four limit cycles can be obtained, is determined.

Matviichuk M.
On the set of periodic intervals of an interval map
Abstract
We investigate the closure of the set of periodic closed intervals for a continuous interval map with respect to Hausdorff metric. We prove that if a nondegenerate interval is limit of periodic ones then either a) it is periodic itself, or b) it is asymptotically degenerate, i.e. its diameter tends to 0 (when iterating under $f$). We present a continuous interval map for which case b) is possible.

Nefedov N. N., Recke L., Schneider K. R.
Asymptotic stability via the Krein–Rutman theorem for singularly perturbed parabolic periodic Dirichlet problems
Abstract
We consider singularly perturbed semilinear parabolic periodic problems and assume the existence of a family of solutions. We present an approach to establish the exponential asymptotic stability of these solutions by means of a special class of lower and upper solutions. The proof is based on a corollary of the Krein–Rutman theorem.

Novokshenov V. Y.
Poles of tritronquée solution to the Painlevé I equation and cubic anharmonic oscillator
Abstract
The distribution of poles of zeroparameter solution to Painlevé I, specified by P. Boutroux as intégrale tritronquée, is studied in the complex plane. This solution has regular asymptotics $−\sqrt{z/6} + O(1)$ as $z \to \infty$, $\arg z < 4\pi/5$. At the sector $\arg z > 4\pi/5$ it is a meromorphic function with regular asymptotic distribution of poles at infinity. This fact together with numeric simulations for $z < const$ allowed B. Dubrovin to make a conjecture that all poles of the intégrale tritronquée belong to this sector. As a step to prove this conjecture, we study the Riemann–Hilbert (RH) problem related to the specified solution of the Painlevé I equation. It is "undressed" to a similar RH problem for the Schrödinger equation with cubic potential. The latter determines all coordinates of poles for the intégrale tritronquée via a Bohr–Sommerfeld quantization conditions.

Osipov A. V., Pilyugin S. Y., Tikhomirov S. B.
Periodic shadowing and $\Omega$stability
Abstract
We show that the following three properties of a diffeomorphism $f$ of a smooth closed manifold are equivalent: (i) $f$ belongs to the $C^1$interior of the set of diffeomorphisms having the periodic shadowing property; (ii) $f$ has the Lipschitz periodic shadowing property; (iii) $f$ is $\Omega$stable.
