Volume 15, Numbers 2-3

Volume 15, Numbers 2-3, 2010
On the 75th birthday of Professor L.P. Shilnikov

Citation: Leonid Pavlovich Shilnikov. On His 75th Birthday, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 2-3, pp. 101-106
Wittig A.,  Berz M.,  Grote J.,  Makino K.,  Newhouse S.
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.
Keywords: Taylor model, invariant manifold, hyperbolicity, homoclinic point
Citation: Wittig A.,  Berz M.,  Grote J.,  Makino K.,  Newhouse S., Rigorous and accurate enclosure of invariant manifolds on surfaces, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 2-3, pp. 107-126
Afraimovich V. S.,  Bunimovich L. A.,  Moreno S. V.
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.
Keywords: global stability, topological pressure, topological Markov chain, dynamical networks
Citation: Afraimovich V. S.,  Bunimovich L. A.,  Moreno S. V., Dynamical networks: continuous time and general discrete time models, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 2-3, pp. 127-145
Belykh I.,  Jalil S.,  Shilnikov A. L.
We study the emergence of in-phase and anti-phase 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 bursting-spiking transition is associated with codimension-one bifurcations of a saddle-node 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 burst-duration 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 anti-phase synchronized bursting.
Keywords: bursting neurons, synchronization, inhibitory networks, burst duration
Citation: Belykh I.,  Jalil S.,  Shilnikov A. L., Burst-duration mechanism of in-phase bursting in inhibitory networks, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 2-3, pp. 146-158
Gelfreich V. G.,  Turaev D. V.
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.
Keywords: homoclinic tangency, wild hyperbolic set, Newhouse phenomenon, Hamiltonian system, area-preserving map, volume-preserving flow, exponentially small splitting, KAM theory
Citation: Gelfreich V. G.,  Turaev D. V., Universal dynamics in a neighborhood of a generic elliptic periodic point, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 2-3, pp. 159-164
Gonchenko S. V.,  Li M.
We study the hyperbolic dynamics of three-dimensional quadratic maps with constant Jacobian the inverse of which are again quadratic maps (the so-called 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 so-called 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 saddle-focus types.
Keywords: quadratic map, Smale horseshoe, hyperbolic set, symbolic dynamics, saddle, saddlefocus
Citation: Gonchenko S. V.,  Li M., Shilnikov’s cross-map method and hyperbolic dynamics of three-dimensional Hénon-like maps, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 2-3, pp. 165-184
Grines V. Z.,  Pochinka O. V.
The paper contains exposition of results devoted to the existence of an energy functions for dynamical systems.
Keywords: Lyapunov function, energy function, Morse–Smale system
Citation: Grines V. Z.,  Pochinka O. V., Energy functions for dynamical systems, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 2-3, pp. 185-193
Lerman L. M.
We study a smooth symplectic 2-parameter unfolding of an almost hyperbolic diffeomorphism on two-dimensional 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 period-2 point exists for all parameter values close to the co-dimension two point whose separatrices intersect transversely the boundary curves of the channel that implies the existence of a quadratic homoclinic tangency for this period-2 point which occurs along a sequence of parameter values accumulating at the co-dimension 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 non-uniformly hyperbolic.
Keywords: symplectic diffeomorphism, torus, Anosov diffeomorphism, almost hyperbolic, degenerate saddle, bifurcation, homoclinic tangency, break of hyperbolicity, $\pi_1$-diffeomorphism
Citation: Lerman L. M., Breaking hyperbolicity for smooth symplectic toral diffeomorphisms, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 2-3, pp. 194-209
Li M.,  Malkin M. I.
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.
Keywords: chaotic dynamics, difference equations, one-dimensional maps, topological entropy, hyperbolic orbits
Citation: Li M.,  Malkin M. I., Approximation of entropy on hyperbolic sets for one-dimensional maps and their multidimensional perturbations, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 2-3, pp. 210-221
Delshams A.,  Gutierrez P.,  Koltsova O. Y.,  Pacha J. R.
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 $(n+2)$-degree-of-freedom 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).
Keywords: hyperbolic KAM tori, transverse homoclinic orbits, Melnikov method
Citation: 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, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 2-3, pp. 222-236
Gardini L.,  Tramontana F.
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 snap-back-repellers. 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 two-dimensional smooth map is given and one with a two-dimensional piecewise linear discontinuous map.
Keywords: snap back repellers, homoclinic orbits in noninvertible maps, orbits homoclinic to expanding points
Citation: Gardini L.,  Tramontana F., Snap-back repellers in non-smooth functions, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 2-3, pp. 237-245
Mira C.
The first part is devoted to a presentation of specific features of noninvertible maps with respect to the invertible ones. When embedded into a three-dimensional invertible map, the specific dynamical features of a plane noninvertible map are the germ of the three-dimensional 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.
Keywords: noninvertible maps, embedding problems, discrete dynamics, global bifurcations, critical sets, basin of attraction, fractal sets
Citation: Mira C., Noninvertible maps and their embedding into higher dimensional invertible maps, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 2-3, pp. 246-260
Anishchenko V. S.,  Astakhov S. V.,  Vadivasova T. E.
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.
Keywords: noisy dynamical systems, entropy, mixing, Kolmogorov entropy, recurrency plot
Citation: Anishchenko V. S.,  Astakhov S. V.,  Vadivasova T. E., Diagnostics of the degree of noise influence on a nonlinear system using relative metric entropy, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 2-3, pp. 261-273
Belykh V. N.,  Pankratova E. V.
We consider a system of two coupled Van der Pol-Duffing 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 co-existence of several chaotic regimes and study the structure of the associated riddled basins.
Keywords: Van der Pol–Duffing oscillator, coupled systems, chaotic oscillations
Citation: Belykh V. N.,  Pankratova E. V., Chaotic dynamics of two Van der Pol–Duffing oscillators with Huygens coupling, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 2-3, pp. 274-284
Dobrokhotov S. Y.,  Minenkov D. S.
The main aim of the paper is to compare various averaging methods for constructing asymptotic solutions of the Cauchy problem for the one-dimensional 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 so-called 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.
Keywords: nonlinear oscillator, averaging, asymptotics, phase shift
Citation: Dobrokhotov S. Y.,  Minenkov D. S., On various averaging methods for a nonlinear oscillator with slow time-dependent potential and a nonconservative perturbation, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 2-3, pp. 285-299
Gelfreich V. G.,  Gelfreikh N. G.
We study normal forms for families of area-preserving maps which have a fixed point with neutral multipliers $\pm 1$ at $\varepsilon = 0$. Our study covers both the orientation-preserving and orientation-reversing 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 non-degeneracy 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.
Keywords: area-preserving map, unique normal form, parabolic fixed point
Citation: Gelfreich V. G.,  Gelfreikh N. G., Unique normal forms for area preserving maps near a fixed point with neutral multipliers, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 2-3, pp. 300-318
Grigorieva E. V.,  Kashchenko S. A.
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.
Keywords: differential-difference equations, lasers, synchronization
Citation: Grigorieva E. V.,  Kashchenko S. A., Dynamics of spikes in delay coupled semiconductor lasers, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 2-3, pp. 319-327
Ilyashenko Y. S.
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.
Keywords: Milnor attractors, thick attractors, ergodicity
Citation: Ilyashenko Y. S., Thick attractors of step skew products, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 2-3, pp. 328-334
Plykin R. V.,  Klinshpont N. E.
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 two-sphere. 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.
Keywords: dynamical systems, strange attractor, centralizer, topological invariant
Citation: Plykin R. V.,  Klinshpont N. E., Strange attractors. Topologic, geometric and algebraic aspects, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 2-3, pp. 335-347
Kuznetsov S. P.
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 Bernoulli-type discrete-step 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.
Keywords: attractor, bifurcation, Smale–Williams solenoid, Lyapunov exponent
Citation: Kuznetsov S. P., Example of blue sky catastrophe accompanied by a birth of Smale–Williams attractor, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 2-3, pp. 348-353
Leonov G. A.,  Kuznetsova O. A.
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.
Keywords: Hilbert’s 16-th problem, small and large limit cycles, Lyapunov quantities, symbolic computation, localization of limit cycles, quadratic system, Lienard system
Citation: Leonov G. A.,  Kuznetsova O. A., Lyapunov quantities and limit cycles of two-dimensional dynamical systems. Analytical methods and symbolic computation, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 2-3, pp. 354-377
Matviichuk M.
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.
Keywords: dynamics of sets, periodic interval, interval map
Citation: Matviichuk M., On the set of periodic intervals of an interval map, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 2-3, pp. 378-381
Nefedov N. N.,  Recke L.,  Schneider K. R.
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.
Keywords: singularly perturbed parabolic Dirichlet problems, exponential asymptotic stability, Krein–Rutman theorem, lower and upper solutions
Citation: Nefedov N. N.,  Recke L.,  Schneider K. R., Asymptotic stability via the Krein–Rutman theorem for singularly perturbed parabolic periodic Dirichlet problems, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 2-3, pp. 382-389
Novokshenov V. Y.
The distribution of poles of zero-parameter 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.
Keywords: Painlevé equation, special functions, distribution of poles, Riemann–Hilbert problem, WKB approximation, Bohr–Sommerfield quantization, complex cubic potential
Citation: Novokshenov V. Y., Poles of tritronquée solution to the Painlevé I equation and cubic anharmonic oscillator, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 2-3, pp. 390-403
Osipov A. V.,  Pilyugin S. Y.,  Tikhomirov S. B.
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.
Keywords: periodic shadowing, hyperbolicity, $\Omega$-stability
Citation: Osipov A. V.,  Pilyugin S. Y.,  Tikhomirov S. B., Periodic shadowing and $\Omega$-stability, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 2-3, pp. 404-417

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