Local instability and localization of attractors. From stochastic generator to Chua's systems

This paper discusses the connection between various instability definitions (namely, Lyapunov instability, Poincaré or orbital instability, Zhukovskij instability) and chaotic movements. It is demonstrated that the notion of Zhukovskij instability is the most adequate for describing chaotic movements. In order to investigate this instability, a new type of linearization is offered and the connection between that and the theorems of Borg, Hartman-Olech, and Leonov is established. By means of new linearization, analytical conditions of the existence of strange attractors for impulse stochastic generators are obtained. The assumption is expressed that an analogous analytical tool may be elaborated for continuous dynamical systems describing Chua's circuits. The paper makes a first step in this direction and establishes a frequency criterion of the existence of positive invariant sets with positive Lebesgue measure for piecewise linear systems, which are unstable in every region of phase space where they are linear.     

Synthesis and structure of mesomorphic 2-cyano-5-[p-alkyl(alkoxy)phenyl]-pyridines

Liquid-crystal 2-cyano-5-[p-alkyl(alkoxy)phenyl]pyridines were synthesized by the reaction of 1-dimethylamino-3-dimethylimmonia-2-[p-alkyl (alkoxy)phenyl]propene perchlorates with 2-acetylfuran and subsequently through 2-furyl-5-[p-alkyl(alkoxy)phenyl]pyrylium perchlorates and 2- (2-furyl)-5-[p-alkyl (alkoxy)phenyl]pyridines by conversion of the latter to 5-alkyl(alkoxy)phenylpyridine-2-carboxylic acids, from which the cyano derivatives were obtained by the usual scheme through the amides. The intermediate arylfurylpyridines and arylpyridine-2-carboxylic acid imides also display liquid-crystal properties.Translated from Khimiya Geterotskilicheskikh Soedinenii, No. 7, pp. 888–891, July, 1980.     

Synthesis of [D-phe3 (NH2); Pro3; D-Ala6]- and [D-Phe2 (NO2); Pro3; D-Ala6]luliberins

In order to study the influence of substituents of the aromatic ring of D-phenyl-alanine on the inhibiting capacity of luliberian analogs, we have synthesized two new analogs: and luliberin. The synthesis was performed by the fragmentary condensation method using 2+(3+5) and 2+(5+3) schemes. A new and convenient method of obtaining the amide of the C-terminal tetrapeptide of the luliberin sequence has been developed. In the condensation of the fragments, both the azide and the carbodimide method of synthesis with the addition of l-hydroxybenzotriazole were used. The guanidino group of arginine was protected by nitration, while the hydroxy groups of serine and of tyrosine were not protected. The complete elimination of the protective groups from the decapeptides was performed by catalytic hydrogenation over Pd on carbon and by anhydrous HF with the addition of anisole at 0°C. The protected octa- and decapeptides were purified by gel filtration on Sephadex LH-20 in ethanol or by preparative thin-layer chromatography on silica gel plates. The final peptides were purified by ion-ex-exchange chromatography on Sephadex CM-25. A. A. Zhdanov Leningrad State University. Translated from Khimiya Prirodnykh Soedinenii, No. 5, pp. 706–714, September–October, 1980.     

Two nontrivial solutions for quasilinear periodic equations

In this paper we study a nonlinear periodic problem driven by the ordinary scalar p-Laplacian and with a Carathéodory nonlinearity. We establish the existence of at least two nontrivial solutions. Our approach is variational based on the smooth critical point theory and using the ``Second Deformation Theorem".

     

Existence of Solutions and of Multiple Solutions for Nonlinear Nonsmooth Periodic Systems

In this paper we examine nonlinear periodic systems driven by the vectorial p-Laplacian and with a nondifferentiable, locally Lipschitz nonlinearity. Our approach is based on the nonsmooth critical point theory and uses the subdifferential theory for locally Lipschitz functions. We prove existence and multiplicity results for the sublinear problem. For the semilinear problem (i.e. p = 2) using a nonsmooth multidimensional version of the Ambrosetti-Rabinowitz condition, we prove an existence theorem for the superlinear problem. Our work generalizes some recent results of Tang (PAMS 126(1998)).     

Wave propagation through a high-velocity elastic layer

We consider theoretical seismographs of a wave field passing through an elastic layer with high propagation velocity. We note the presence of a low-frequency part of the seismogram for longitudinal waves. It is shown that the propagation time of low-frequency vibrations is equal to that of longitudinal disturbances screened by the layer.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, AN SSSR, Vol. 173, pp. 134–141, 1988.     

Multilayer graphene waveguides

We study dispersion properties of TM-polarized electromagnetic waves guided by a multilayer graphene metamaterial. We demonstrate that both dispersion and localization of the guided modes can be efficiently controlled by changing the number of layers in the structure. Remarkably, we find that in the long wavelength limit, the dispersion of the fundamental mode of the N-layer graphene structure coincides with the dispersion of a plasmon mode supported by a single graphene layer, but with N times larger conductivity. We also compare our exact dispersion relations with the results provided by the effective media model.     

Investigation of the dispersion equations of a free,transversally isotropic,elastic layer

To investigate the propagation of normal waves in a free, transversally isotropic, elastic layer it is necessary to determine the roots of the corresponding dispersion equation. It is shown that with increasing wave number the roots move monotonically along the axis of the phase velocity. The initial and final points of the roots are found. Examples of computations of the roots are given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 99, pp. 85–103, 1980.     

Electrochemical Hydrogenation of α-Arylvinylphosphonic Acids on Platinum and Palladium Modified with Adatoms

A general characteristic of cathodic reduction of -phenylvinylphosphonic acid and its methyl- and chloro-parasubstituted derivatives on Pt/Pt, Pd/Pt, and Pd/Pt modified with Cd and Cu adatoms, is given. At effective pressures of about 10³–10⁴ Pa the processes occur at high rates and at some potentials are limited by the reactant diffusion from the bulk solution. In the presence of Cd and Cu adatoms two reduction waves (presumably, single-electron) may be isolated, i.e. the hampering by adatoms is more pronounced for a second successive stage of the process. At low effective hydrogen pressures the hydrogenation rate is higher on Pd electrodes, which possess a larger sorption capacity. Potential intervals where the reduction of substituted reactants may be complicated by destruction, are found. It is shown that -phenylvinylphosphonic acid undergoes a complete reduction with a current efficiency of 100% in both hydrochloric- and sulfuric-acid solutions. Conditions where the reduction of -(4-chlorophenyl)vinylphosphonic acid proceeds largely without destruction are found. Prospects for use of electrochemical reduction as a preparative method of a selective production of -arylethylphosphonic acids are considered.